Lectures on the FTAP Notes complementing Delbaen and Schachermayer's book "The Mathematics of Arbitrage" Harry van Zanten September 25, 2007 (this version) use at your own risk this text undoubtedly contains many errors please report them to harry@cs.vu.nl ii Preface These lecture notes treat various versions of the so-called "fundamental theorem of asset pricing". Many students are familiar with statements about models for financial markets saying that "absence of arbitrage" and the existence of an "equivalent martingale measure" are in some sense equivalent. The purpose of these notes is to present and explain mathematical results making such statements precise. We start with the relatively simple situation of models defined on a finite underlying probability space, cf. Harrison and Pliska (1981). In this case a mathematically precise and economically satisfactory fundamental theorem can be derived using the separating hyperplane theorem. We then move to general continuous time models, treating the theorem of Kreps (1981). The basic idea behind this theorem is in fact similar to the finite case. However, since the problem is now infinite-dimensional, it is technically much more involved and it is necessary to involve topological considerations in the definition of "no arbitrage", called "no free lunch" by Kreps. Although Kreps' theorem is satisfactory from the mathematical perspective, the use of the weak -topology in the definition of "no free lunch" destroys the economic interpretation of the result. We finally treat the fundamental theorem of Delbaen and Schachermayer (1994), who work in the setting of asset prices modelled by locally bounded semimartingales, and trading strategies modelled by predictable processes. In this setting Kreps' definition of "no free lunch" can be replaced by the condition of "no free lunch with vanishing risk", which does not involve an unnatural topology, and has a clear economic interpretation. The proof of the fundamental theorem of Delbaen and Schachermayer (1994) is technically very involved, and we only sketch the main arguments in these notes. In particular, we explain the connection to the result of Kreps (1981). The text is mainly based on Chapters 2, 5 and 8 and 9 of Delbaen and Schachermayer (2006). The necessary results from functional analysis are treated in the appendix, and are taken from Conway (1990) or Rudin (1991). Readers are assumed to be familiar with basic topology, measure theoretic probability theory, martingale theory and stochastic integration theory, the latter up to the level of stochastic integration of locally bounded predictable processes relative to general semimartingales. Amsterdam, fall 2006 HvZ iv Contents Preface iii 1 A simple example 1 1.1 Exercises 3 2 Finite underlying probability spaces 5 2.1 Description of the model and basic definitions 5 2.2 Fundamental theorem of asset pricing 7 2.3 Single period versus multiperiod models 8 2.4 Completeness 9 2.5 Change of numerair 11 2.6 No-arbitrage pricing 12 2.7 Example: binomial model 13 2.8 Exercises 15 3 The Kreps-Yan theorem 17 3.1 Description of the model 17 3.2 Kreps-Yan theorem 19 3.3 Discussion 20 3.4 Exercises 22 4 The general theorem 23 4.1 Preliminaries on stochastic integration 23 4.2 No free lunch with vanishing risk and the FTAP 26 4.3 Sketch of proof of the fundamental theorem 27 4.3.1 The relatively easy half 27 4.3.2 The much more difficult half 29 4.4 Example: Itô processes 30 4.5 Exercises 32 A Elements of functional analysis 33 A.1 Separating hyperplane theorem 33 A.2 Topological vector spaces 35 A.3 Hahn-Banach theorem 37 A.4 Dual space 39 A.5 Exercises 44 B Elements of martingale theory 45 B.1 Basic definitions 45 vi Contents B.2 Theorems 46 B.3 Exercises 48 References 48 1 A simple example Consider a world with two time points, t = 0 (today) and t = 1 (tomorrow). In this world there exists a bank where money can be deposited or borrowed at zero interest and a stock is traded. The value S0 of the stock at time 0 equals 1 and the value S1 at time t = 1 equals the value u with probability p (0, 1) and d < u with probability 1-p, respectively. Note that the stock price process can be viewed as a stochastic process defined on an underlying probability space (, F, P) with consisting of just two elements, one corresponding to the stock price going up, one to the price going down, F the power set of and P the probability measure that gives probability p to the event of the price going up and 1 - p to the event of the price going down. A trader in this world can form a portfolio today consisting of a number of units of money in the bank, call this 0, and a number of stocks, call this 0. Clearly, this portfolio is worth V0 = 0 + 0 at time 0. When tomorrow comes, the portfolio will have the new, random value V1 = 0 + 0S1. We say that there exists an arbitrage opportunity in this world if there exists a portfolio (0, 0) as above such that the associated value process V satisfies V0 = 0, V1 0 and P(V1 > 0) > 0. Clearly, this corresponds to a possibility of making a risk-free profit. Proposition 1.0.1. There exist no arbitrage opportunities in this world if and only if d < 1 < u. Proof. If d < 1 < u then q = (1 - d)/(u - d) belongs to (0, 1) and hence the probability measure Q under which the stock price moves up or down according to the probability q instead of p is equivalent (i.e. mutually absolutely continuous) to the underlying probability measure P (see Exercise 1). Now let (0, 0) be a portfolio such that V0 = 0 and V1 0. Then by construction we have EQV1 = (0 + 0u)q + (0 + 0d)(1 - q) = 0 + 0 = V0 = 0. Hence, since V1 0, we have Q(V1 > 0) = 0. But since the measures P and Q are equivalent, it follows that P(V1 > 0) = 0 as well. To prove the converse statement, suppose for instance that 1 d < u. Then the stock always performs at least as good as money in the bank and there 2 A simple example is a positive probability that it performs strictly better. Borrowing money from the bank and investing it in stock then yields an arbitrage. Specifically, consider the portfolio 0 = -1, 0 = 1. The corresponding values process satisfies V0 = 0 and V1 is either d-1 or u-1, which is strictly positive with positive probability in this case. The case d < u 1 can be handled similarly. In the proof of the proposition we noted that in the case of no-arbitrage, i.e. when d < 1 < u, the new underlying probability measure Q under which the stock goes up with probability q = (1 - d)/(u - d) is equivalent to the original probability measure P. Observe that EQS1 = uq + d(1 - q) = S0. In other words, the stock price process S = (S0, S1) is a martingale under Q (relative to the trivial filtration F0 = {, }, F1 = P()). Note that any other equivalent measure Q is fully described by specifying a probability q (0, 1) with which the stock goes up (Exercise 1 again). For such a measure Q we have EQ S1 = uq + d(1 - q ) = q (u - d) + d. Hence if Q has the property that EQ S1 = S0, then (1 - d)/(u - d) = q (0, 1) which is the same as saying that d < 1 < u. We just proved that this implies absence of arbitrage opportunities. We conclude that the condition for no-arbitrage can be reformulated in terms of the existence of certain probability measures. Proposition 1.0.2 (Fundamental theorem of asset pricing O). There exist no arbitrage opportunities in this world if and only there exists a probability measure Q equivalent to the original probability measure P such that the stock price process S = (S0, S1) satisfies EQS1 = S0. For obvious reasons a probability measure Q as in the proposition is called an equivalent martingale measure. Using this terminology the result asserts that absence of arbitrage is equivalent to the existence of an equivalent martingale measure. It turns out that (the appropriate version of) this result, often called the fundamental theorem of asset pricing, is true in a very general setting. This is of great interest, since it relates a fundamental economic notion (arbitrage) to an important mathematical concept (martingales). As a result, martingale theory plays a central role in the modelling of financial markets and pricing of derivatives. In these notes we discuss the fundamental theorem of asset pricing in increasingly general settings. 1.1 Exercises 3 1.1 Exercises 1. Show that two probability measures on a finite set are equivalent (i.e. mutually absolutely continuous) if and only if they give positive probability to the same singletons. 4 A simple example 2 Finite underlying probability spaces 2.1 Description of the model and basic definitions Consider a finite probability space (, F, P), with = {1, . . . , n}, F = P() and pi = P({i}) > 0 for every i. Suppose that on this probability space we are given a filtration (Ft)t=0,...,T and a (d + 1)-dimensional adapted stochastic process S = (S (0) t , . . . , S (d) t )t=0,...,T , where T is some finite positive integer. Assume that FT = F. We think of the components of S as the price processes of d + 1 different financial assets, measured relative to the price of the 0-th asset, called a numerair. Since we measure prices relative to the price of the numerair, the price process S(0) equals 1 at all times. A portfolio is a (d + 1)-dimensional, predictable process = ( (0) t , . . . , (d) t )t=1,...,T . We think of (i) t as the number of assets of type i that is in the portfolio in the time interval (t - 1, t]. The requirement that is predictable means that at each time t-1, the portfolio is constructed using only the information available up to that time, i.e. the trader can not look into the future. The value process associated with a portfolio is the process V = (Vt)t=0,...,T defined by V0 = d i=0 (i) 1 S (i) 0 , Vt = d i=0 (i) t S (i) t , t 1. (2.1) Clearly V is an adapted process. The value V0 is called the initial value of the portfolio. A special role is played by portfolios that do not involve injections or withdrawals of money after time 0. Consider such a portfolio with value Vt-1 at time t - 1. Just after t - 1 the portfolio is rebalanced, and the new value equals d i=0 (i) t S (i) t-1 = Vt - d i=0 (i) t (S (i) t - S (i) t-1). 6 Finite underlying probability spaces If no money is injected or withdrawn this should equal Vt-1, hence Vt - Vt-1 = d i=0 (i) t (S (i) t - S (i) t-1) = d i=1 (i) t (S (i) t - S (i) t-1) (we use that S(0) equals 1 at all times). Definition 2.1.1. A portfolio is called self-financing if its value process V satisfies Vt - Vt-1 = d i=1 (i) t (S (i) t - S (i) t-1) for all t 1. We use the usual notation f(t) = f(t) - f(t - 1) for a (possibly vectorvalued) function f on the integers. Moreover, we write v, w for the Euclidean inner product of two vectors in Rd . Using that notation the preceding display reads Vt = t, St and a self-financing portfolio satisfies the relation Vt = V0 + t u=1 u, Su . If we define the process S by ( S)0 = 0 and ( S)t = t u=1 u, Su for t 1, we can write Vt = V0 + ( S)t for a self-financing portfolio. If we compare the definition of a self-financing portfolio with (2.1) we see that for such a portfolio it holds that (0) 1 = V0 - d i=1 (i) 1 S (i) 0 and (0) t = - d i=1 (i) t S (i) t-1, t 2. Hence, if we specify the initial value V0 and ((1) , . . . , (d) ), the process (0) describing the holdings in the numerair asset is completely determined by the requirement that the portfolio is self-financing. 2.2 Fundamental theorem of asset pricing 7 2.2 Fundamental theorem of asset pricing In this setting the definition of an arbitrage opportunity is as follows. Definition 2.2.1. An arbitrage opportunity is a self-financing portfolio whose value process V satisfies V0 = 0, VT 0 and P(VT > 0) > 0. To prepare for the proof of the theorem below it is useful to reformulate this in more geometric terms. Define the collection of random variables K = K(S) = {( S)T : predictable}. Note that K is the set of all possible pay-offs of self-financing portfolios with zero initial value. Denoting the collection of all integrable nonnegative random variables on (, F, P) by L + , absence of arbitrage is the same as the requirement that K L + = {0}. In our setting of finite we can identify collections of random variables with subsets of of Rn : simply identify a random variable X with the vector of possible realizations (X(1), . . . , X(n)). For instance L + corresponds to the set {(x1, . . . , xn) : x1 0, . . . , xn 0}. This way the requirement KL + = {0} of no-arbitrage translates into a geometric statement about subsets of Rn . By L we denote the collection of all bounded random variables. Since is finite, bounded just means finite-valued, so that L can be identified with all of Rn . We will see, as in the preceding chapter, that absence of arbitrage is equivalent to the existence of an equivalent martingale measure, which is defined as follows. Definition 2.2.2. A probability measure Q on (, F) is called equivalent martingale measure if it is equivalent to P (i.e. Q P and P Q) and S is a (d-dimensional) martingale with respect to Q. The collection of equivalent martingale measures is denoted by Me = Me (S). Theorem 2.2.3 (Fundamental theorem of asset pricing I). There are no arbitrage opportunities in this model if and only if there exists an equivalent martingale measure. Proof. Suppose first that there exists a martingale measure Q and let be a self-financing portfolio whose value process V satisfies V0 = 0 and VT 0. Since is finite is bounded, hence V = V0 + S is a Q-martingale (see Exercise 1). In particular, EQVT = EQV0 = 0, so VT = 0 with Q-probability one, but then also P-almost surely. Now assume that no arbitrage opportunities exist, so that K L + = {0}. Let A be the convex hull of the elements 1{1}, . . . , 1{n} in L . This is a convex, compact subset of L , disjoint from K by assumption. Since the latter is a linear subspace it is closed and convex, and hence we can apply the 8 Finite underlying probability spaces separating hyperplane theorem. This yields a vector q Rn and , R such that q, f < q, h (2.2) for all f K and h A. Since K is a linear space we can take = 0 in this display, i.e. q, f 0 < q, h for all f K and h A (see Exercise 2). It follows that for every i we have qi = q, 1{i} > 0, hence we can renormalize q such that it becomes a vector of strictly positive probabilities adding up to 1. The last display then remains true, but with suitably normalized. The corresponding probability measure Q, defined by Q({i}) = qi, is equivalent to P and satisfies EQf 0 for all f K. But since K is a linear space this implies that in fact EQf = 0 for all f K. By Exercise 3 if follows that S is a Q-martingale. 2.3 Single period versus multiperiod models Recall our setting of a finite underlying probability space (, F, P) on which we have a filtration (Ft)t=0,...,T and a d-dimensional adapted stochastic process S = (S (1) t , . . . , S (d) t )t=0,...,T describing the discounted asset prices. If T 2 this is called a multiperiod model. It turns out that that absence of arbitrage in such a model is equivalent to absence of arbitrage in all the single-period sub-models. Theorem 2.3.1. There are no arbitrage opportunities in the full model if and only if for every t, the one-period model (St, St+1), with respect to the filtration (Ft, Ft+1), admits no arbitrage opportunities. Proof. Suppose all the one-period models are free of arbitrage. Then by the fundamental theorem there exist probability measures Qt on (, Ft+1) such that Qt is equivalent to P on Ft+1 and EQt (St+1 | Ft) = St. By the lemma following the theorem we may assume that Qt|Ft = P|Ft . Now define the process L by L0 = 1 and Lt = dQ0 dP dQt-1 dP , and define the measure Q by dQ = LT dP. Then Q Me (Exercise 4), hence the full model is free of arbitrage by the fundamental theorem. The following lemma applies to the general, multiperiod model, but in the proof of the theorem it is applied only to single period models. Lemma 2.3.2. Suppose there is no arbitrage. Let Q Me and define Zt = EP(dQ/dP | Ft) and Lt = Zt/Z0. Then the measure Q defined by dQ = LT dP belongs to Me and satisfies Q |F0 = P|F0 . 2.4 Completeness 9 Proof. Since Z is a martingale we have EP(ZT | Ft) = Zt for all t T. Hence for A Ft and t T, Q(A) = A ZT dP = A Zt dP. It follows that Zt = (dQ|Ft )/(dP|Ft ). The fact that Q is equivalent to P now implies that Zt > 0 for all t, and in particular that the process L is well defined. Since Z is a positive P-martingale and Z0 is F0-measurable, L is a positive Pmartingale as well and therefore Q is a probability measure equivalent to P. The fact that S is a Q-martingale implies that SZ is a P-martingale (check!). Since Z0 is F0-measurable, SL is a P-martingale as well. Using the fact that Lt = (dQ |Ft )/(dP|Ft ), we obtain that S is a Q -martingale (check!), hence Q Me . Finally, the fact that L1 = 1 implies that Q |F0 = P|F0 . 2.4 Completeness The FTAP does not say how many equivalent martingale measures there are in the absence of arbitrage. We will see below that this is related to the notion of completeness. Definition 2.4.1. We call f L attainable if f = a+(S)T for some a R and predictable process . The model is called complete if every claim f L is attainable. So an attainable contingent claim f is a random pay-off at time T that can be realized by following a self-financing strategy requiring some initial capital a. For the proof of the following theorem it is useful to introduce, in addition to the set K, the set of random variables C = {f L : there exists a g K such that g f}. This set is a cone1 containing K and it is easy to see that K L + = {0} if and only if C L + = {0} (see Exercise 5). Moreover, under no-arbitrage it holds that K = C (-C) (Exercise 6). Lemma 2.4.2. For any probability measure Q we have that S is a Q-martingale if and only if EQg 0 for all g C. 1Recall that a subset C of a vector space is called a cone if for all x C and a 0, it holds that ax C. 10 Finite underlying probability spaces Proof. Take g C, say g f for f K. Then if S is Q-martingale we have EQf = 0 (see Exercise 1) and hence EQg 0. If EQg 0 for all g C then for all f K it holds that EQf = 0, since f C and -f C for f K. Hence, by Exercise 3, S is a Q-martingale. Theorem 2.4.3. Assume there are no arbitrage opportunities. Then K = {f L : EQf = 0 for all Q Me }. Proof. The set C is convex and closed (see Exercise 7) and hence, by the Bipolar Theorem, equals its own bipolar C00 . Since C is closed under multiplication with positive scalars we have (see the appendix) C0 = {q Rn : g, q 0 for all g C}. Hence, by the Lemma preceding the theorem, the collection Ma of probability measure Q such that S is a Q-martingale is contained in C0 , hence cone(Ma ) C0 . By considering the elements -1{i} C we see that every q C0 has nonnegative coordinates and hence is a nonnegative multiple of a probability distribution. By the lemma again this probability measure belongs to Ma . We conclude that cone(Ma ) = C0 . Now C0 is closed under multiplication with positive scalars and hence C = C00 = {g Rn : g, q 0 for all q C0 }. Combined with the preceding observations we obtain C = {g Rn : EQg 0 for all Q Ma }. By Exercise 8 we have that Me is dense in Ma and hence C = {g Rn : EQg 0 for all Q Me }. (2.3) The proof is completed by using the fact that K = C (-C) (Exercise 9). Corollary 2.4.4 (Completeness). Assume there are no arbitrage opportuni- ties. (i) The model is complete if and only if the equivalent martingale measure is unique. (ii) In case of completeness the representation f = a + f0 with a R and f0 K of a claim f L is unique. Proof. (i). Suppose first that Me = {Q} and take f L . By Theorem 2.4.3 f - EQf K, hence f is attainable. Conversely, suppose we have Q1 = Q2 in Me . Then there exists an f L such that EQ1 f = EQ2 f. If this f were attainable, there would exist an a R such that f - a K. By Theorem 2.4.3 this would imply that EQ1 f = a = EQ2 f, a contradiction. (ii). Exercise 10. 2.5 Change of numerair 11 2.5 Change of numerair Recall that in our model we have d + 1 traded assets and the processes S(0) , . . . , S(d) are the prices of these assets relative to the price of the 0-th asset, the so-called numerair. Intuitively, absence or presence of arbitrage should not depend on the choice of numerair. In this section we prove that this is indeed the case. Any asset with a strictly positive price at all times could be taken as a numerair. More generally, we shall allow any self-financing portfolio of assets with a strictly positive value at all times. Let be a predictable process and consider the value process V = 1 + S. Assume that almost surely Vt > 0 for all t. We can view this portfolio as a traded asset and use it to express the value of our d + 1 original assets. The new value process ~S = ( ~S(0) , . . . , ~S(d) ) is given by ~S(i) = S(i) V , i = 0, . . . , d. Theorem 2.5.1 (Change of numerair). Suppose the model S admits no arbitrage opportunities. Then the model ~S admits no arbitrage opportunities either. It holds that Q Me (S) if and only the measure ~Q defined by d~Q = VT dQ belongs to Me ( ~S). Proof. We claim that that K( ~S) = V -1 T K(S). To see this, first observe that, for every i, ~S (i) t = 1 Vt S (i) t - ~S (i) t-1Vt . It follows that for a given predictable process , we have ( ~S)T = t ft Vt , with ft an Ft-measurable element of K(S), for every t. By the lemma following the theorem it holds that ft/Vt = gt/VT for certain gt K(S). Hence, we have the inclusion K( ~S) V -1 T K(S). The converse inclusion follows by symmetry, by considering the model ~S and taking 1/V as numerair. It now follows from the definition of arbitrage that the model S is free of arbitrage if and only if this holds for ~S. To complete the proof, take an equivalent probability measure Q. By Lemma 2.4.2 and Exercise 6 it holds that Q Me (S) if and only if EQf = 0 for all f K(S). By the first part of the proof, this holds if and only if EQVT f for all f K( ~S), which is the same as saying that the measure ~Q defined by d~Q = VT dQ belongs to Me ( ~S). 12 Finite underlying probability spaces Lemma 2.5.2. For f an Ft-measurable element of K(S) and t T, it holds that (VT /Vt)f K(S). Proof. Observe that VT Vt f = f + T s=t+1 f Vt Vs = f + ( S)T , where s = f Vt s1{st+1}. Since f is Ft-measurable the process is predictable, and it follows that (VT /Vt)f K(S). 2.6 No-arbitrage pricing Suppose that in our market we can buy the claim f L at price a at time t = 0. Then the collection of all claims that we can attain with 0 initial cost changes from K to Kf,a = span(K {f - a}). In case of no-arbitrage it should hold, as before, that Kf,a L + = {0}. This leads us to the following definition. Definition 2.6.1. We call a R an arbitrage free price for the claim f L if Kf,a L + = {0}. Observe that if the claim f L is attainable at price a, i.e. f - a K, then Kf,a = K. In the absence of arbitrage we have K L + = {0} and hence in this case a is an arbitrage-free price for f. Moreover, any other value b = a is not an arbitrage-free price for f (Exercise 11). Theorem 2.6.2 (No-arbitrage pricing). Assume absence of arbitrage and let f L . Define I = {EQf | Q Me }. The set I is the collection of all arbitrage-free prices for f. Either I = {a}, in which case f is attainable at price a, or I is a bounded, open interval, in which case f is not attainable. Proof. Define (f) = inf{EQf | Q Me }, (f) = sup{EQf | Q Me }. Suppose (f) = (f) = a. Then by Theorem 2.4.3, f - a K, which means that f is attainable at price a. Hence, by the remarks preceding the theorem, a is the unique arbitrage-free price for f. 2.7 Example: binomial model 13 Now assume (f) < (f). Since f is bounded and Me is convex, I = {EQf | Q Me } is a bounded subinterval of R. Suppose a I. Then there is a Q Me such that EQ(f - a) = 0. This implies that Kf,a L + = {0} (check). Hence, a is an arbitrage-free price for f. Conversely, suppose that Kf,a L + = {0}. Then by repeating the proof of Theorem 2.2.3 with Kf,a in the place of K we find a Q Me such that EQg = 0 for all g Kf,a . In particular, we see that EQ(f - a) = 0, hence a I. It remains to show that I is an open interval. Set a = (f), the right boundary point of I, and consider the claim f - a. By definition we have EQ(f - a) 0 for all Q Me . Since we have the representation (2.3) for the set C, it follows that f - a C. Hence there exists a g K such that g f - a. Now suppose that a I. Then there exists a Q Me such that EQ f = a and hence EQ (g - (f - a)) = 0, so that g = f - a. But this means that f - a K, i.e. f is attainable at price a. Theorem 2.4.3 implies that I reduces to a singleton in that case, which leads to a contradiction. We conclude that the right endpoint of I does not belong to I. The left endpoint can be handled similarly (or by considering the claim -f). 2.7 Example: binomial model Consider a world with n time points, t = 0, . . . , n. In this world there exists a bank where money can be deposited or borrowed at interest rate r > 0 and a stock is traded. We denote the bank account process by B, so Bt = (1 + r)t . The value X0 of the stock at time 0 equals 1 and given the stock has value Xt at time t, the value Xt+1 at time t + 1 equals uXt with probability p (0, 1) and dXt with probability 1-p, respectively, where d < u are certain given numbers. This model is called the binomial model. Let us choose the bank account process B as numerair and denote the discounted price processes by S(0) and S(1) , so S(0) 1 and S (1) t = Xt/Bt. Then under the objective probability measure P described above we have S (1) 0 = 1 and given S (1) t we have that S (1) t+1 equals (u/(1 + r))S (1) t with probability p and (d/(1 + r))S (1) t with probability 1 - p. We want to investigate the existence of arbitrage opportunities in this model. According to Theorem 2.3.1, it suffices to consider the one-period model we studied in Chapter 1. Proposition 1.0.1 (applied with d/(1 + r) in the place of d and u/(1 + r) in the place of u) then implies that the binomial model is free of arbitrage if and only if d < 1 + r < u. The considerations following Proposition 1.0.1 show that the one-period model admits a unique martingale measure, described by changing the probability with which the stock moves up from p to q = (1 + r - d)/(u - d). This implies that the full, multi-period model also admits only one martingale measure Q. Hence, by Corollary 2.4.4, the model is complete. In this complete model every claim f L is attainable and by Theorem 2.6.2 its arbitrage-free price is given by the expectation of f under the martingale measure. Recall that all of this is still relative to the chosen numerair, the bank 14 Finite underlying probability spaces account process B. Hence, a claim f should be interpreted as a pay-off of at time T of f units of the bank account process. In ordinary money terms, this is a pay-off of fBT = f(1 + r)T euros at time T. At time 0 this has the value of EQf units of the bank account process. But B0 equals one euro, and hence a pay-off of f(1 + r)T euros at time T is worth EQf euros at time 0. In other words, a pay-off of f euros at time T is worth EQ(1 + r)-T f euros at time 0. Putting this together we obtain the following result. Proposition 2.7.1. The binomial model is free of arbitrage if and only d < 1 + r < u. In this case the model is complete and the arbitrage-free value at time 0 of a claim paying f L euros at time T is EQ(1 + r)-T f, where Q is the probability measure obtained by changing the probability with which the stock price moves up from p to q = (1 + r - d)/(u - d). 2.8 Exercises 15 2.8 Exercises 1. If S is a (multi-dimensional) martingale and is a (multi-dimensional) bounded, predictable process, then S is a martingale. 2. In the proof of Theorem 2.2.3, show that the fact that K is a linear space implies that we can take = 0 in (2.2). 3. If EQ( S)T = 0 for every bounded, predictable process , then S is a Q-martingale. 4. In the proof of Theorem 2.3.1, show that Q Me . 5. Show that K L + = {0} if and only if C L + = {0}. 6. Show that under no-arbitrage, K = C (-C). 7. Let K be a linear subspace of Rn and L+ = {x Rn : x1 0, . . . , xn 0}. (a) For v1, . . . , vm Rn , show that the convex cone C = { civi : c1 0, . . . , cm 0} is closed. (Hint: Write C = {ax : a 0, x H}, where H = { civi : c1 0, . . . , cm 0, ci = 1} is the convex hull of the vi, and first prove that H is compact.) (b) Show that K+L+ is closed. (Hint: Say dim(K) = k and let f1, . . . , fn be an orthonormal basis of Rn such that f1, . . . , fk is an orthonormal basis of K. Using (a), show that the coordinates of the points of K + L+ relative to this basis form a closed subset of Rn .) 8. Show that under no-arbitrage, Me is dense in Ma . (Here we identify probability measures on with points in Rn again.) 9. Supply the details of the last part of the proof of Theorem 2.4.3. 10. Prove Corollary 2.4.4.(ii). 11. Assume absence of arbitrage. Show that for a claim f L that is attainable at price a, the value a is the unique arbitrage-free price. 12. Consider a one-period model with a bank with zero interest and a stock which has value 1 at time 0 and value s1, s2 or s3, respectively, at time 1, with probabilities p1, p2 or p3, respectively. Assume that s1 > s2 > s3 and the pi's are non-zero and add up to 1. (i) Give conditions on the values s1, s2, s3 characterizing absence of ar- bitrage. (ii) In case of absence of arbitrage, investigate whether the model is complete or not. 16 Finite underlying probability spaces 3 The Kreps-Yan theorem 3.1 Description of the model Let S = (St)t[0,T ] be a (d + 1)-dimensional, cadlag, adapted, locally bounded stochastic process, with T > 0 a fixed time horizon, defined on a probability space (, F, P) endowed with a filtration (Ft)t[0,T ] satisfying the usual conditions. As in the preceding chapter, we think of S = (S(0) , . . . , S(d) ) as describing the value of d+1 financial assets, expressed relative to a chosen numerair, which is the 0-th asset. In particular, we assume that S(0) 1. By the classical theorems on regularity of continuous-time martingales, the usual conditions on the filtration imply that (local) martingales have a cadlag modification. Since the basic theorems of martingale theory are valid for rightcontinuous martingales, the usual conditions ensure that we may apply the classical theorems to the martingales that we encounter. Observe that our setup includes the discrete-time case, simply take the process S (and the filtration) to be constant on the intervals [t - 1, t) for integers t. If the underlying probability space is finite the process S is necessarily uniformly bounded, and hence the present setup includes the setting of the preceding chapter. A first important question is which trading strategies we want to allow in this model. At the very least we shall allow strategies in which the asset portfolio is only rebalanced at a finite number of stopping times, in a predictable manner. Definition 3.1.1. A d-dimensional process is called a simple trading strategy if it is of the form t = n i=1 i1(i-1,i](t), where 0 = 0 n T are finite stopping times and the i are ddimensional, Fi-1 -measurable random variables. The strategy is called admissible if, in addition, the stopped process Sn and the random variables 1, . . . , n are uniformly bounded. 18 The Kreps-Yan theorem The interpretation of the definition is clear: (j) i is the number of assets of type j in the portfolio between times i-1 and i. As in the preceding chapter we define the stochastic process S by ( S)t = n i=1 i, Sit - Si-1t , t [0, T]. As before, S should be interpreted as the value process of a self-financing portfolio starting with 0 initial capital and following the trading strategy , the adjustments of the positions in assets 1 to d being financed by taking the appropriate amount from the "bank account", modelled by the numerair asset 0. Our first notion of no-arbitrage in this setting is the requirement that we can not make a risk-free profit by following a simple, admissible strategy. We proceed analogously to the finite case and first define the space Ks = {( S)T | simple, admissible} of pay-offs that can be achieved with 0 initial capital, following a simple, admissible self-financing trading strategy. Definition 3.1.2. We say that S satisfies the condition of no-arbitrage with simple strategies if Ks L + (, F, P) = {0}. A sufficient condition for the absence of arbitrage with simple strategies is the existence of a so-called equivalent local martingale measure. This is, by definition, a probability measure Q equivalent to the objective measure P such that S is a local martingale under Q. Proposition 3.1.3. If there exists an equivalent local martingale measure, S admits no arbitrage with simple strategies. Proof. Let Q be an equivalent local martingale measure. We first show that for every simple, admissible strategy it holds that EQ( S)T = 0. By Definition 3.1.1 it suffices to show that if 0 T are stopping times such that S is bounded and X is a bounded, d-dimensional, F-measurable random variable, then EQ X, S - S = 0. This can be derived from the optional stopping theorem (Exercise 1). Now suppose we have f Ks , f 0, say f = ( S)T for a simple, admissible . Then by the first part of the proof we have EQf = 0. Since f is nonnegative it follows that f vanishes Q-a.s. and hence also P-a.s., since P and Q are equivalent. 3.2 Kreps-Yan theorem 19 The converse of this proposition is, unfortunately, not true. To have the existence of a (local) martingale measure in this general continuous-time setting, the absence of simple arbitrages is not strong enough. Example 3.1.4. Consider a process S = (St)t[0,1] with S0 = 1 and that is constant except for jumps at times tn = 1 - (n + 1)-1 for n = 1, 2, . . .. At time tn the process S has a jump of magnitude 3-n Zn, where Z1, Z2, . . . are independent random variables with P(Zn = 1) = 1-P(Zn = -1) = 1/2+n for certain numbers n (-1/2, 1/2). Since the process S is uniformly bounded, it is a martingale under a measure Q if it local martingale (check!). But there is only one probability measure under which S is a martingale, namely the measure Q under which Q(Zn = 1) = 1 - Q(Zn = -1) = 1/2. Hence, by Example B.2.4, there exists no equivalent local martingale measure if we choose the n such that 2 n = . However, this model does satisfy the condition of no-arbitrage with simple strategies. To see that, first observe that if a there is a simple admissible arbitrage strategy, then there is simple arbitrage strategy of the form = h1(,], for stopping times 1 and a bounded Fmeasurable random variable h (Exercise 2). Moreover, we only have to consider stopping times that take values in the collection of tn's. Such a strategy has payoff V = h(S -S). Now observe that on the event An = { = tn-1, tn} we have sign(S - S) = sign(Zn) = Zn, so sign(V ) = sign(h)Zn. By assumption sign(V ) 0, so we get that sign(h)1An Zn 0. Note that An Ftn-1 and hence, by definition of F, sign(h)1An is Ftn-1 measurable. So sign(h)1An and Zn are independent and in view of the preceding display, it follows that sign(h)1An = 0 a.s. (check) . Hence h = 0 on every event An and therefore h = 0 on the event { > } = nAn. 3.2 Kreps-Yan theorem As in Chapter 2 we can introduce the cone Cs = {f L (, F, P) : there exists a g Ks such that g f}. Then (see Exercise 5 of Chapter 2) no-arbitrage with simple integrands is equivalent to Cs L + (, F, P) = {0}. In the preceding section we remarked that this condition is not strong enough to guarantee the existence of an equivalent martingale measure. It turns out we have to replace the cone Cs by its closure C with respect to the weak -topology on L (, F, P), the latter space viewed as the dual of L1 (, F, P). Definition 3.2.1. We say that S satisfies the condition of no free lunch if C L + (, F, P) = {0}. 20 The Kreps-Yan theorem Clearly this condition is stronger than the condition of no-arbitrage with simple strategies. It gives us the following version of the fundamental theorem of asset pricing. Theorem 3.2.2 (Fundamental theorem of asset pricing II). The process S satisfies the condition of no free lunch if and only if there exists an equivalent local martingale measure. Proof. Suppose first that there exists an equivalent local martingale measure Q. Then by the first part of the proof of Proposition 3.1.3 it holds that EQf 0 for all f Cs . Since the map f EQf is weak -continuous (check), the inequality EQf 0 holds in fact for every f C. It follows that if f C, f 0, then EQf = 0, so that f vanishes Q-a.s. and hence also P-a.s.. Now suppose that S satisfies the condition of no free lunch. For (0, 1), define B = {f L : 0 f 1, Ef }. The set B is a weak -closed subset of the unit ball of L , the latter space viewed as dual of L1 (Exercise 4). Hence, by Alaoglu's theorem it is weak -compact. Clearly, it is also convex. By the separation theorem there exists a g L1 and , R such that sup fC Egf < inf hB Egh. Since 0 C we have 0. Since C is a cone, it follows that Egf 0 for all f C, hence sup fC Egf 0 < inf hB Egh. Since C contains all negative functions in L we have g 0 a.s.. Also observe that 1 B, so that Eg > 0, and therefore g does not vanish almost surely. We renormalize g such that Eg = 1. For every positive integer n we now define the probability measure Qn by dQn = g2-n dP and Q = anQn, for a sequence an of positive numbers such that an = 1. Note that if P(A) > 2-n then 1A B2-n and hence Qn(A) > 0. It follows that P is absolutely continuous with respect to Q (check). The fact that Q is absolutely continuous relative to P is immediate, and hence P and Q are equivalent. To complete the proof observe that if f C, then EQf = anEQn f = anEg2-nf 0. This implies that S is a local martingale under Q (Exercise 3). 3.3 Discussion We saw in this chapter that in the general continuous-time setting, absence of arbitrage with simple strategies is not sufficient for the existence of a (local) martingale measure. The collection of pay-offs that can be attained with simple strategies is somehow to small, and it turned out to be necessary to take the 3.3 Discussion 21 weak -closure of this set. While this is completely satisfactory from a mathematical point of view, we should observe that it destroys the economic meaning of Theorem 3.2.2. Since taking the weak -closure is not a very intuitive operation, the weak -closure of a collection of pay-offs of simple strategies is not a set that can for instance be interpreted as a collection of pay-offs of "more complex" strategies. To obtain an economically meaningful result, we would prefer to replace the weak -topology by a stronger, more intuitive one. It turns out that this is possible if we are willing to restrict ourselves to asset prices that are semimartingales. Then the class of simple strategies can be enlarged in a natural way, using the theory of integration with respect to semimartingales. Taking the closure with respect to weak -topology can then be replaced by taking the closure in the norm-topology of L , which is much more satisfactory from the economic perspective. 22 The Kreps-Yan theorem 3.4 Exercises 1. Complete the first part of the proof of Proposition 3.1.3. 2. Show that if there exists a simple arbitrage strategy, there also exists an arbitrage strategy of the form = h1(,] with 1 stopping times and h a bounded, F-measurable random variable. (Hint: use induction on the number of stopping times in the given simple strategy.) 3. Show that if the equivalent measure Q satisfies EQf 0 for all f C, then it is an equivalent local martingale measure. 4. Show that the set B defined in the proof of Theorem 3.2.2 is a weak closed subset of the unit ball of (L1 ) . 4 The general theorem 4.1 Preliminaries on stochastic integration In this chapter we will assume that the asset price process S is a one-dimensional semimartingale, defined on some filtered probability space (, F, (Ft), P) satisfying the usual conditions. We will consider all processes on a finite time interval [0, T], with the time horizon T > 0 fixed. As before, S is interpreted as the value of an asset relative to a numerair asset, the numerair asset itself has the constant value 1. We will interpret a predictable process as a trading strategy, t denoting the number of non-numerair assets that we hold at time t. If is a simple process of the form = i1(ti-1,ti](t), for 0 = t0 < < tn = T a deterministic partition of [0, T] and i bounded and Fti-1 -measurable, then, as explained in Chapter 2, the process ( S)t = i(Stit - Sti-1t), t [0, T], can be interpreted as the value process of a self-financing portfolio starting with 0 initial capital and following the trading strategy , the adjustments of the position of the non-numerair assets being financed by taking the appropriate amount from the "bank account", modelled by the numerair asset. As the notation suggests, S is exactly the stochastic integral process of the locally bounded, predictable process relative to the semimartingale S. Hence, using stochastic integration theory we can now go beyond simple strategies. However, to retain an economically meaningful theory we have to verify that for non-simple predictable processes, S can still be interpreted as the value process associated to the trading strategy . Since we are now considering strategies that can involve continuous trading, we should consider approximations to make this precise. For a fine enough partition 0 = t0 < < 24 The general theorem tn = T of [0, T], the strategy is well approximated by the simple strategy ti-1 1(ti-1,ti] and the value process associated to this simple strategy equals ti-1 (Sti - Sti-1). Hence, we want the latter process to be a good approximation of the integral process S. Thanks to the continuity property of the stochastic integral this is indeed the case. Lemma 4.1.1. Let be a left-continuous process. Then if 0 = tn 0 < < tn kn = T is a sequence of partitions of [0, T] with mesh tending to 0, we have sup t[0,T ] tn i-1 (Stn i t - Stn i-1t) - ( S)t P 0. Proof. For every n, define the process n by n = tn i-1 1(tn i-1,tn i ]. Then n is left-continuous and adapted, hence locally bounded and predictable. Since is left-continuous it holds that n on [0, T] × , and we have |n t | sup st |s|. The process on the right-hand side of the display is adapted and left-continuous, hence predictable and locally bounded (cf. Exercise 5.53 in Van der Vaarťs notes). The conclusion of the lemma now follows from Lemma 5.52 of Van der Vaarťs notes. Unfortunately, the definition of the stochastic integral of locally bounded predictable processes with respect to semimartingales is still not general enough for the next version of the FTAP. To extend the integral we first endow the space of semimartingales with a topology. We denote by S(P) the space of all P-semimartingales on our fixed filtered probability space. For X S(P) we define X S(P) = sup |H|1 E(|(H X)T | 1), where the supremum is over all predictable processes H that are uniformly bounded by 1. It is easy to see that S(P) satisfies the triangle inequality (check). It follows that we can define a metric d on S(P) by setting d(X, Y ) = X - Y S(P). The topology that this metric generates on S(P) is called the semimartingale topology. Observe that Xn X in this topology if and only if (H Xn )T P (H X)T for all uniformly bounded predictable processes H (Exercise 1). We can now use the semimartingale topology to extend the definition of the stochastic integral. For a predictable process H and a positive integer n the 4.1 Preliminaries on stochastic integration 25 process H1{|H|n} is bounded and predictable. Hence if X is a semimartingale the stochastic integral process H1{|H|n} X is well defined in the sense of integration of locally bounded predictable processes (see for instance Van der Vaarťs notes). If X S(P), the sequence of processes H1{|H|n} X belongs to S(P) as well. If the sequence has a limit in S(P) (relative to the semimartingale topology) we say that H is X-integrable and we denote the limit semimartingale by H X. Observe that by Lemma 5.52 of Van der Vaart, the new definition of H X coincides with the old one if H is locally bounded (check). Since the FTAP involves changes of measure, it is useful to investigate how stochastic integrals depend on the underlying probability measure. We write (H X)P if we want to emphasize the dependence of the integral process on P. For a simple predictable process it is clear that the stochastic integral does not depend on the underlying measure at all. Now let H be a nonnegative, bounded, predictable process and X a P-semimartingale and let Q be equivalent to P. A general version of Girsanov's theorem says that for equivalent probability measures P and Q, the spaces S(P) and S(Q) of P- and Q-semimartingales coincide, hence X is a Q-semimartingale as well. Now by standard measure theory there exist a sequence of Hn of simple predictable processes, independent of P, such that Hn H on [0, ) × . By Lemma 5.52 of Van der Vaart we have that (Hn X)t P (H X)P t for all t 0. Hence, there exists a sequence kn such that (Hkn X)t (H X)P t , P-a.s.. Repeating the argument with Q instead of P we see that kn has a further subsequence ln such that (Hln X)t (H X)Q t , Q-a.s.. But P and Q are equivalent, so (H X)P t = (H X)Q t almost surely (relative to P or Q). We conclude that for bounded predictable H and P and Q equivalent, (H X)P and (H X)Q are indistinguishable. It can be shown that if P and Q are equivalent, the semimartingale topologies induced on S(P) = S(Q) by P and Q coincide. For a predictable process H we just observed that H1{|H|n} X does not depend on the probability measure. It follows that whether or not a predictable process H is X-integrable only depends on the equivalence class of the underlying probability measure P, and the same holds for the integral processes H X. Some care should be taken with integrands that are not locally bounded. For the extended integral it is for instance no longer true that the integral with respect to a local martingale is again a local martingale. Example 4.1.2. Suppose we have a standard exponential random variable and, independent of , a standard Bernoulli variable B, i.e. P(B = -1) = P(B = 1) = 1/2. Define the process M by Mt = B1{t}. Then M is a martingale relative to its natural filtration (Ft) (Exercise 3). Now define the deterministic process H by Ht = 1/t for t > 0. Then it holds that (H1{|H|n} M)t = 0, t < B 1{1/n}, t . It follows that H1{|H|n} M converges in the semimartingale topology to the 26 The general theorem process X given by Xt = 0, t < B , t , and X = H M by definition (check!). Observe however that E|Xt| = E 1 1{t} = t 0 1 x e-x dx = , hence X is not a martingale. It can be shown that X is not a local martingale either (Exercise 4). 4.2 No free lunch with vanishing risk and the FTAP Replacing the general asset price process of the preceding chapter by a semimartingale will allow us to replace the economically meaningless condition of no free lunch by the condition of no free lunch with vanishing risk. As discussed in the preceding section, we think of a predictable process as describing a self-financing trading strategy. We will assume that a trader has a finite credit line, in the sense that her wealth always stays above some deterministic (but possibly very negative) number. This is formalized by the following definition. Definition 4.2.1. An a-admissible strategy is an S-integrable predictable process that satisfies (S) -a. An admissible strategy is a predictable process that is a-admissible for some a > 0. In order to formulate the condition of no free lunch with vanishing risk we introduce the sets K = {( S)T : admissible}, which is a convex cone in the space L0 of finite-valued random variables (it is not a linear space in general, since admissibility is a one-sided restriction), and C = {f L : there exists a g K such that g f}. By C we denote in this chapter the closure of C with respect to the norm-topology of L . Definition 4.2.2. We say that S satisfies the condition of no free lunch with vanishing risk if C L + = {0}. Observe that this condition has a clear economic interpretation. If S does not satisfy the condition, there exists for every small enough > 0 an admissible strategy (depending on ) such that for the pay-off we have (S)T > - and P(( S)T > 0) > 0 (Exercise 2). Hence, if we are willing to take an arbitrarily 4.3 Sketch of proof of the fundamental theorem 27 small, but positive loss, we have a positive probability of receiving a strictly positive pay-off. We can now formulate the following version of the fundamental theorem of asset pricing. The proof is discussed in the next section. Theorem 4.2.3 (FTAP IIIa). If S = (St)t[0,T ] is a bounded, real-valued semimartingale, then there exists an equivalent martingale measure if and only if S satisfies the condition of no free lunch with vanishing risk. If S is only locally bounded, the martingale measure has to be replaced by a local martingale measure. Corollary 4.2.4 (FTAP IIIb). If S = (St)t[0,T ] is a locally bounded, realvalued semimartingale, then there exists an equivalent local martingale measure if and only if S satisfies the condition of no free lunch with vanishing risk. Proof. The sufficiency of no free lunch with vanishing risk follows from the preceding theorem. Indeed, suppose it holds and let n be stopping times such that |Sn | Kn, with Kn deterministic numbers. Define the new process ~S by ~S = S1[0,1] + n2 2-n 1 Kn 1(n-1,n] S. Then ~S is a bounded semimartingale. Moreover, it satisfies the condition of no free lunch with vanishing risk (Exercise 5). Hence, by the theorem, there exists an equivalent probability measure Q such that ~S is a Q-martingale. But then the original process S is a Q-local martingale (Exercise 6). The converse statement is proved as in Theorem 4.2.3, see the next section. 4.3 Sketch of proof of the fundamental theorem 4.3.1 The relatively easy half We noted above that if M is a local martingale and H is M-integrable, then H M is not necessarily a local martingale anymore. The proof of the fact that the existence of a martingale measure is sufficient for no free lunch with vanishing risk uses a characterization of the martingality of H M. To explain this characterization it is useful to first note that a local martingale M is in fact locally uniformly integrable. Indeed, let n be a localizing sequence for M, i.e. n a.s. and every Mn is a martingale. Then n = n n also satisfies 28 The general theorem n a.s. and every Mn is a uniformly integrable martingale. Moreover, if we now consider the stopping time n = n inf{t : |Mt| > n} we have that sup tn |Mt| n + |Mn |. Since n n and Mn is UI, the right-hand side of the display is integrable. So we have proved the following useful lemma. Lemma 4.3.1. A local martingale M is locally uniformly integrable. Moreover, there exists a localizing sequence n such that E sup tn |Mt| < for all n. Now consider a local martingale M and an M-integrable predictable process H and suppose that H M is a local martingale. For a localizing sequence n such that E suptn |(H M)t| < we have supt |(H M)n t | 2 suptn |(H M)t|, so (H M)n Zn, where Zn = -2 suptn |(H M)t|. So we see that there exists a localizing sequence n and integrable random variables Zn such that (H M)n Zn for all n. It turns out that the converse is true as well. Theorem 4.3.2. Let M be a local martingale and H a predictable processes that is M-integrable. Then H M is a local martingale if and only if there exists a localizing sequence n and integrable random variables Zn such that (H M)n Zn for all n. We can now prove that the existence of an equivalent martingale measure implies there is no free lunch with vanishing risk. Suppose there exists an equivalent martingale measure Q and let be an a-admissible strategy. By the observations in the preceding section the integral process S does not depend on the underlying measure. Under Q the process S is a local martingale. Now consider the stopping times n = inf{t : ( S)t n}. Then n and by the admissibility of we have ( S)n t = ( S)n t - ( S)n t- -(n + a). Hence, by the preceding theorem, S is a Q-local martingale. Together with the a-admissibility this implies that S is in fact a Q-supermartingale (Exercise 7). It follows that for every f C we have EQf 0 and then also EQf 0 for every f C. This implies that every f C L + vanishes Q-a.s., but then also P-a.s.. 4.3 Sketch of proof of the fundamental theorem 29 4.3.2 The much more difficult half The essential step in the proof of the fact that no free lunch with vanishing risk is sufficient for the existence of a martingale measure is the following theorem. Theorem 4.3.3. If the bounded semimartingale S satisfies the condition of no free lunch with vanishing risk, then the cone C is weak -closed. Indeed, if we have this result we can argue as in the proof of Theorem 3.2.2 to find a probability measure Q equivalent to P such that EQf 0 for all f C. It then follows that EQf = 0 for all f K. Since S is bounded it is integrable. Moreover, for s t and A Fs the predictable process = 1A×(s,t] is admissible, so ( S)T K and hence EQ1A(St - St) = EQ( S)T = 0, which shows that S is a Q-martingale. The proof of Theorem 4.3.3 is long and difficult. The first difficulty that arises is that the weak -topology is in general not metrizable. This implies that to show that a set is weak -closed, it is in general not enough to consider converging sequences (in fact, one should consider nets). However, it turns out that in the present context the situation is not that complicated, since we can use the following consequence of the so-called Krein-Smulian theorem. Theorem 4.3.4. Let (E, E, ) be a finite measure space and C L () a convex cone. Suppose that for each uniformly bounded sequence fn in C that converges in measure to a function f, it holds that f C. Then C is weak - closed. So to prove Theorem 4.3.3 it suffices to consider a sequence hn in C such that |hn| 1 for all n and hn as h for some h L , and show that h belongs to C. To prove that h C we have to find an f0 K such that h f0. To that end it turns out to be useful to consider the set D = {f : there exist 1-admissible n such that (n S)T as f, f h}. It can be shown that this set contains a maximal element f0. So the random variable f0 dominates h and is the almost sure limit of elements (n S)T of K, n 1-admissible. The remaining task is to show that f0 belongs to K itself. The first step is the observation that convergence of n S at the terminal time T in fact implies convergence for all time points. To see this one first shows that as n, m , sup t[0,T ] |(n S)t - (m S)t| P 0. (4.1) The proof of this fact uses the 1-admissibility of the n and the maximality of f0. Next we want to apply some results on the semimartingale topology, 30 The general theorem but the preceding display does not imply that n S is a Cauchy sequence in the semimartingale topology. A long and technical proof however shows that for every n there exits a n in the convex hull of the processes n, n+1, . . . such that n S is a Cauchy sequence in the semimartingale topology. The semimartingale topology can be shown to be complete, so that we now have n S Z for some semimartingale Z. Moreover, Mémin's theorem shows that the semimartingale Z must necessarily be of the form Z = S for some S-integrable predictable process . Observe that since n is a convex combination of n, n+1, . . ., the process n is 1-admissible. Since the convergence in the semimartingale topology implies that (n S)t P ( S)t for all t [0, T], it follows that is 1-admissible as well. The fact that n is a convex combination of n, n+1, . . . also implies that the almost sure limit of (n S)T equals the almost sure limit of (n S)T , which is f0 (Exercise 8). Combined with the previous observations we conclude that f0 = ( S)T , so indeed f0 K. 4.4 Example: Itô processes Suppose we have, on some filtered probability space (, F, (Ft), P) satisfying the usual conditions, continuous adapted processes B and X satisfying Bt = exp t 0 rs ds , dXt = tXt dt + tXt dWt, where W is a standard Brownian motion and r, and are locally bounded predictable processes. All processes are indexed by [0, T] for some fixed time horizon T > 0. We think of B as describing a bank account process with (continuous, stochastic) interest rate rt, and X as describing the value of a stock with local return rate t and (possibly stochastic) volatility t. We use B as numerair, putting S = X/B. Integration by parts then gives the stochastic differential equation dSt = (t - rt)St dt + tSt dWt for the discounted process S (check). Now suppose that the Sharp ratio t = t - rt t is uniformly bounded by a deterministic constant for all t [0, T]. Then by the classical Girsanov theorem, there exists a probability measure Q equivalent to P under which the process Bt = Wt + t 0 s ds, t [0, T], 4.4 Example: Itô processes 31 is a Brownian motion. Combining the definition of B with the SDE for S we get dSt = tSt dBt. In particular, we see that S is a Q-local martingale. Hence, by Corollary 4.2.4, this model satisfies the condition of no free lunch with vanishing risk. Observe that the classical Black-Scholes model corresponds to the special case that r, and are deterministic and independent of time. The condition on the Sharp ratio is then trivially satisfied, so we recover the well-known fact that the Black-Scholes model is free of arbitrage (in the sense of no free lunch with vanishing risk). 32 The general theorem 4.5 Exercises 1. Show that Xn P X if and only if E(|Xn - X| 1) 0. 2. Show that if S does not satisfy the condition of no free lunch with vanishing risk, there exists for every > 0 small enough an admissible strategy with a pay-off satisfying ( S)T > - and P(( S)T > 0) > 0. 3. Show that the process M defined in Example 4.1.2 is a martingale. 4. Show that the process X in Example 4.1.2 is not a local martingale. 5. Show that the process ~S in the proof of Corollary 4.2.4 satisfies the condition of no free lunch with vanishing risk. 6. Show that the process S in the proof of Corollary 4.2.4 is a Q-local mar- tingale. 7. Show that a local martingale that is bounded from below by a deterministic number is a supermartingale. 8. Show that in Section 4.3.2, we have the a.s. convergence (n S)T f0. A Elements of functional analysis A.1 Separating hyperplane theorem Let v Rn and R be given and consider the set H = {x Rn : v, x = }. For x H we have v, x - (/ v 2 )v = 0, so H = v + (/ v 2 )v. The complement of H consists of the two sets {x : v, x < } and {x : v, x > } on the two "sides" of the hyperplane. The following theorem says that for two disjoint, convex sets, one compact and one closed, there exists two "parallel" hyperplanes such that the sets lie strictly one different sides of those hyperplanes. The assumption that one of the sets is compact can not be dropped (see Exercise 1) Theorem A.1.1 (Separating hyperplane theorem). Let K and C be disjoint, convex subsets of Rn , K compact and C closed. There exist v Rn and 1, 2 R such that v, x < 1 < 2 < v, y for all x K and y C. Proof. Consider the function f : K R defined by f(x) = inf{ x - y : y C}, i.e. f(x) is the distance of x to C. The function f is continuous (check) and since K is compact, there exists x0 K such that f attains its minimum at x0. Let yn C be such that x0 - yn f(x0). By the parallelogram law we have yn - ym 2 2 = yn - x0 2 ym - x0 2 2 = 1 2 yn - x0 2 + 1 2 ym - x0 2 yn + ym 2 - x0 2 . 34 Elements of functional analysis By convexity (yn + ym)/2 C, so that (yn + ym)/2 - x0 f(x0). Hence, we have yn - ym 2 2 1 2 yn - x0 2 + 1 2 ym - x0 2 - f2 (x0). The right-hand side of this display converges to 0 as n, m . So the yn form a Cauchy sequence and hence they converge to some y0 Rn . Since C is closed, y0 C. Let v = y0 - x0. Since K and C are disjoint, v = 0. It follows that 0 < v 2 = v, y0 - x0 = v, y0 - v, x0 . It remains to show that v, x v, x0 and v, y0 v, y for all x K and y C. Take y C. Since C is convex, the line segment y0 + (y - y0), [0, 1], belongs to C. Since y0 minimizes the distance to x0, we have y0 - x0 y0 - x0 + (y - y0) for every . By squaring this we find that 0 2 y0 - x0, y - y0 + 2 y - y0 2 . Dividing by and then letting 0 gives v, y - y0 0, as desired. A similar argument shows that v, x v, x0 for x K. The polar C0 of a set C Rn is defined as C0 = {y Rn : x, y 1 for all x C}. Note that in the special case that C is closed under multiplication with positive scalars, we have C0 = {y Rn : x, y 0 for all x C} (check). For a given x, the set C0 x = {z : x, z 0} is the set of all vectors that lie on the same side of x as -x. The polar is in this case the intersection of all the sets C0 x for x C. To illustrate the bipolar theorem geometrically, consider a V -shaped set: C the union of two rays emanating from the origin. Then one readily sees that the polar of the polar of C precisely equals the convex hull of C. The general result is as follows. Theorem A.1.2 (Bipolar theorem). Let C Rn contain 0. Then the bipolar C00 = (C0 )0 equals the closed convex hull of C. Proof. It is clear that C00 is a closed, convex set containing C, so the closed convex hull A of C is a subset of C00 . Suppose that the converse inclusion does not hold. Then there exists a point x0 C00 that is not in A. By the separating hyperplane theorem there then exists a vector v Rn and 1, 2 R such that x0, v > 1 > 2 > y, v for all y A. Since 0 C A we have 1 > 0. Dividing by 1 shows there exists a vector v Rn such that x0, v > 1 > y, v for all y A. The second inequality implies that v C0 , and then the first one implies that x0 C00 , which gives a contradiction. A.2 Topological vector spaces 35 A.2 Topological vector spaces A vector space X is called a topological vector space if it is endowed with a topology which is such that every point of X is a closed set and the addition and scalar multiplication operations are continuous. It is easy to see that translation by a fixed vector and multiplication by a nonzero scalar are homeomorphisms of a topological vector space. This implies in particular that the topology is translation-invariant, meaning that a set E X is open if and only if each of its translates x + E is open. Topological vector spaces have nice separation properties. Combined with the fact that points are closed sets, the next theorem implies for instance that they are always Hausdorff. Theorem A.2.1. Suppose that K and C are disjoint subsets of a topological vector space X, K compact and C closed. Then there exits a neighborhood V of 0 such that K + V and C + V are disjoint. Proof. The continuity of addition implies that for every neighborhood W of 0 there exist neighborhoods V1 and V2 of 0 such that V1 + V2 W (check). Now put U = V1 V2 (-V1) (-V2). Then U is symmetric (i.e. U = -U) and U + U W. Applying the same procedure to the neighborhood U we see that for every neighborhood W of 0 there exists a symmetric neighborhood U such that U + U + U W (etc.). Pick an x K. Then X\C is an open neighborhood of x. By translation invariance and the preceding paragraph there exists a symmetric neighborhood Vx of 0 such that x + Vx + Vx + Vx does not intersect C. By the symmetry of Vx this implies that x + Vx + Vx and C + Vx are disjoint (check). Since K is compact, it is covered by finitely many sets x1 + Vx1 , . . . , xn + Vxn . Put V = Vx1 Vxn . Then K + V (xi + Vxi + V ) (xi + Vxi + Vxi ) and none of the terms in the last union intersects C + V . The following lemma implies that if V is a neighborhood of 0 in a topological vector space X, then for every x X it holds that x rV if r is large enough. A set V with this property is called absorbing. Lemma A.2.2. Suppose V is a neighborhood of 0 in a topological vector space X and rn is a sequence of positive numbers tending to infinity. Then rnV = X. Proof. Fix x X. Then since V is open in X and x from R to X is continuous, { : x V } is open in R. The set contains 0, and hence it contains 1/rn for n large enough. This completes the proof. 36 Elements of functional analysis For an arbitrary absorbing subset A (for instance a neighborhood of 0) of a topological vector space X we define the Minkowsky functional A : X [0, ) by A(x) = inf{t > 0 : x/t A}. Note that A is indeed finite-valued, since A is absorbing. The following lemma collects properties that we need later. Lemma A.2.3. Let A be a convex, absorbing subset of a topological vector space X and let A be its Minkowsky functional. (i) A(x + y) A(x) + A(y) for all x, y X. (ii) A(tx) = tA(x) for all x X and t 0. Proof. For x, y X and > 0, consider t = A(x) + , s = A(y) + . Then by definition of A, x/t A and y/s A. Hence, the convex combination x + y s + t = t s + t x t + s s + t y s belongs to A as well. This proves (i). The proof of (ii) is easy. For the proof of the following characterization of continuous linear functionals we need the notion of a balanced neighborhood. A set B X is said to be balanced if B B for every scalar R with || 1. Lemma A.2.4. Every neighborhood of 0 contains a balanced neighborhood of 0. Proof. Let U be a neighborhood of 0. Since scalar multiplication is continuous, there exists a > 0 and a neighborhood V of 0 in X such that V U whenever || < . Then W = ||< V is a balanced neighborhood of 0. A linear map : X R is called a linear functional on the space X. A linear functional on X is called bounded on a subset A X if there exists a number K > 0 such that |x| K for all x A. Theorem A.2.5. Let be a nontrivial linear functional on a topological vector space X. Then is continuous if and only if is bounded on a neighborhood of 0. Proof. Suppose is continuous. Then the null space N = {x X : x = 0} is closed. Since is nontrivial, there exists x X\N. By Theorem A.2.1 there exists a balanced neighborhood V of 0 such that x+V and N are disjoint. Then (V ) is a balanced subset of R. Suppose it is not bounded. Then since it is balanced, it most be all of R. In particular, there then exists a y V such that A.3 Hahn-Banach theorem 37 y = -x. But then x + y N, a contradiction. Hence, (V ) is bounded, i.e. is bounded on V . Conversely, suppose that |x| M for all x V . For r > 0, put W = (r/M)V . Then for x W, say x = (r/M)y for y V , we have |x| = (r/M)|y| r. Hence, is continuous at 0. By translation invariance, it is continuous everywhere. A.3 Hahn-Banach theorem The proof of the following version of the Hahn-Banach theorem relies on the axiom of choice, in the form of the Hausdorff maximality theorem: Every nonempty partially ordered set P contains a totally ordered subset Q which is maximal with respect to the property of being totally ordered. A proof of this fact can for instance be found in Rudin (1987), pp. 395­396. Theorem A.3.1 (Hahn-Banach theorem). Suppose X is a (real) vector space and p : X R satisfies p(x + y) p(x) + p(y) and p(tx) = tp(x) for x, y X and t 0. Then if f is a linear functional on a subspace M of X such that f(x) p(x) for all x M, f extends to a linear functional on the whole space X such that -p(-x) x p(x) for all x X. Proof. Suppose M is a proper subspace of X and pick x1 X\M. For x, y M we have f(x) + f(y) = f(x + y) p(x + y) p(x - x1) + p(y + x1), hence f(x) - p(x - x1) p(y + x1) - f(y). So there exists an such that f(x) - p(x - x1), f(y) + p(y + x1) (A.1) for all x, y M. Now let M1 be the vector space spanned by M and x1. An element of M1 is of the form x+x1 for some R. So we can extend f to M1 by setting f1(x + x1) = f(x) + . Then f1 is a well-defined linear functional on M1 and the inequalities in (A.1) imply that f1(x) p(x) for all x M1 (check). Let C be the collection of pairs (M , f ), where M is a subspace of X containing M and f is a linear extension of f to M such that f p on M . Put an ordering on C by saying that (M , f ) (M , f ) if M M and f |M = f . This is a partial ordering and C is not empty. Hence, by the Hausdorff maximality theorem, we can extract a maximal totally ordered subcollection C . Let ~M be the union of all M for which (M , f ) C . Then 38 Elements of functional analysis ~M is a subspace of X (check). If x ~M then x M for some M such that (M , f ) C . We then put x = f (x). This defines a linear function on ~M and we have that p on ~M (check). If ~M were a proper subspace of X the construction of the preceding paragraph would give us a further extension of , contradicting the maximality of C . Hence, ~M = X. This completes the proof, upon noting that p implies that -p(-x) -(-x) = x for all x X. Before we use the Hahn-Banach theorem to prove the infinite-dimensional version of the separating hyperplane theorem we introduce some more concepts and notation. A topological vector space X is called locally convex if for every neighborhood V of 0 there exists a convex neighborhood U of 0 such that U V . The space of continuous linear maps from X to R is denoted by X . It is called the dual of X, and is treated in more detail in the next section. Theorem A.3.2 (Separation theorem). Let A and B be disjoint, nonempty, convex subsets of a topological vector space X. (i) If A is open there exist X and R such that x < y for every x A and y B. (ii) If X is locally convex, A is compact and B is closed, there exist X and 1, 2 R such that x < 1 < 2 < y for every x A and y B. Proof. (i). Pick a0 A and b0 B and put x0 = b0-a0. Define C = A-B+x0 and note that C is a convex, open neighborhood of 0. Let C be the Minkowsky functional of C. Let M be the linear subspace generated by x0 and define a linear functional f on M by putting f(x0) = . Since A and B are disjoint, x0 C so we have C(x0) 1 and hence, for 0, f(x0) = C(x0) = C(x0). For < 0 we have f(x0) < 0 C(x0). By Lemma A.2.3 and the Hahn-Banach theorem, Theorem A.3.1, the functional f extends to a linear functional on X, and the extension satisfies x C(x) for all x X. In particular 1 on C, so that || 1 on the neighborhood C -C of 0. By Theorem A.2.5 this implies that is continuous, i.e. X . Now for a A and b B we have that a - b + 1 = (a - b + x0) C(a - b + x0) < 1, since a - b + x0 C and C is open (Exercise 2), so a < b. It follows that (A) and (B) are disjoint, convex subsets of R, the first one lying on the left of the second one. Since A is open and is nonconstant, (A) is open as well A.4 Dual space 39 (Exercise 3). Letting be the right end point of (A) completes the proof of (i). (ii). By Theorem A.2.1 and the local convexity of X there exists a convex neighborhood V of 0 such that (A + V ) B = . By the proof of part (i) there exists X such that (A+V ) and (B) are disjoint, convex subsets of R, the first one lying on the left of the second one, the first one being open. Moreover, (A) is a compact subset of (A + V ). The proof is now easily completed. Corollary A.3.3. If X is a locally convex topological vector space, X separates the points of X. Proof. given distinct points x, y X, apply the separation theorem with A = {x} and B = {y}. For x X and X we define, in analogy with the finite-dimensional situation, x, = x. The polar C0 of a set C X is defined as C0 = { X : x, 1 for all x C}. Similarly, the bipolar is defined as C00 = (C0 )0 = {x X : x, 1 for all C0 }. Theorem A.3.4 (Bipolar theorem). The bipolar C00 of a subset C of a locally convex topological vector space X equals the closed convex hull of C. Proof. It is clear that C00 is a convex set containing C, so the closed convex hull A of C is a subset of C00 . Suppose that the reverse inclusion does not hold. Then there exists a point x0 C00 that is not in A. By the separation theorem there then exists a functional X such that x0 > 1 > y for all y A (check). The second inequality implies that C0 , and then the first one implies that x0 C00 , which is a contradiction. A.4 Dual space The dual of a topological vector space X is the space X of continuous linear functionals on X. By Theorem A.2.5 this is the same as the space of linear functionals that are bounded on a neighborhood of 0. It is easy to see that if the topology on X is induced by a norm , a linear functional belongs to X if and only if the unit ball in X is mapped into a bounded subset of R. In that case we define the norm of by = sup x 1 |(x)| 40 Elements of functional analysis and we have the relation |(x)| x for every x X. Example A.4.1. Let (E, E, ) be a measure space with a finite measure, p [1, ) and X = Lp (E, E, ). Consider a continuous linear functional on X. Then the map : E R defined by (B) = (1B) is a signed measure (note that the finiteness of implies that is well-defined). Indeed, if Bn are disjoint elements of E and B = Bn, then 1knBk 1B in Lp . Since is continuous, this implies that is countably additive. If (B) = 0 then 1B vanishes in Lp and hence (B) = 0, so . Hence, by the Radon-Nikodym theorem, there exists a g L1 such that (1B) = (B) = B g d for all B E. By linearity we then have (f) = fg d (A.2) for all simple functions f. Every bounded measurable function f is the uniform limit of simple functions and since is finite, uniform convergence implies convergence in Lp . It follows that (A.2) holds for all f L . Suppose that p > 1 and let q be the conjugate exponent. For En = {x : |g(x)| n} we have, since g is bounded on En and is continuous and hence bounded, En |g|q d = En |g|q-1 sign(g)g d = (1En |g|q-1 sign(g)) En |g|q d 1/p . It follows that En |g|q d 1/q (A.3) and letting n shows that g Lq . If p = 1 then for every B E we have B g d = |(1B)| (B). But this implies that |g| a.e. (indeed: if not there would exist an > 0 such that the set B = {x : |g(x)| > + } has positive -measure, leading to a contradiction), hence g L . So in all cases the function g in (A.2) belongs to Lq . We proved already that (A.2) holds for all bounded functions f. Now is continuous on Lp by assumption and H¨olders inequality implies that the right-hand side is continuous for f Lp as well. This shows that the relation holds in fact for all f Lp . Uniqueness of g is easy to prove. We conclude that we may identify the dual of Lp with Lq . Moreover, using (A.3) it is easy to see that for (Lp ) given by (A.2), we have = g Lq (Exercise 4). Let X be a topological vector space with dual X . Every point x X induces a linear functional on X , defined by x. The weak -topology of X is the weakest (i.e. smallest) topology making all these maps continuous. A.4 Dual space 41 The following theorem states that X with the weak -topology is a locally convex topological vector space. This implies for instance that we can apply the separation theorem to it. In general, the space X endowed with the weak topology is not a Banach space. (In fact, it is not even metrizable if X is an infinite-dimensional Banach space.) Theorem A.4.2. The dual X of a topological vector space X, endowed with the weak -topology, is a locally convex topological vector space. Its dual is given by { x : x X}. Proof. Denote by fx be the linear functional x. If = in X , there exists an x X such that fx = fx . Hence, in R there exist disjoint neighborhoods U of fx and U of fx . Since fx is continuous, f-1 x (U) and f-1 x (U ) are disjoint neighborhoods of and . This shows that X is Hausdorff, and in particular that points are closed. To show that the weak -topology is translation invariant, consider an open base set U = { : x1 B1, . . . , xn Bn} and X . Then + U = { : x1 B1 + x1 . . . , xn Bn + xn} is an open base set as well. It follows that the topology is translation invariant. Note that the open sets V of the form V = { : |x1| < r1, . . . , |xn| < rn} (A.4) for x1, . . . , xn X and r1, . . . , rn > 0 form a local base at 0. Every such set V is convex, balanced and absorbing (check). In particular, X is a locally convex space. For the set V in the preceding display we have V/2 + V/2 = V and hence addition is continuous at (0, 0). As for scalar multiplication, suppose that V for some scalar R and X . By Exercise 2, there exists t > 0 such that t < 1/|| and tV . For > 0 and tV we have that (+) (+)tV . Hence, since V is balanced, (+) V for all such that ||t+||t 1. Since ||t < 1 there is a nonempty interval around 0 of satisfying this condition. Hence, scalar multiplication is continuous. It remains to identify the dual of X (endowed with the weak -topology). If x X, the linear map (x) is weak -continuous by definition of the weak topology. Conversely, let f : X R be weak -continuous. By Theorem A.2.5, f is bounded on a neighborhood of 0, and hence also on a base set V of the form (A.4). This implies that f vanishes on the set N = { : x1 = = xn = 0} (Exercise 5). Now N is the kernel of the linear map : X Rn defined by () = (x1, . . . , xn). It follows that the linear map F : (X ) R given by F(()) = f() is well defined (check). We can extend F to a linear functional on Rn . It is then necessarily of the form F(z1, . . . , zn) = izi for certain real numbers i. In particular, f() = F(x1, . . . , xn) = ixi. So indeed, f() = x, with x = ixi. 42 Elements of functional analysis If X is a Banach space its dual X is endowed with a norm, and the unit ball in X is the set { X : |x| x for all x X}. In the normtopology this set is not compact in general (think of an infinite-dimensional Hilbert space). In the weak -topology however, it is always compact. Theorem A.4.3 (Banach-Alaoglu). The unit ball of the dual of a Banach space is weak -compact. Proof. Denote the Banach space by X and let B be the unit ball in its dual. By Tychonov's theorem, P = xX[- x , x ] is compact (relative to the product topology). We can view P as a collection of functions on X, with f P if and only if |f(x)| x for all x X. As such, we have B X P. Hence, B inherits two topologies: the weak -topology from X and the product topology from P. These two topologies on B coincide. To see this, take 0 B . The sets of the form V1 = { X : |x1 - 0x1| < r1, . . . , |xn - 0x1| < rn} and V2 = {f P : |f(x1) - 0x1| < r1, . . . , |f(xn) - 0x1| < rn} form a local base for the weak -topology and, respectively, the product topology at 0. Since B X P we have V1 B = V2 B and hence the two relative topologies coincide. Next we show that B is closed in P. Take f0 in the closure of B (with respect to the product topology). For x, y X, , R and > 0 we have that the set U = {f P : |f(x)-f0(x)| < , |f(y)-f0(y)| < , |f(x+y)-f0(x+y)| < } is an open neighborhood of f0. Hence, there exist an f U B . Since f is linear we have f0(x + y) - f0(x) - f0(y) = (f0 - f)(x + y) - (f0 - f)(x) - (f0 - f)(y) and hence |f0(x + y) - f0(x) - f0(y)| (1 + || + ||). Since was arbitrary, it follows that f0 is linear. By definition of P we have that |f0(x)| x for every x X, so indeed f0 B . The proof is now completed upon noting that by the preceding paragraph, B is compact with respect to the product topology. But by the first part of the proof, the latter topology coincides on B with the weak -topology. A.4 Dual space 43 Example A.4.4. Although the weak -topology has some nice properties according to Theorem A.4.2, it is good to note that it is typically "strange". Consider for instance a finite measure on the line and view L () as the dual of L1 (). Then from the form of the local base at 0 given in the proof of the theorem one sees that a sequence fn in L converges in the weak -topology to 0 if fng d 0 for every g L1 . By dominated convergence, this holds for instance for fn = 1(-n,n)c . This sequence does however not converge to 0 in the ordinary, uniform topology on L . More generally, to say that a function f L belongs to the weak -closure of a set C L does not necessarily mean that f is well-approximated by elements of C in a uniform or any other intuitively reasonable way. 44 Elements of functional analysis A.5 Exercises 1. Give an example which shows that the separation theorem does not hold in general if the assumption of compactness of one of the sets in dropped. 2. Suppose that C is an open neighborhood of 0 in a topological vector space and let C be its Minkowsky functional. Show that for all x C it holds that C(x) < 1. 3. Show that a non-constant linear functional on a topological vector space maps open sets to open sets. 4. In Example A.4.1, show that for the functional on Lp defined by (A.2) we have = g Lq . 5. In the last part of the proof of Theorem A.4.2, show that the functional f vanishes on the set N. B Elements of martingale theory B.1 Basic definitions Let (, F, P) be a probability space. A collection of Rd -valued random variables X = (Xt)tT indexed by a set T R is called a (d-dimensional) stochastic process. We call the process continuous (or cadlag), it its trajectories t Xt() are continuous (or cadlag). The process is called bounded if there exists a finite number K such that a.s. Xt K for all t. A filtration is a collection (Ft)tT of sub--fields of F such that Fs Ft for all s t. It is said to satisfy the usual conditions if it is right-continuous, i.e. s>tFs = Ft for all t and F0 contains all the P-null sets in F. A process X is called adapted to (Ft) is for every t, Xt is Ft-measurable. For a process X and t T we define FX t to be the -field generated by the collection of random variables {Xs : s t}. The filtration (FX t ) is called the natural filtration of the process X. It is the smallest filtration to which it is adapted. A process X = (Xt)t[0,T ] is called progressively measurable relative to the filtration (Ft)t[0,T ] if for all t, the map (, s) Xs() on × [0, t] is Ft B([0, t])-measurable. A [0, ]-valued random variable is called a stopping time relative to the filtration (Ft) if { t} Ft for every t. If is a stopping time and X a process, the stopped process X is defined by X t = Xt. A localizing sequence is a sequence of stopping times n increasing a.s. to infinity. A process X is said to have a property P locally if there exists a localizing sequence n such that for every n, the stopped process Xn has the property P. A process M is called a martingale relative to the filtration (Ft) if every Mt is integrable and for all s t it holds that E(Mt | Fs) = Ms a.s.. In accordance with the previously introduced notation the process M is called a local martingale if there exists a localizing sequence n such that for every n, the stopped process Mn is a martingale. Every martingale is a local martingale, but not vice versa.. 46 Elements of martingale theory B.2 Theorems For a filtration (Ft) and a stopping time we define F = {A F : A { t} Ft for all t}. The set F is always a -field and should be thought of as the collection of events describing the history before time . Theorem B.2.1 (Optional stopping theorem). Let M be a cadlag, uniformly integrable martingale. Then for all stopping times , E(M | F) = M. Theorem B.2.2 (Kakutani's theorem). Let X1, X2, . . . be independent nonnegative random variables with mean 1. Define M0 = 1 and Mn = X1X2 Xn. It holds that M is uniformly integrable if and only if (1 - E Xn) < . If M is not uniformly integrable, then Mn 0 a.s.. Corollary B.2.3. Let X = (X1, X2, . . .) and Y = (Y1, Y2, . . .) be two sequences of independent random variables. Assume Xi has a positive density fi with respect to a dominating measure , and Yi has a positive density gi with respect to . Then the laws of the sequences X and Y are equivalent probability measures on (R , B(R )) if and only if n i=1 ( fi - gi)2 d < . If the laws are not equivalent, they are mutually singular. Proof. Let (, F) = (R , B(R )) and Z = (Z1, Z2, . . .) the coordinate process on (, F), so Zi() = i. Let Fn F be the -field generated by Z1, . . . , Zn. Since the densities fi and gi are all positive, the distributions PX and PY of the sequences X and Y are equivalent on Fn. For A Fn we have PX(A) = A Mn dPY , where the Radon Nikodym derivative is defined by Mn = n i=1 fi(Zi)/gi(Zi). Observe that under PY , the process M is a martingale to which the preceding theorem applies. It is readily verified that the measures PX and PY are equivalent on the whole -field F if and only if M is uniformly integrable with B.2 Theorems 47 respect to PY (Exercise 1). Hence, by the preceding theorem, the measures are equivalent if and only if n i=1 1 - figi d < . The proof of the first part is completed by noting that ( fi - gi)2 d = 2 - 2 figi d. We noted that if PX and PY are not equivalent, then M is not uniformly integrable relative to PY . Hence, by the preceding theorem, Mn 0, PY a.s.. We can reverse the roles of X and Y , which amounts to replacing M by 1/M. Then we find that if PX and PY are not equivalent, 1/Mn 0, PXa.s.. It follows that for the event A = {Mn 0} we have PY (A) = 1 and PX(A) = 0. Example B.2.4. Let X = (X1, X2, . . .) and Y = (Y1, Y2, . . .) be two sequences of independent random variables. Suppose that P(Xi = 1) = P(Xi = -1) = 1/2 and P(Yi = 1) = 1 - P(Yi) = -1 = 1/2 + i for some i (-1/2, 1/2). By the corollary, applied with the counting measure, fi(1) = fi(-1) = 1/2, gi(1) = 1-gi(-1) = 1/2+i, the laws of the sequences X and Y are equivalent if and only if ( 1/2 - 1/2 + i)2 + ( 1/2 - 1/2 - i)2 < . By Taylor's formula the function h(x) = ( 1/2 - 1/2 + x)2 + ( 1/2 - 1/2 - x)2 behaves like a multiple of x2 near x = 0 (check!). It follows that the sequences are equivalent if and only if 2 i < . 48 Elements of martingale theory B.3 Exercises 1. In the proof of Corollary B.2.3, show that the measures PX and PY are equivalent on the whole -field F if and only if M is uniformly integrable. References Conway, J. B. (1990). A course in functional analysis, volume 96 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition. Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300(3), 463­520. Delbaen, F. and Schachermayer, W. (2006). The mathematics of arbitrage. Springer Finance. Springer-Verlag, Berlin. 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