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1 Local martingales and Girsanov’s theorem The purpose of this section is to formulate and prove Girsanov’s theorem on absolutely continuous changes of measure for discrete time processes. 1.1 Local martingales As always we have a probability space (Ω, F, P) on which all random variables are defined. Before defining local martingales, we relax the notion of conditional expectation of a random variable X by dropping the requirement that E |X| < ∞. Suppose X ≥ 0 and that G is a sub-σ-algebra of F. Then there exists a G-measurable random variable X̂ such that E X1G = E X̂1G for all G ∈ G (Exercise 1.1), we adopt again the notation X̂ = E [X|G] for any version X̂. If X is such that E [X + |G] < ∞ a.s. or E [X − |G] < ∞ a.s., we define X̂ = E [X + |G] − E [X − |G], the generalized conditional expectation of X given G, denoted (again) E [X|G]. Note that also here we have different versions, and that these are a.s. equal. In what follows we work with a fixed filtration F, without explicitly mentioning it further. Definition 1.1 Let X be an adapted process. If there exists a sequence of n stopping times T n such that T n → ∞ and such that the stopped processes X T are all martingales, we call the process X a local martingale. The sequence (T n ) is called a fundamental or localizing sequence. It is obvious that any martingale is a local martingale, but the converse is not n true, see Exercise 1.2. Note also that the definition implies that X0 = X)T belongs to L1 (Ω, F, P). There exist alternative definitions for local martingales that also allow for non-integrable X0 . Proposition 1.2 A real valued process X is a local martingale iff the following two assertions are hold true. (i) X0 ∈ L1 (Ω, F0 , P) and (ii) E [|Xn+1 | |Fn ] < ∞ a.s. and E [Xn+1 |Fn ] = Xn for all n in the sense of generalized conditional expectations. Proof Suppose X is a local martingale and (T n ) a localizing sequence. Then n Tn (i) immediately follows and we also have E [XkT |Fk−1 ] = Xk−1 , which yields 1{T n >k−1} E [Xk |Fk−1 ] = Xk−1 1{T n >k−1} . Letting n → ∞ gives E [Xk |Fk−1 ] = Xk−1 . From this generalized martingale property we see that the conditional expectation E [Xk |Fk−1 ] is finite a.s. As a consequence, one automatically has E [|Xk | |Fk−1 ] < ∞ a.s. Pm+1 Conversely we assume (i) and (ii). Let T n = inf{m : k=1 E [|Xk | |Fk−1 ] ≥ n}. Then T n is a stopping time and E |XkTn | ≤ E |X0 | + n < ∞, so XkTn ∈ L1 (Ω, F, P). Since {T n ≥ k} ∈ Fk−1 we also have from (ii) with k instead of n n Tn n + 1 that E [XkT |Fk−1 ] = Xk−1 , which yields the martingale property of X T . 1 Proposition 1.3 The following results on martingale transforms hold true. (i) If Y is a predictable process and M a martingale, then X = Y · M is a local martingale. (ii) If Y is a predictable process and M a local martingale, then Y · M is a local martingale. (iii) If X is a local martingale, then there exists a predictable process Y and martingale M such that X = Y · M . Proof (i) Let T n = inf{k ≥ 0 : |Yk+1 | > n}. Then (T n ) is a fundamental n sequence of stopping times and Y T is a bounded process for every n. Moren over, the stopped processes M T are martingales as well, and one easily verifies n n n n X T = Y T · M T . It follows from RESULT that X T is a martingale for every n and so X is a local martingale. (ii) follows from (iii). Indeed, M being a local martingale, must be of the form M = Y 0 · M 0 for a predictable process Y 0 and a martingale M 0 . This implies X = Y · (Y 0 · M 0 ) = Y Y 0 · M 0 and the result follows from (i). n (iii) Let Ank = {E [Xn | |Fn−1 ∈ [k, S k + 1). For every n, the Ak are P disjoint and by Proposition 1.2 we have P( k Ank = Ω) = 1. Put un := k≥0 (k + −3 1) ∆Xn 1Ank . Then un ∈ Fn , E |un | < ∞ and E [un |Fn−1 ] = 0. Thus (Mn ) Pn P with Mn = i=1 is a martingale with M0 = 0. Put then Yn = k≥0 (k +1)3 1Ank and we see that ∆Xn = Yn ∆Mn . Proposition 1.4 Suppose X is a nonnegative local martingale such that E X0 < ∞, then it is a true martingale. Proof We first show that E Xk < ∞ for all k. Let (T n ) be a localizing sequence. n By Fatou’s lemma and by the martingale property of X T we have n E Xk ≤ lim inf E XkT = E X0 . n→∞ Pk n Furthermore, XkT ≤ i=0 Xi , of which we now know that this upper bound has finite expectation. Hence we can apply the Dominated Convergence Theorem n Tn in E [XkT |Fk−1 ] = Xk−1 to get E [Xk |Fk−1 ] = Xk−1 . 1.2 Quadratic variation If X and Y are two stochastic processes, their optional quadratic covariation process [X, Y ] is defined as X [X, Y ]t = ∆Xs ∆Ys , s≤t where ∆X0 = ∆Y0 = 0. For X = Y we write [X] instead of [X, Y ]. Note the integration by parts formula Xt Yt = X0 Y0 + (X− · Y )t + (Y− · X)t + [X, Y ]t , 2 where the (predictable) process X− at time t equals Xt−1 , defined for t ≥ 1. If X and Y are square integrable martingales, we have encountered in Section ?? their predictable covariation process hX, Y i, which was the unique predictable process such that XY − hX, Y i is a martingale too. It follows from the above integration by parts formula, recall that (X− · Y ) and (Y− · X) are martingales, that also [X, Y ] − hX, Y i is a martingale. In fact, hX, Y i is also the unique predictable process that, subtracted from [X, Y ], yields a martingale, already known from Corollary ??, E [∆Xt ∆Yt |Ft−1 ] = ∆hX, Y it . (1.1) This property carries over to the case where X and Y are local martingales under the extra conditions E [∆Xt2 |Ft−1 ] < ∞ and E [∆Yt2 |Ft−1 ] < ∞, Exercise 1.3, and under this condition, XY − hX, Y i is a local martingale. 1.3 Measure transformation In this section we suppose that P and Q are probability measures on (Ω, F). Their restrictions to Fn are denoted Pn and Qn respectively. We will always assume that P0 = Q0 , which is automatically the case if F0 is trivial. We start with a definition whose contents are supposed to be in force throughout this section. See Exercise ?? for the more restrictive setting with Q P. Definition 1.5 The probability measure Q is called locally absolutely continuous w.r.t. the probability measure P if Qn Pn for all n ≥ 0. In this case we can define the Fn -measurable random variables (Radon-Nikodym derivatives, n densities) Zn = dQ dPn . The process Z = (Zn )n≥0 is called the density process. One easily verifies that Z is a martingale under the probability measure P. Note that Q(Zn > 0) = 1, but P(Zn > 0) may be less than 1, unless Pn ∼ Qn , in which case P(F ) = EQ Z1n 1F for F ∈ Fn . Furthermore, on the set {Zn = 0} also Zn−1 = 0, as follows from the martingale property of the nonnegative process Z. Hence we can define Zn /Zn−1 with the understanding that it is put equal to zero on {Zn−1 = 0}. If M is martingale under P, it is usually not a martingale under Q. We shall see below how to transform M parallel to the measure change from P to Q such that the changed process is a martingale again, or a local martingale in a more general setting. The following lemma generalizes Equation ?? to conditional expectations. Lemma 1.6 Let Q P with Radon-Nikodym derivative Z := dQ dP and G a sub-σ-algebra of F. Suppose X ∈ L1 (Ω, G, Q). Then Q(E [Z|G] > 0) = 1 and a.s. under Q EQ X = Z E [XZ|G] = E [X ]. E [Z|G] E [Z|G] 3 (1.2) Proof Observe first that Q(E [Z|G] = 0) = EQ 1{E [Z|G]=0} = E Z1{E [Z|G]=0} = E E [Z1{E [Z|G]=0} |G] = E E [Z|G]1{E [Z|G]=0} |G] = 0. Take G ∈ G and consider the following chain of equalities. E [XZ1G ] = EQ [X1G ] = EQ EQ [X1G |G] = E (ZEQ [XG]1G ) = E E ([ZEQ [X|G]|G]1G ) = E (E [Z|G]EQ [X|G]1G ) . So, E [XZ1G ] = E (E [Z|G]EQ [X|G]1G ). Since E [Z|G]EQ [X|G] ∈ G, the just proved property Q(E [Z|G] > 0) = 1 and G is arbitrary, Equation (1.2) follows. Lemma 1.7 Let Q be locally absolutely continuous w.r.t. P with density process Z and let T be a stopping time. Then for F ∈ FT it holds that Q(F ∩ {T < ∞}) = E ZT 1F ∩{T <∞} . Proof Exercise 1.4. Proposition 1.8 Let X be an adapted process and assume Q locally absolutely continuous w.r.t. P with density process Z. (i) The process X is a martingale under Q iff the process XZ is a martingale under P. (ii) The process X is a local martingale under Q if the process XZ is a local martingale under P. (iii) If moreover P is also locally absolutely continuous w.r.t. Q, then the process XZ is a local martingale under P if the process X is a local martingale under Q. Proof To prove (i) we use Lemma 1.6 and recall that E [Zn |Fn−1 ] = Zn−1 . Note that EQ |Xn | < ∞ iff E |Xn |Zn < ∞. We have EQ [Xn |Fn−1 ] = E [Xn Zn |Fn−1 ] , Zn−1 from which the assertion immediately follows. (ii) Let (T n ) be a localizing sequence for XZ. Then P(limn→∞ T n < ∞) = 0 and then also Q(limn→∞ T n < ∞) = 0 by virtue of Lemma 1.7. Because n n n XkT Zk = XkT ZkT + XT n (Zk − ZT n )1{k≥T n } 4 (1.3) and both(!) terms on the right hand side are martingales (Exercise ??) under n P, we can apply part (i) to deduce that X T is a martingale under Q. Hence X is a local martingale under Q. (iii) Let X be a local martingale under Q and (T n ) be a localizing sequence for it. Then Q(limn→∞ T n < ∞) = 0 and by the argument in the proof of (ii) also P(limn→∞ T n < ∞) = 0, since P is given to be locally absolutely continuous w.r.t. Q. Similar to Equation (1.3) we have n n n XkT ZkT = XkT Zk − XT n (Zk − ZT n )1{k≥T n } , again with two martingales under P on the right hand side, and so XZ is a local martingale under P. Theorem 1.9 1 Let M be a local martingale under P and let Q be locally absolutely continuous w.r.t. P. The processes MQ = M − 1 · [M, Z] Z M̃ Q := M − 1 · hM, Zi Z− and are well defined under Q and (i) M Q is a local martingale under Q, and (ii) if moreover E [|∆Mn |Zn |Fn−1 ] < ∞2 a.s. for all n, then also M̃ Q is a local martingale under Q. Proof The property Q(Zn > 0) = 1 allows for division by Zn a.s. under Q. (i) Let us compute ∆MnQ = ∆Mn − ∆Mn ∆Zn Zn ∆Mn (Zn − ∆Zn ) Zn ∆Mn = Zn−1 . Zn = Hence, by Lemma 1.6 with Z = Zn , G = Fn−1 and Z E [Z|G] EQ [∆MnQ |Fn−1 ] = E [∆Mn |Fn−1 ] = 0. (ii) Recall that ∆hM, Zin = E [∆Mn ∆Zn |Fn−1 ] and Q ∆(M̃ Q Z)n = Zn−1 ∆M̃nQ + M̃n−1 ∆Zn + ∆M̃nQ ∆Zn . 1 maybe 2 maybe martingale case only E [|∆Mn |∆Zn |Fn−1 ] < ∞, seems to be equivalent 5 = Zn Zn−1 we get Hence ∆hM, Zin Q ∆Zn + ∆M̃nQ ∆Zn ) + M̃n−1 Zn−1 ∆hM, Zin Q = Zn−1 ∆Mn + (M̃n−1 − )∆Zn + ∆Mn ∆Zn − ∆hM, Zin . Zn−1 ∆(M̃ Q Z)n = Zn−1 (∆Mn − The first two terms are obviously local martingale differences under P, whereas the same holds true for the difference ∆M̃nQ ∆Zn − ∆hM, Zin under the stated assumption. The second assertion of Theorem 1.9 is the most appealing one, since it allows to write M = M̃ Q + 1 · hM, Zi, Z− where the first term is a local martingale under Q and the second term is predictable. This gives the Doob decomposition of M under Q. 1.4 Exercises 1.1 Consider a probability space (Ω, F, P). Suppose X ≥ 0 and that G is a sub-σ-algebra of F. Then there exists a G-measurable random variable X̂ such that E X1G = E X̂1G for all G ∈ G. 1.2 Consider a probability space (Ω, F, P) and let {An , n ≥ 1} be a measurable partition of Ω with P(An ) = 2−n . Let (Zn ) be a sequence of random Pn variables, independent of the An , with P(Zn = ±2n ) = 21 . Put Yn = i=1 1Aj Zj for n ≥ 1, Y∞ , the a.s. limit of the Yn , as well as X0 = 0 and Xn = Y∞ if n ≥ 1, T n = ∞ · 1{∪1≤k≤n Ak } . (a) Show that Y∞ is well defined and finite a.s. What is it? n (b) Show that XkT = Yn for k ≥ 1. n (c) Show that X T is a martingale for every n. (d) Show that X1 is not integrable (and hence X is not a martingale). 1.3 Prove Equation (1.1) for local martingales X and Y satisfying E [∆Xt2 |Ft−1 ] < ∞ and E [∆Yt2 |Ft−1 ] < ∞. 1.4 Prove Lemma 1.7. 1.5 Show that the process {XT n (Zk − ZT n )1{k≥T n } , k ≥ 0} in the proof of Proposition 1.8 is a martingale under P. 6