Chapter 3. Stochastic calculus. 3.1 Decomposition of martingales

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1 Chapter 3 Stochastic calculus In this chapter we investigate the stochastic calculus for processes which may have jumps as well as a continuous component. If X is not a continuous process, it is no longer true that X t^tn is a bounded process when T N =inf{t : X t N}, sincetherecouldbealargejumpattimet N. We investigate stochastic integrals with respect to square integrable (not necessarily continuous) martingales, Itô s formula, and the Girsanov transformation. We prove the reduction theorem that allows us to look at semimartingales that are not necessarily bounded. We will need the Doob-Meyer decomposition, which can be found in Chapter 16 of Bass, Stochastic Processes. Thatinturndependsonthedebutand section theorems. A simpler proof than the standard one for the debut and section theorems can be found in the Arxiv: Decomposition of martingales We assume throughout this chapter that {F t } is a filtration satisfying the usual conditions. This means that each F t contains every P-null set and \ "> F t+" = F t for each t. Let us with a few definitions and facts. The predictable -field is the -field of subsets of [, 1) generatedbythecollectionofbounded,leftcontinuous processes that are adapted to {F t }. A stopping time T is predictable and 35

2 36 CHAPTER 3. STOCHASTIC CALCULUS predicted by the sequence of stopping times T n if T n " T,andT n <T on the event (T >). A stopping time T is totally inaccessible if P(T = S) = for every predictable stopping time S. The graph of a stopping time T is [T,T]={(t,!) :t = T (!) < 1}. IfX t is a process that is right continuous with left limits, we set X t =lim s!t,s<t X s and X t = X t X t.thus X t is the size of the jump of X t at time t. Let s look at some examples. If W t is a Brownian motion and T =inf{t : W t =1}, thent n =inf{t : W t =1 (1/n)} are stopping times that predict T. On the other hand, if P t is a Poisson process (with parameter 1, say, for convenience), then we claim that T =inf{t : P t =1} is totally inaccessible. To show this, suppose S is a stopping time and S n " S are stopping times such that S n <Son (S >). We will show that P(S = T )=. Todothat, it su ces to show that P(S ^ N = T )=foreachpositiveintegern. Since P t t is a martingale, E P Sn^N = E (S n ^ N). Letting n!1,weobtain (by monotone convergence) that E P (S^N) = E (S ^ N). We also know that E P S^N = E (S ^ N). Therefore E P (S^N) = E P S^N.SinceP has increasing paths, this implies that P (S^N) = P S^N,andweconcludeP(S^N = T )=. In this chapter we will assume througout for simplicity that every jump time of whichever process we are considering is totally inaccessible. The general case is not much harder, but the di erences are only technical. AsupermartingaleZ is of class D if the family of random variables: is uniformly integrable. {Z T : T afinitestoppingtime} Theorem 3.1 (Doob-Meyer decomposition) Let {F t } be a filtration satisfying the usual conditions and let Z be a supermartingale of class D whose paths are right continuous with left limits. Then Z can be written = M t A t in one and only one way, where M and A are adapted processes whose paths are right continuous with left limits, A has continuous increasing paths and A 1 =lim t!1 A t is integrable, and M is a uniformly integrable martingale Suppose A t is a bounded increasing process whose paths are right continuous with left limits. Recall that a function f is increasing if s<timplies

3 3.1. DECOMPOSITION OF MARTINGALES 37 f(s) apple f(t). Then trivially A t is a submartingale, and by the Doob-Meyer decomposition, there exists a continuous increasing process A e t such that A t At e is a martingale. We call At e the compensator of A t. If A t = B t C t is the di erence of two increasing processes B t and C t, then we can use linearity to define A e t as B e t Ct e. We can even extend the notion of compensator to the case where A t is complex valued and has paths that are locally of bounded variation by looking at the real and imaginary parts. We will use the following lemma. A 1 =lim t!1 A t. For any increasing process A we let Lemma 3.2 Suppose A t has increasing paths that are right continuous with left limits, A t apple K a.s. for each t, and let B t be its compensator. Then E B1 2 apple 2K 2. Proof. If M t = A t B t,thenm t is a martingale, and then E [M 1 M t F t ]=. We then write E B1 2 =2E =2E (B 1 B t ) db t =2E E [A 1 E [B 1 A t F t ] db t apple 2KE =2KE B 1 =2KE A 1 apple 2K 2. B t F t ] db t db t From the lemma we get the following corollary. Corollary 3.3 If A t = B t C t, where B t and C t are increasing right continuous processes with B = C =, a.s., and in addition B and C are bounded, then E sup ea 2 t < 1. t

4 38 CHAPTER 3. STOCHASTIC CALCULUS Proof. By a proposition, E e B 2 1 < 1 and E e C 2 1 < 1, andso E sup t ea 2 t apple E [2 sup t eb 2 t +2sup t ec 2 t ] apple 2E e B E e C 2 1 < 1. We are done. Akeyresultisthefollowingorthogonality lemma. Lemma 3.4 Suppose A t is a bounded increasing right continuous process with A =, a.s., A e t is the compensator of A, and M t = A t At e. Suppose N t is a right continuous square integrable martingale such that ( N t )( M t )= for all t. Then E M 1 N 1 =. Proof. By Lemma 3.3, M is square integrable. Suppose H(s,!) =K(!)1 (a,b] (s) with K being F a measurable. Since M t is of bounded variation, we have (this is a Lebesgue-Stieltjes integral here) E H s dm s = E [K(M b M a )] = E [KE [M b M a F a ]]=. We see that linear combinations of such H s generate the predictable -field. Thus by linearity and taking limits, E R 1 H s dm s =ifh s is a predictable process such that E R 1 H s dm s < 1. In particular, since N s is left continuous and hence predictable, E R 1 N s dm s =,providedwecheck integrability: E N s dm s apple E by the Cauchy-Schwarz inequality. = E [(sup r (sup N r ) dm s r N r )(A 1 + e A 1 )] < 1 By hypothesis, E R 1 N s dm s =,soe R 1 N s dm s =. Ontheother hand, using Proposition 1.17, we see E M 1 N 1 = E N 1 dm s = E N s dm s =.

5 3.1. DECOMPOSITION OF MARTINGALES 39 The proof is complete. If we apply the above to N t^t,wehavee M 1 N T =. Ifwethencondition on F T, E [M T N T ]=E [N T E [M 1 F T ]]=E [N T M 1 ]=. (3.1) The reason for the name orthogonality lemma is that by (3.1), M t N t is a martingale. This implies that hm,ni t (which we will define soon, and is defined similarly to the case of continuous martingales) is identically equal to. Let M t be a square integrable martingale with paths that are right continuous and left limits, so that E M1 2 < 1. Foreachi 2 Z, lett i1 =inf{t : M t 2[2 i, 2 i+1 )}, T i2 =inf{t >T i1 : M t 2[2 i, 2 i+1 )}, andsoon;i can be both positive and negative. Since M t is right continuous with left limits, for each i, T ij!1as j!1.weconcludethatm t has at most countably many jumps. We relabel the jump times as S 1,S 2,... so that each S k is totally inaccessible, the graphs of the S k are disjoint, M has a jump at each time S k and only at these times, and M Sk is bounded for each k. Wedo not assume that S k1 apple S k2 if k 1 apple k 2,andingeneralitwouldnotbepossible to arrange this. and If S i is a totally inaccessible stopping time, let A i (t) = M Si 1 (t Si ) (3.2) M i (t) =A i (t) e Ai (t), (3.3) where A e i is the compensator of A i. A i (t) istheprocessthatisuptotime S i and then jumps an amount M Si ;thereafteritisconstant.weknowthat ea is continuous. Note that M M i has no jump at time S i. Theorem 3.5 Suppose M is a square integrable martingale and we define M i as in (3.3). (1) Each M i is square integrable. (2) P 1 i=1 M i(1) converges in L 2. (3) If M c t = M t P 1 i=1 M i(t), then M c is square integrable and we can find a version that has continuous paths.

6 4 CHAPTER 3. STOCHASTIC CALCULUS (4) For each i and each stopping time T, E [M c T M i(t )] =. Proof. (1) If S i is a totally inaccessible stopping time and we let B t = ( M Si ) + 1 (t Si ) and C t =( M Si ) 1 (t Si ),then(1)followsbycorollary3.3. (2) Let V n (t) = P n i=1 M i(t). By the orthogonality lemma (Lemma 3.4), E [M i (1)M j (1)] = if i 6= j and E [M i (1)(M 1 V n (1)] = if i apple n. We thus have nx E M i (1) 2 = E V n (1) 2 i=1 h apple E M 1 i 2 V n (1) + E Vn (1) 2 h i 2 = E M 1 V n (1)+V n (1) = E M 2 1 < 1. Therefore the series E P n i=1 M i(1) 2 converges. If n>m, E [(V n (1) h V m (1)] 2 = E nx i=m+1 i 2 M i (1) = nx i=m+1 E M i (1) 2. This tends to as n, m!1,sov n (1) isacauchysequenceinl 2,and hence converges. (3) From (2), Doob s inequalities, and the completeness of L 2,therandom variables sup t [M t V n (t)] converge in L 2 as n! 1. Let Mt c = lim n!1 [M t V n (t)]. There is a sequence n k such that sup (M t V nk (t)) Mt c!, a.s. t We conclude that the paths of Mt c are right continuous with left limits. By the construction of the M i, M V nk has jumps only at times S i for i>n k. We therefore see that M c has no jumps, i.e., it is continuous. (4) By the orthogonality lemma and (3.1), E [M i (T )(M T V n (T )] = if T is a stopping time and i apple n. Lettingn tend to infinity proves (4).

7 3.2. STOCHASTIC INTEGRALS Stochastic integrals If M t is a square integrable martingale, then Mt 2 is a submartingale by Jensen s inequality for conditional expectations. Just as in the case of continuous martingales, we can use the Doob-Meyer decomposition to find a predictable increasing process starting at, denoted hmi t,suchthatmt 2 hmi t is a martingale. Let us define [M] t = hm c i t + X sapplet M s 2. (3.4) Here M c is the continuous part of the martingale M as defined in Theorem 3.5. As an example, if M t = P t t,wherep t is a Poisson process with parameter 1, then M c t =and [M] t = X sapplet P 2 s = X sapplet P s = P t, because all the jumps of P t are of size one. In this case hmi t = t; thisfollows from Proposition 3.6 below. In defining stochastic integrals, one could work with hmi t,buttheprocess [M] t is the one that shows up naturally in many formulas, such as the product formula. Proposition 3.6 M 2 t [M] t is a martingale. Proof. By the orthogonality lemma and (3.1) it is easy to see that hmi t = hm c i t + X i hm i i t. Since Mt 2 hmi t is a martingale, we need only show [M] t hmi t is a martingale. Since [M] t hmi t = hm c i t + X M s 2 hm c i t + X hm i i t, sapplet i it su ces to show that P i hm P ii t Pi sapplet M i(s) 2 is a martingale.

8 42 CHAPTER 3. STOCHASTIC CALCULUS By an exercise M i (t) 2 =2 M i (s ) dm i (s)+ X sapplet M i (s) 2, (3.5) where the first term on the right hand side is a Lebesgue-Stieltjes integral. If we approximate this integral by a Riemann sum and use the fact that M i is a martingale, we P see that the first term on the right in (3.5) is a martingale. Thus Mi 2 (t) sapplet M i(s) 2 is a martingale. Since Mi 2 (t) hm i i t is a martingale, summing over i completes the proof. If H s is of the form H s (!) = nx K i (!)1 (ai,b i ](s), (3.6) i=1 where each K i is bounded and F ai measurable, define the stochastic integral by nx N t = H s dm s = K i [M bi^t M ai^t]. i=1 Very similar proofs to those in the Brownian motion case (see Chapter 1 of Bass, Stochastic Processes), show that the left hand side will be a martingale and (with [ ] insteadofh i), Nt 2 [N] t is a martingale. If H is P-measurable and E R 1 H 2 s d[m] s < 1, approximateh by integrands Hs n of the form (3.6) so that E (H s H n s ) 2 d[m] s! and define Nt n as the stochastic integral of H n with respect to M t. By almost the same proof as that of the construction of stochastic integrals with respect to Brownian motion, the martingales Nt n converge in L 2. We call the limit N t = R t H s dm s the stochastic integral of H with respect to M. A subsequence of the N n converges uniformly over t, a.s., and therefore the limit has paths that are right continuous with left limits. The same arguments as those for Brownian motion apply to prove that the stochastic integral is a martingale and [N] t = H 2 s d[m] s.

9 3.3. ITÔ S FORMULA 43 Aconsequenceofthislastequationisthat 3.3 Itô s formula 2 E H s dm s = E Hs 2 d[m] s. (3.7) We will first prove Itô s formula for a special case, namely, we suppose X t = M t + A t,wherem t is a square integrable martingale and A t is a process of bounded variation whose total variation is integrable. The extension to semimartingales without the integrability conditions will be done later in the chapter (in Section 3.5) and is easy. Define hx c i t to be hm c i t. Theorem 3.7 Suppose X t = M t + A t, where M t is a square integrable martingale and A t is a process with paths of bounded variation whose total variation is integrable. Suppose f is C 2 on R with bounded first and second derivatives. Then f(x t )=f(x )+ + X sapplet f (X s ) dx s [f(x s ) f(x s ) f (X s ) X s ]. f (X s ) dhx c i s (3.8) Proof. The proof will be given in several steps. Set S(t) = f (X s ) dx s, Q(t) = 1 2 f (X s ) dhx c i s, and J(t) = X sapplet [f(x s ) f(x s ) f (X s ) X s ]. We use these letters as mnemonics for stochastic integral term, quadratic variation term, and jump term, resp. Step 1: Suppose X t has a single jump at time T which is a totally inaccessible stopping time and there exists N>suchthat M T + A T applen a.s.

10 44 CHAPTER 3. STOCHASTIC CALCULUS Let C t = M T 1 (t T ) and let e C t be the compensator. If we replace M t by M t C t + e C t and A t by A t + C t b Ct,wemayassumethatM t is continuous. Let B t = X T 1 (t T ). Set X b t = X t B t and A b t = A t B t. Then bx t = M t + A b t and X b t is a continuous process that agrees with X t up to but not including time T. We have X b s = X b s and Xs b =ifsapple T. By Ito s formula for continuous processes, f( b X t )=f( b X )+ = f( b X )+ + X sapplet f ( b X s ) d b X s f ( b X s ) d b X s f ( b X s ) dhmi s f ( b X s [f( b X s ) f( b X s ) f ( b X s ) b Xs ], ) dh e X c i s since the sum on the last line is zero. For t<t, b Xt agrees with X t.attime T, f(x t )hasajumpofsizef(x T ) f(x T ). The integral with respect to bx, S(t), will jump f (X T ) X T, Q(t) doesnotjumpatall,andj(t) jumps f(x T ) f(x T ) f (X T ) X T. Therefore both sides of (3.8) jump the same amount at time T,andhenceinthiscasewehave(3.8)holdingfor t apple T. Step 2: Suppose there exist times T 1 <T 2 < with T n!1,eacht i is a totally inaccessible stopping time stopping time, for each i, thereexists N i > suchthat M Ti and A Ti are bounded by N i,andx t is continuous except at the times T 1,T 2,...LetT =. Fix i for the moment. Define Xt = X (t Ti ) +,definea t and Mt similarly, and apply Step 1 to X at time T i + t. WehaveforT i apple t apple T i+1 f(x t )=f(x Ti )+ f (X s ) dx s + 1 f (X 2 s ) dhx c i s T i T i + X [f(x s ) f(x s ) f (X s ) X s ]. Thus for any t we have T i <sapplet f(x Ti+1^t) =f(x Ti^t)+ X + Z Ti+1^t T i^t T i^t<sapplet i+1^t f (X s ) dx s Z Ti+1^t T i^t f (X s [f(x s ) f(x s ) f (X s ) X s ]. ) dhx c i s

11 3.3. ITÔ S FORMULA 45 Summing over i, wehave(3.8)foreacht. Step 3: We now do the general case. As in the paragraphs preceding Theorem 3.5, we can find stopping times S 1,S 2,... such that each jump of X occurs at one of the times S i and so that for each i, thereexistsn i > such that M Si + A Si applen i. Moreover each S i is a totally inaccessible stopping time. Let M be decomposed into M c and M i as in Theorem 3.5 and let A c t = A t 1 X i=1 A Si 1 (t Si ). Since A t is of bounded variation, then A c will be finite and continuous. Define and M n t = M c t + A n t = A c t + nx i=1 nx M i (t) i=1 A Si 1 (t Si ), and let Xt n = Mt n + A n t. We already know that M n converges uniformly over t tom in L 2. If we let Bt n = P n P i=1 ( A S i ) + 1 (t Si ) and Ct n = n i=1 ( A S i ) 1 (t Si ) and let B t =sup n Bt n, C t =sup n Ct n,thenthefactthat A has paths of bounded variation implies that with probability one, Bt n! B t and Ct n! C t uniformly over t anda t = B t C t.inparticular,wehave convergence in total variation norm: E d(a n t ) A t )!. We define S n (t), Q n (t), and J n (t) analogouslytos(t), Q(t), and J(t), resp. By applying Step 2 to X n, we have f(x n t )=f(x n )+S n (t)+q n (t)+j n (t), and we need to show convergence of each term. We now examine the various terms. Uniformly in t, X n t converges to X t in probability, that is, P(sup Xt n X t >")! t

12 46 CHAPTER 3. STOCHASTIC CALCULUS as n!1for each ">. Since R t dhm c i s < 1, bydominatedconvergence f (X n s ) dhm c i s! f (X s ) dhm c i s in probability. Therefore Q n (t)! Q(t) in probability. Also, f(x n t )! f(x t ) and f(x )! f(x ), both in probability. We now show S n (t)! S(t). Write We see that f (X n s ) da n s f (X s h = f (Xs n ) da s ) da n s f (X n s ) da s i h + f (Xs n ) da s f (X s = I n 1 + I n 2. I1 n applekf k 1 da n s da s! ) da s i as n!1,whilebydominatedconvergence, I n 2 also tends to. We next look at the stochastic integral part of S n (t). f (X n s The L 2 norm of I n 3 E ) dm n s f (X s h = f (Xs n ) dm n s h + f (X s = I n 3 + I n 4. is bounded by f (X n s ) f (X s ) 2 d[m n ] s apple E ) dm s ) dm n s f (X s which goes to by dominated convergence. Also 1X I4 n = f (X s ) dm i (s), i=n+1 ) dm n s f (X s i ) dm s i f (X n s ) f (X s ) 2 d[m] s,

13 3.3. ITÔ S FORMULA 47 so using the orthogonality lemma (Lemma 3.4), the L 2 norm of I4 n than 1X 1X kf k 2 1 E [M i ] 1 applekf k 2 1 E M i (1) 2, i=n+1 which goes to as n!1. i=n+1 is less Finally, we look at the convergence of J n. The idea here is to break both J(t) andj n (t) intotwoparts,thejumpsthatmightberelativelylarge (jumps at times S i for i apple N where N will be chosen appropriately) and the remaining jumps. Let N>1bechosenlater. J(t) J n (t) = X sapplet = X [f(x s ) f(x s ) f (X s ) X s ] X [f(xs n ) f(xs n ) f (Xs n ) Xs n ] sapplet {i:s i applet} = X [f(x Si ) f(x Si ) f (X Si ) X Si ] X {i:s i applet} {i>n:s i applet} X {i>n:s i applet} + X {iapplen,s i applet} = I N 5 I n,n 6 + I n,n 7. [f(x n S i ) f(x n S i ) f (X n S i ) X n S i ] [f(x Si ) f(x Si ) f (X Si ) X Si ] [f(x n S i ) f(x n S i ) f (X n S i ) X n S i ] n [f(x Si ) f(x Si ) f (X Si ) X Si ] o [f(xs n i ) f(xs n i ) f (XS n i ) XS n i ] By the fact that M and A are right continuous with left limits, M Si apple 1/2 and A Si apple 1/2 ifi is large enough (depending on!), and then X Si apple1, and also X Si 2 apple 2 M Si 2 +2 A Si 2 apple 2 M Si 2 + A Si.

14 48 CHAPTER 3. STOCHASTIC CALCULUS We have and I N 5 applekf k 1 X i>n,s i applet I n,n 6 applekf k 1 X n i>n,s i applet ( X Si ) 2 ( X Si ) 2. Since P 1 i=1 M S i 2 apple [M] 1 < 1 and P 1 i=1 A S i < 1, thengiven">, we can choose N large such that P( I N 5 + I n,n 6 >") <". Once we choose N, wethenseethati n,n 7 tends to in probability as n!1, since X n t converges in probability to X t uniformly over t. We conclude that J n (t) convergestoj(t) inprobabilityasn!1. This completes the proof. 3.4 The reduction theorem Let M be a process adapted to {F t }.IfthereexiststoppingtimesT n increasing to 1 such that each process M t^tn is a uniformly integrable martingale, we say M is a local martingale. IfeachM t^tn is a square integrable martingale, we say M is a locally square integrable martingale. We say a stopping time T reduces aprocessm if M t^t is a uniformly integrable martingale. Lemma 3.8 (1) The sum of two local martingales is a local martingale. (2) If S and T both reduce M, then so does S _ T. (3) If there exist times T n!1such that M t^tn is a local martingale for each n, then M is a local martingale. Proof. (1) If the sequence S n reduces M and the sequence T n reduces N, then S n ^ T n will reduce M + N. (2) M t^(s_t ) is bounded in absolute value by M t^t + M t^s. Both { M t^t } and { M t^s } are uniformly integrable families of random variables.

15 3.4. THE REDUCTION THEOREM 49 Now use that the sum of two uniformly integrable families is uniformly integrable. (3) Let S nm be a family of stopping times reducing M t^tn and let Snm = S nm ^ T n. Renumber the stopping times into a single sequence R 1,R 2,... and let H k = R 1 _ _R k. Note H k "1.ToshowthatH k reduces M, we need to show that R i reduces M and use (2). But R i = Snm for some m, n, so M t^ri = M t^snm^t n is a uniformly integrable martingale. Let M be a local martingale with M =. Wesaythatastoppingtime T strongly reduces M if T reduces M and the martingale E [ M T F s ]is bounded on [,T), that is, there exists K>suchthat sup E [ M T F s ] apple K, apples<t a.s. Lemma 3.9 (1) If T strongly reduces M and S apple T, then S strongly reduces M. (2) If S and T strongly reduce M, then so does S _ T. (3) If Y 1 is integrable, then E [E [Y 1 F T ] F S ]=E [Y 1 F S^T ]. Proof. (1) Note E [ M S F s ] apple E [ M T F s ] by Jensen s inequality, hence S strongly reduces M. (2) It su ces to show that E [ M S_T F t ]isboundedfort<t,sinceby symmetry the same will hold for t<s.fort<t this expression is bounded by E [ M T F t ]+E [ M S 1 (S>T) F t ]. The first term is bounded since T strongly reduces M. Forthesecondterm, if t<t, 1 (t<t ) E [ M S 1 (S>T) F t ]=E [ M S 1 (S>T) 1 (t<t ) F t ] apple E [ M S 1 (t<s) F t ] = E [ M S F t ]1 (t<s), which in turn is bounded since S strongly reduces M.

16 5 CHAPTER 3. STOCHASTIC CALCULUS (3) Let Y t be the right continuous version of E [X F t ]. We thus need to show that E [Y S F T ]=Y S^T.TherighthandsideisF S^T measurable and F S^T F T.WethusneedtoshowthatifA 2F T,then E [Y S ; A] =E [Y S^T ; A]. Let B =(S apple T ). We will show E [Y S ; A \ B] =E [Y S^T ; A \ B] (3.9) and E [Y S ; A \ B c ]=E [Y S^T ; A \ B c ] (3.1) Adding (3.9) and (3.1) will achieve our goal. Since Y S = Y S^T on B, therighthandsideof(3.9)isequaltoe [Y S ; A\B] as required. For (3.1), S>Ton B c,sos = S _ T on B c. Also A \ B c 2F T F S_T. Since Y is a martingale, E [Y S ; A \ B c ]=E [Y S_T ; A \ B c ]=E [Y T ; A \ B c ]=E [Y S^T ; A \ B c ], which is (3.1) Lemma 3.1 If M is a local martingale with M =, then there exist stopping times T n "1that strongly reduce M. Proof. Let R n "1be a sequence reducing M. Let S nm = R n ^ inf{t : E [ M Rn F t ] m}. Arrange the stopping times S nm into a single sequence {U n } and let T n = U 1 _ _U n.inviewoftheprecedinglemmas,weneedtoshowu i strongly reduces M, whichwillfollowifs nm does for each n and m.

17 3.4. THE REDUCTION THEOREM 51 Let Y t = E [ M Rn F t ], where we take a version whose paths are right continuous with left limits. Y is bounded by m on [,S nm ). By Jensen s inequality for conditional expectations and Lemma 3.9 E [ M Snm 1 (t<snm) F t ] apple E [ E [ M Rn F Snm ] 1 (t<snm) F t ] = E [ E [ M Rn 1 (t<snm) F Snm ] F t ] = E [ M Rn 1 (t<snm) F Snm^t] = Y Snm^t1 (t<snm) = Y t 1 (t<snm) apple m. We used that 1 (t<snm) is F Snm^t measurable; to see that we have by Lemma 3.9(3) that E [1 (t<snm) F Snm^t] =E [E [1 (t<snm) F Snm ] F t ]=E [1 (t<snm) F t ] We are done. =1 (t<snm). Our main theorem of this section is the following. Theorem 3.11 Suppose M is a local martingale. Then there exist stopping times T n " 1 such that M t^tn = Ut n + Vt n, where each U n is a square integrable martingale and each V n is a martingale whose paths are of bounded variation and such that the total variation of the paths of V n is integrable. Moreover, Ut n = UT n n and Vt n = VT n n for t T n. The last sentence of the statement of the theorem says that U n and V n are both constant from time T n on. Proof. It su ces to prove that if M is a local martingale with M =and T strongly reduces M, thenm t^t can be written as U + V with U and V of the described form. Thus we may assume M t = M T for t T, M T is integrable, and E [ M T F t ]isbounded,saybyk, on[,t). Let A t = M T 1 (t T ) = M t 1 (t T ),let e A be the compensator of A, letv = A e A,andletU = M A+ e A.ThenV is a martingale of bounded variation. We compute the expectation of the total variation of V.LetB t = M + T 1 (t T ) and C t = M T 1 (t T ). Then the expectation of the total variation of A is

18 52 CHAPTER 3. STOCHASTIC CALCULUS bounded by E M T < 1 and the expectation of the total variation of e A is bounded by E e B 1 + E e C 1 = E B 1 + E C 1 apple E M T < 1. We need to show U is square integrable. Note M t A t = M t 1 (t<t ) = E [M 1 F t ] 1 (t<t ) = E [E [M 1 F T _t ] F t ] 1 (t<t ) = E [M T _t F t ] 1 (t<t ) = E [M T F t ] 1 (t<t ) apple E [ M T F t ]1 (t<t ) apple K. Therefore it su ces to show e A is square integrable. Our hypotheses imply that E [M + T F t]isboundedbyk on [,T), hence E [B 1 B t F t ]isbounded,andsoeb e 1 2 < 1. Similarly, E C e 1 2 < 1. Since A = B C, thena e = B e C,anditfollowsthatsupt e At e is square integrable. 3.5 Semimartingales We define a semimartingale to be a process of the form X t = X + M t + A t, where X is finite, a.s., and is F measurable, M t is a local martingale, and A t is a process whose paths have bounded variation on [,t]foreacht. If M t is a local martingale, let T n be a sequence of stopping times as in Theorem We set M c t^t n =(U n ) c t for each n and [M] t^tn = hm c i t^tn + X sapplet^t n M 2 s. It is easy to see that these definitions are independent of how we decompose of M into U n + V n and of which sequence of stopping times T n strongly reducing M we choose. We define hx c i t = hm c i t and define [X] t = hx c i t + X sapplet X 2 s. We say an adapted process H is locally bounded if there exist stopping times S n "1and constants K n such that on [,S n ]theprocessh is bounded

19 3.5. SEMIMARTINGALES 53 by K n. If X t is a semimartingale and H is a locally bounded predictable process, define R t H s dx s as follows. Let X t = X + M t + A t. If R n = T n ^ S n,wherethet n are as in Theorem 3.11 and the S n are the stopping times used in the definition of locally bounded, set R t^r n H s dm s to be the stochastic integral as defined in Section 3.2. Define R t^r n H s da s to be the usual Lebesgue-Stieltjes integral. Define the stochastic integral with respect to X as the sum of these two. Since R n "1,thisdefines R t H s dx s for all t. One needs to check that the definition does not depend on the decomposition of X into M and A nor on the choice of stopping times R n. We now state the general Itô formula. Theorem 3.12 Suppose X is a semimartingale and f is C 2. Then f(x t )=f(x )+ f (X s ) dx s f (X s + X [f(x s ) f(x s ) f (X s ) X s ]. sapplet ) dhx c i s Proof. First suppose f has bounded first and second derivatives. Let T n be stopping times strongly reducing M t,lets n =inf{t : R t da s n}, let R n = T n ^ S n,andletxt n = X t^rn A Rn. Since the total variation of A t is bounded on [,R n ), it follows that X n is a semimartingale which is the sum of a square integrable martingale and a process whose total variation is integrable. We apply Theorem 3.7 to this process. Xt n agrees with X t on [,R n ). As in the proof of Theorem 3.7, by looking at the jump at time R n, both sides of Itô s formula jump the same amount at time R n,andsoitô s formula holds for X t on [,R n ]. If we now only assume that f is C 2,we approximate f by a sequence f m of functions that are C 2 and whose first and second derivatives are bounded, and then let m!1; we leave the details to the reader. Thus Itô s formula holds for t in the interval [,R n ]andfor f without the assumption of bounded derivatives. Finally, we observe that R n!1,soexceptforanullset,itô sformulaholdsforeacht. The proof of the following corollary is similar to the proof of Itô s formula.

20 54 CHAPTER 3. STOCHASTIC CALCULUS Corollary 3.13 If X t =(X 1 t,...,x d t ) is a process taking values in R d such that each component is a semimartingale, and f is a C 2 function on R d, then f(x t )=f(x ) X sapplet dx i,j=1 i (X 2 j (X s h f(x s ) f(x s ) where hy,zi t = 1 2 [hy + Zi t hy i t hzi t ]. ) dx i s ) dh(x i ) c, (X j ) c i s dx i=1 If X and Y are real-valued semimartingales, i (X s ) Xs i [X, Y ] t = 1 2 ([X + Y ] t [X] t [Y ] t ). (3.11) The following corollary is the product formula for semimartingales with jumps. Corollary 3.14 If X and Y are semimartingales of the above form, X t Y t = X Y + X s dy s + Y s dx s +[X, Y ] t. Proof. Apply Theorem 3.12 with f(x) =x 2.Sinceinthiscase we obtain f(x s ) f(x s ) f (X s ) X s = X 2 s, X 2 t = X 2 +2 X s dx s +[X] t. (3.12) Applying (3.12) with X replaced by Y and by X + Y and using gives our result. X t Y t = 1 2 [(X t + Y t ) 2 X 2 t Y 2 t ]

21 3.6. THE GIRSANOV THEOREM The Girsanov theorem Let P and Q be two equivalent probability measures, that is, P and Q are mutually absolutely continuous. Let M 1 be the Radon-Nikodym derivative of Q with respect to P and let M t = E [M 1 F t ]. The martingale M t is uniformly integrable since M 1 2 L 1 (P). Once a non-negative martingale hits zero, it is easy to see that it must be zero from then on. Since Q and P are equivalent, then M 1 >, a.s., and so M t never equals zero, a.s. Observe that M T is the Radon-Nikodym derivative of Q with respect to P on F T. If A 2F t,wehave using that M is a martingale. Q(A) =E P [M 1 ; A] =E P [M t ; A], Theorem 3.15 Suppose X is a local martingale with respect to P. D t is a local martingale with respect to Q, where X t D t = 1 M s d[x, M] s. Then Note that in the formula for D, weareusingalebesgue-stieltjesintegral. Proof. Since E Q [X t D t ; A] =E P [M t (X t D t ); A] ifa 2F t and the same with t replaced by s, itsu cestoshowthatm t (X t D t ) is a local martingale with respect to P. ByCorollary3.14, d(m(x D)) t =(X D) t dm t + M t dx t M t dd t + d[m,x D] t. The first two terms on the right are local martingales with respect to P. Since D is of bounded variation, the continuous part of D is zero, hence Thus [M,D] t = X sapplet M s D s = M s dd s. M t (X t D t )= localmartingale +[M,X] t M s dd s. Using the definition of D shows that M t (X t D t )isalocalmartingale.