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L7-Stochastic-Calculus.Pdf Introduction and Techniques Lecture 7 in Financial Mathematics UiO-STK4510 Autumn 2015 Teacher:S. Ortiz-Latorre Stochastic Calculus 1 Itô Processes Let ( ; ;P ) be a complete probability space. Throughout this lecture W will denote a Wiener F process (possibly dW -dimensional) de…ned on ( ; ;P ) : The reference …ltration F will be the mini- F mal augmented …ltration generated by W , i.e. FW : De…nition 1 A stochastic process X = Xt t [0;T ] that can be written as f g 2 t t Xt = X0 + gsds + hsdWs; (1) Z0 Z0 2 2 2 where h; g L and X0 L ( ; 0;P ) is called a L -Itô process. 2 a;T 2 F Remark 2 Note that, actually, as 0 = ?; the random variable X0 is a constant. Moreover, one can prove that the representationF (1) isf unique,g in the sense that if X can also be written as t t Xt = X0 + gs0 ds + hs0 dW; Z0 Z0 2 2 where h0; g0 L and X0 L ( ; 0;P ); then ht(!) = h0 (!) and gt(!) = g0(!); P -a.e and 2 a;T 0 2 F t t X0 = X0 : An obvious extension of the previous de…nition is when W is a dW -dimensional Brownian motion, 2 X0 is a dX -dimensional random vector with components in L ( ; 0;P ); h is a dX dW matrix and 2 F g is a dX vector with components in La;T . Then, X is a dX -dimensional vector with components given by t dW t i i i i;j j Xt = X0 + gsds + hs dWs ; i = 1; :::; dX : (2) 0 j=1 0 Z X Z From a notational point of view, sometimes it is convenient to write the previous expressions in di¤erential form dW i i i;j j dXt = gsds + hs dWs ; i = 1; :::; dX : j=1 X However, keep in mind that the di¤erential form is only a handy notation for the the integral form, which is the one with a sound mathematical meaning. 2 0 0 Remark 3 The de…nition of an L -Itô process can be extended to X0 L ( ; 0;P ); h La;T and T 2 F 2 g measurable and adapted process such that 0 gt dt < ;P -a.s.. We call such a process an Itô process. j j 1 R 1 Last updated: November 29, 2015 2 The Itô Formula If we only could compute Itô integrals using their de…nition, the concept will be of limited applica- bility. The Itô integral has been a success because one can develop a calculus for it. That is, given an Itô process X and a regular function f; one can prove that the process Yt = f(t; Xt) is still an Itô process and we can …nd the explict representation of Y as an Itô process. 1;2 @f @f Theorem 4 (Itô Formula) Let f C ([0;T ] R) ; that is, f : R+ R R such that @t ; @x @2f 2 ! and @x2 are continuous functions and t t Xt = X0 + gsds + hsdWs; 0 t T: Z0 Z0 an L2-Itô process. Assume that T 2 2 2 2 @f @f 2 @ f E (t; Xt) + gt (t; Xt) + ht (t; Xt) dt < ; (3) @t @x @x2 1 "Z0 ( ) # and T @f 2 E ht (t; Xt) dt < : (4) @x 1 "Z0 # Then, t @f @f 1 @2f t @f f (t; X ) = f(0;X ) + (s; X ) + g (s; X ) + h2 (s; X ) ds + h (s; X )dW ; t 0 @t s s @x s 2 s @x2 s s @x s s Z0 Z0 for 0 t T and it is also an L2-Itô process. Proof. We just give the heuristics for the case Xt = Wt: Let n = 0 = t0 < t1 < < tn 1 < f tn = t be a sequence of partitions of the interval [0; t]: From a Taylor expansion of f around the g points in the partition n we get n 1 f(Wt) f(W0) = f(Wt ) f(Wt ) i+1 i i=0 X n 1 n 1 1 2 = f 0(Wt )(Wt Wt ) + f 00(Wt )(Wt Wt ) i i+1 i 2 i i+1 i i=0 i=0 X X n 1 1 2 + (f 00(i) f 00(Wt )) (Wt Wt ) 2 i i+1 i i=0 X =: A1 + A2 + A3: where i is a random variable in the random interval [min(Wti ;Wti+1 ); max(Wti ;Wti+1 )]: The term 2 2 t A1 converges in L to the Itô integral, the term A2 converges in L to 0 f 00(Ws)ds and the term 2 A3 converges in L to zero. Some remarks are in order: R Remark 5 1. Sometimes we will write the Itô formula in di¤erential form, which reads @f @f 1 @2f @f df(t; X ) = (t; X ) + g (t; X ) + h2 (t; X ) dt + h (t; X )dW t @t t t @x t 2 t @x2 t t @x t t @f @f 1 @2f =: (t; X )dt + (t; X )dX + (t; X )(dX )2 ; @t t @x t t 2 @x2 t t where to write the last line we have used the so called Itô’s product rule dt dWt dt 0 0 : dWt 0 dt This last expression is useful for mnemonic reasons and can be given a rigourous meaning using the concept of covariation between processes. 2 Last updated: November 29, 2015 0 2. If we consider the stochastic integral for processes in La;T ; that is, for an Itô process the previous theorem holds without requiring the integrability conditions (3) and (4) : n Example 6 Let f(x) = x ; n 2 and Xt = Wt. Then, we obtain t t n n 1 n(n 1) n 2 W = n W dW + W ds: t s s 2 s Z0 Z0 Theorem 7 Let W be a dW -dimensional Brownian motion and X a dX -dimensional Itô process such as in (2). Let f be a RdY - valued function with components in C1;2([0;T ]; RdX ): Then the process Yt = f(t; Xt) has the following di¤erential expression d d @f k X @f 1 X @2f dY k = f i(0;X )+ (t; X )dt+ (t; X )dXi + (t; X )dXidXj; k = 1; :::; d ; t 0 @t t @x t t 2 @x @x t t t Y i=1 i i;j=1 i j X X i j 2 i i with the Itô product rules dWt dWt = ijdt and (dt) = dWt dt = dtdWt = 0: As a consequence of this multidimensional Itô formula we can deduce the following useful result. Proposition 8 (Integration by parts formula) Let X and Y be two Itô processes.Then, d(XtYt) = XsdYs + YsdXs + dXsdYs: Example 9 Let’s …nd the Itô expression for the process tWt: Thanks to the integration by parts formula we get d(tWt) = tdWt + Wtdt + dtdWt = tdWt + Wtdt; which written in integral form reads t t t t tWt = 0W0 + sdWs + Wsds = sdWs + Wsds: Z0 Z0 Z0 Z0 3 The Martingale Representation Theorem W We assume that the sigma algebra is equal to T = T : This assumption is crucial and the results presented in this section do not holdF true, in general,F F if we allow to contain more information that the one generated by W up to time T: The martingale representationF theorem is one of the cornerstones of stochastic calculus and essentially says that any martingale, having enough integra- bility, with respect to the Brownian …ltration can be expressed in an unique way as an stochastic integral with respect to this Brownian motion. This result paves the way of the martingale methods in option pricing. The following is a technical lemma that we state without proof. Lemma 10 The linear span of random variables of the Dóleans exponential (or stochastic exponen- tial) type, i.e., random variables of the form T T 1 2 exp htdWt h dt ; 2 t Z0 Z0 ! 2 2 for a deterministic process h L ([0;T ]) ; is dense in L ( ; T ;P ): 2 F 2 Theorem 11 (Itô Representation Theorem) Let F L ( ; T ;P ): Then, there exists a unique 2 2 F h La;T such that 2 T F = E[F ] + htdWt: (5) Z0 3 Last updated: November 29, 2015 Proof. The idea is to prove …rst the result for F being of Dóleans exponential type and then extend it to any square integrable F by a density argument. Let h L2 ([0;T ]) deterministic. Assume that 2 T T 1 2 F = exp htdWt h dt ; 2 t Z0 Z0 ! and consider the process t t 1 2 Xt = exp hsdWs h ds : 2 s Z0 Z0 Applying Itô’sformula to the exponential function we obtain 1 2 1 2 dXt = YthtdWt + Yth + Yth dt = YthtdWt; 2 t 2 t or t Xt = 1 + XshsdWs; t [0;T ]: 2 Z0 In particular, T F = 1 + XthtdWt; Z0 and taking expectations we get that E[F ] = 1 because T T T t 2 2 2 2 2 E Xtht dt = E Xt ht dt = exp hs ds ht dt " 0 j j # 0 j j j j 0 0 j j j j Z Z h i Z Z t T 2 2 exp hs ds ht dt < : j j j j 1 Z0 Z0 2 and, therefore, the expectation of the Itô integral is zero (note that we have used that E Xt = j j t 2 exp hs ds ; see exercises of List 2).
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