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). For arbitrary F let Fn n 1 be a sequenceh of lineari 0 j j f g  2 combinationsR of Dóleans exponentials approximating F in L ( ; T ;P ). By linearity, the property F n 2 (5) also holds for Fn n 1: Then, for each n we have that there exists h La;T such that f g  2 T n Fn = E[Fn] + ht dWt: (6) Z0 Using (6) ; the linearity of the expectation, that the integrals has zero mean and the Itô isometry we get that

2 T 2 n m E[(Fn Fm) ] = E E[Fn] E[Fm] + (ht ht ) dWt 2 3 Z0 ! 4 T 5 2 n m 2 = E[Fn Fm] + E ht ht dt j j "Z0 # T n m 2 E ht ht dt ;  j j "Z0 # 2 As Fn n 1 is a convergent sequence in L ( ; T ;P ) we have that Fn n 1 is Cauchy and, hence, f g  F f g  T n m 2 2 ht ht dt E[(Fn Fm) ] 0; j j  n;m! "Z0 # !1 n 2 Therefore, we have proved that the sequence h n 1 is a Cauchy sequence in L ( [0;T ]) which is a complete metric space. Consequently, itf willg converge to a process h L2 ( [0;T ]). We can choose a version of h which is adapted, because there exists a subsequence2hn which converges to h

4 Last updated: November 29, 2015 for almost all (!; t) [0;T ]. Therefore, ht( ) is t-measurable for a.e. t [0;T ] (that is true because the …ltration2 satis…es the usual conditions)
andF changing h in set of measure2 zero on [0;T ] we can get h adapted. Finally, using Itô’sisometry again we get that

T T 2 2 n F = L - lim Fn = L - lim E[Fn] + ht dWt = E[F ] + htdWt: Z0 ! Z0 The uniqueness is also an easy consequence of Itô isometry and the fact that a positive process, say T u; such that E[ 0 usds] = 0 must be P  -a.e. zero. From the Itô’srepresentation theorem one can show the martingale representation theorem. R

Theorem 12 (Martingale Representation Theorem) Let M = Mt t [0;T ] be a square inte- 2 f g 2 grable F-martingale. Then, there exists a unique h La;T such that 2 t Mt = E[M0] + hsdWs: Z0 2 Proof. Note that Mt = E[MT t]: Applying Itô’srepresentation theorem to MT L ( ; T ;P ) we jF 2 T 2 F have that there exists a unique h La;T such that MT = E[M0] + hsdWs: Hence, 2 0 T R Mt = E[MT t] = E E[M0] + hsdWs t jF jF " Z0 # T = E[M0] + E hsdWs t jF "Z0 # t = E[M0] + hsdWs; Z0 where in the last equality we have used that the stochastic integral, as a process, is an F-martingale.

3 Example 13 Consider the random variable WT : Then, by example (6) we have that

T T 3 2 WT = 3 Wt dWt + 3 Wtdt: Z0 Z0 Moreover, by example (9) we have that

T T T Wtdt = TWT tdWt = (T t) dWt: Z0 Z0 Z0 Therefore, we can conclude that

T 3 2 W = 3 W + (T t) dWt T t Z0
All the results of this section also hold when the …ltration is generated by a multidimensional Brownian motion. They can even be extended to random variables that are not square integrable 0 using the Itô integral for processes in La;T ; but then we loss the unicity in the representation.

4 Girsanov’sTheorem

Girsanov’s theorem basically states that if we add a drift term to a Brownian motion then the new process does not change much in the sense that it is a Brownian motion but under a di¤erent probability measure. The prof of Girsanov’s theorem is based on the following characterization of Brownian motion that we state without proof.

5 Last updated: November 29, 2015 Theorem 14 (Lévy’scharacterization of Brownian motion) Let X = Xt t R+ be a real X f g 2 valued process with continuous paths and F = F the minimal augmented …ltration generated by X. The following two statements are equivalent:

1. X is a F-Brownian motion. 2. X and X2 t are both F-martingales. The following lemma is a version of the Bayes’ theorem. The proof is similar to the proof of exercise 32. in List 1.

Lemma 15 Let Q and P be two probability measures on some measurable space ( ; ) such that Q P: Let X L1 ( ; ;Q) and a sub--algebra of : Then, F  2 F H  F F dQ dQ EQ[X ]EP = EP X : jH dP jH dP jH     Theorem 16 (Girsanov) Let Y be an Itô process of the form

dYt = gtdt + dWt;Y0 = 0; 0 t T:   Set t t 1 2 Mt = exp gsdWs g ds ; 0 t T 2 s    Z0 Z0  dQ Assume that M is a martingale with respect to P and de…ne the measure Q on T by = MT : F dP Then Q is a probability measure on T and Y is a Brownian motion under Q: F Proof. The idea is to use Lévy’s characterization theorem. We have to check that Y and Y 2 t are martingales under Q. De…ne Kt = MtYt: By the integration by parts formula we have that

dKt = MtdYt + YtdMt + dMtdYt

= Mt(gtdt + dWt) Yt (MtgtdWt) dWt (MtgtdWt) = Mt(1 Ytgt)dWt:

Hence, Kt is a martingale under P . By Lemma 15, we get that

EP [MtYt s] EP [Kt s] Ks EQ[Yt s] = jF = jF = = Ys; jF EP [Mt s] Ms Ms jF so we have proved that Y is a martingale under Q: The proof that Y 2 t is a martingale under Q is similar. The delicate point in Girsanov’sTheorem is the assumption that M is a martingale. In order to check this assumption one can use the following result.

Lemma 17 (Novikov’s) Let M be as in Girsanov’s theorem. If

T 1 2 E exp gt dt < ; 2 j j 1 " Z0 !# then M is a martingale and E[Mt] = E[M0] = 1:

5 Stochastic Di¤erential Equations (SDEs)

i A particularly useful type of Itô processes are solutions of SDEs. Let b (t; x); ij(t; x); i = 1; :::; dX ; j = dX 1; :::; dW ; be Borel measurable functions from R+ R to R: De…ne the drift vector b(t; x) = i  b (t; x) i=1;:::;dX and the dispersion matrix (t; x) = ij(t; x) i=1;:::;dX ;j=1;:::;dW : The goal of this fsection isg to give a meaning to the stochastic di¤erentialf equationg

dXt = b(t; Xt)dt + (t; Xt)dWt; (7)

6 Last updated: November 29, 2015 written componentwise as

dW i i j dXt = b (t; Xt)ds + ij(t; Xt)dWt ; i = 1; :::; dX ; j=1 X 1 dW 1 dX where W = Wt ; :::; Wt is Brownian motion and X = Xt ; :::; Xt is a suitable stochastic process with continuousf processg with continuous sample paths,f the "solution"g of the equation. The drift vector b(t; x) and the dispersion matrix (t; x) are the coe¢ cients of the equation and the T dX dX -matrix a(t; x) := (t; x) (t; x) with elements 

dW aij(t; x) = ik(t; x)jk(t; x); k=1 X is called the di¤usion matrix.

De…nition 18 Let ( ; ;P ) be a probability space with a dW -dimensional Brownian motion W F de…ned on it and consider F the augmented …ltration of W: A strong solution, on ( ; ;P ) and F with respect to the …xed Brownian motion W; of the equation (7) is a process X = Xt t [0;T ] with continuous sample paths and satisfying f g 2

1. X is F-adapted. d 2. X0 = x R : 2 t i 2 3. P b (s; Xs) + ij(s; Xs) ds < = 1 holds for every i = 1; :::; dX ; j = 1; :::; dW and 0 j j 1 t R[0;T ]:  2 4. The integral version of (7) holds, i.e,

t t Xt = x + b(s; Xs)ds + (s; Xs)dWs;P -a.s.,. Z0 Z0 The following theorem gives su¢ cient conditions on b and  to guarantee the existence and uniqueness of a solution of equation (7) : From now on will denote the euclidean norm of the the appropriate dimension. k
k

Theorem 19 (Existence and Uniqueness) Let the coe¢ cients b and  be continuous functions satisfying

1. (Lipschitz coe¢ cients) b(t; x) b(t; y) + (t; x) (t; y) K x y ; k k k k  k k 2 2 2 2. (Linear growth) b(t; x) + (t; x) K2(1 + x ); for all x R; k k k k  k k 2

dX for all t R+; x; y R and a constant K > 0 (where denotes the Euclidean norm of suitable dimension).2 Then,2 for any T > 0 there exists a (uniquek
k up to indistinguishability) strong solution X of (7) in the interval [0;T ]: Moreover, this solution X satis…es for …xed m 1 

2m Ct 2m E sup Xs < Ce 1 + x ; 0 s t k k k k       for all t [0;T ] and a suitable constant C. 2 5.1 The Markov Property of the Solution of a SDE s;x s;x We shall denote by X = Xt t [s;T ] the unique solution of equation (7) starting from x at time s: That is, Xs;x satis…es f g 2

t t s;x s;x s;x X = x + b(u; X )du + (u; X )dWu; t [s; T ]: t u u 2 Zs Zs

7 Last updated: November 29, 2015 Moreover, we will denote by X = Xx = X0;x the solution of equation (7) starting from x at time 0. Using the uniqueness of the solution of equation (7) one can deduce the following ‡ow property of the solution s;Xx Xx = X s ;P -a.s. t [s; T ]: t t 2 From the ‡ow property one can prove the following theorem.

Theorem 20 (Markov property of SDEs) Assume that the hypothesis of Theorem 19 hold. Then, X is a Markov process with respect to the …ltration F: Furthermore, for any Borel measurable function f such that E[ f(Xt) ] < ; t [0;T ] we have j j 1 2

E[f(Xt) s] = (Xs); jF s;x where s t and (x) = E[f(Xt )]: The previous equality is often written as  s;x E [f(Xt) s] = E [f(Xt )] x=Xs : jF j Remark 21 If X is time homogenous, that is, the coe¢ cients b(t; x) = b(x) and (t; x) = (x) do not depend on time, then the markov property can be written as

x E[f(Xt) s] = E f(Xt s) x=Xs : jF j   5.2 The Feynman-Kac Representation Associated to the coe¢ cients (b; ) of an SDE we can de…ne the following second order di¤erential operator

d d d 1 X X @2f X @f i 2 dX Atf(x) = aij(t; x) (x) + b (t; x) (x); f C (R ); 2 @x @x @x 2 i=1 j=1 i j i=1 i X X X where the matrix a is the di¤usion matrix a = (t; x)T (t; x): We call this operator the characteristic operator of (b; ): We now consider the following Cauchy problem corresponding to At:

Problem 22 Find a function v(t; x) : [0;T ] RdX R with  ! @v dX (t; x) + k(t; x)v(t; x) = Atv(t; x); on [0;T ) R @t  d v(T; x) = f(x); for x R X ; 2

dX dX where k : [0;T ] R R+ is continuous and f : R R is also continuous and satisfy  ! ! f(x) L(1 + x 2); L > 0;  1 or f(x) 0: j j  k k   Theorem 23 (Feynman-Kac Representation) Let v : [0;T ] RdX R be a continuous solution  ! of the Cauchy problem (22) with v C1;2 [0;T ) RdX and satisfying the following polynomial growth condition 2  sup v(t; x) M(1 + x 2) with
M > 0;  1: 0 t T j j  k k    Assume that At is the characteristic operator of (b; ) satisfying the hyphotesis of Theorem 19. Then v(t; x) admits the following stochastic representation

T T x x t;x t;x v(t; x) = E f(XT ) exp k(s; Xs )ds t = E f(XT ) exp k(s; Xs )ds " t ! F # " t !# Z Z

d on [0;T ] R X : In particular, such a solution is unique. 

8 Last updated: November 29, 2015 Proof. We sketch the main idea. As v(t; x) is C1;2 we can apply Itô’sformula to t exp k(s; Xs)ds v(t; Xt)  Z0  to get that the process t Mt := exp k(s; Xs)ds v(t; Xt)  Z0  t s @v exp k(r; Xr)dr (s; Xs) + Asv(s; Xs) k(s; Xs)v(s; Xs) ds; @t Z0  Z0    is equal to the stochastic integral

dW t s dX @v j exp k(r; Xr)dr (s; Xs)ij(s; Xs)dW : @x s j=1 0 0 i=1 i X Z  Z  X As a consequence, if

T s @v 2 E exp k(r; Xr)dr (s; Xs)ij(s; Xs) ds < ; i = 1; :::; dX ; j = 1; :::; dW ; @x 1 "Z0  Z0  i #

then Mt is a martingale and if v(t; x) satis…es the Cauchy problem we get that t Mt = exp k(s; Xs)ds v(t; Xt);  Z0  is a martingale. Hence, by the martingale property of Mt we can write

t T exp k(s; Xs)ds v(t; Xt) = E exp k(s; Xs)ds v(T;XT ) t ; 0 " 0 ! F #  Z  Z which yields

T v(t; Xt) = E exp k(s; Xs)ds v(T;XT ) t " t ! F # Z T t;x t;x = E f(X ) exp k(s; Xs )ds ; T " Zt !# by the Markov property of X = Xx: Remark 24 Note that in the previous theorem we have assumed that there exists a solution to the Cauchy problem satisfying certain regularity conditions and polynomial growth. If we can show that such solution exists then it is given by the above expectation and it is unique. This is done by calculating the above expectation, which depends on the parameters (t; x); and showing that it solves the PDE and it satis…es the required regularity and growth conditions. However, the previous theorem does not resolve, in general, wether or not a solution to the Cauchy problem actually exists because the previous approach may fail, that is, the above expectation may fail to have the required regularity or growth condition. Another interesting use of this stochastic representation formula is to numerically compute the solution (when it does exist) by Monte Carlo methods.

References

[1] I. Karatzas and S.E. Shreve. Brownian Motion and Stochastic Calculus. Second Edition. Springer, New York. 1998. [2] D. Nualart. Stochastic Processes. Unpublished notes. You can …nd it at https://www.math.ku.edu/~nualart. [3] B. Øksendal. Stochastic Di¤erential Equations. An Introduction with Applications. Springer, Berlin. 2010.

9 Last updated: November 29, 2015