## HT H0 f Ht

O I 2 E aij ((yi - Yi(t)) o (M(t)W(t) + (t) + (yj - Yj(t)) o Mi(t)W(t) + Vi(t) dt, (7.47) where Mj is the jth row of the matrix M. Proof By the Wick chain rule (see Theorem 6.13) we get *xdkt ( - 2 E aij (yi - Yi(t)) o (yj - Yj(t)) ) ) dt o 2 E aii( (yi - Yi(t)) o fr (t) + (yj - Yj (t)) o (t) ) dt o I 2 aij (y - Yi(t)) o (tj(t)W(t) + Mj(t)) + (yj Yj(t)) o (t)W(t) + Mt))) ) dt. We can now prove the main result of this section. Theorem 7.18. Suppose that a 0,T > 1 is a deterministic function...

## X2gXZ

Y2H(Sy(V(t)))(C) y2g(y, Z) J e iyXd g(X, Z)dX. Once again by a Levy version of Theorem 5.12, we see that the integral (14.13) is well-defined in (S)*. Finally, by using the inverse Levy-Hermite transform and Lemma 14.3 we obtain f eiyXSy(V(t))dy f(X) eiXn(t) (14.14) Thus, we have proved relation (14.1) for h(y) eiXy. Since (14.1) still holds for linear combinations of such functions, the general case is attained by a well-known density argument. The continuity of y i Sy(n(t)) is also a direct...

## Fn T U Xn T Xvn T yxdpyxdp

With this expression for Fn(T) in hands, we can see that if Fn (T) e 0 2 then a portfolio n is optimal if and only if it satisfies the following equation (M(s) - p(s) - o-2(s)n(s))E Fn(T) FS + a(s)E Ds(Fn(T)) FS r e(s,z)E Da,z(Fn(T)) Fs - n(s)02(s,z)E Fn(T) FS 16.3 Optimal Portfolio under Partial Information in an Anticipating Environment On the probability space (Q, F,P), (see Sect. 12.5) we consider filtrations E Et,t G 0,T , F Ft,t G 0,T , and G Gt,t G 0,T such that Et QFt gt...

## Xm sy Infnsy m

Dt,zXm s, y N 6s, dy - Dt, zX s,y N 6s,dy the proof follows by the closedness of Dt,z. In this section we state and prove a jump diffusion version of the Clark-Ocone formula see Theorem 4.1 . For this result we refer to, for example, 153 . F E F i I E Dt,zF Ft N dt, dz , 12.20 where we have chosen a predictable version of the conditional expectation process E DtzF Ft , t gt 0. Proof The proof is similar to the one for the Brownian motion case Theorem 4.1 . Let us consider the chaos expansion...

## Xt Xt6Xt Xt32tdt Xt3 tDt Xtdt atXtdt 3tXt5W t 3 tXtDt X t

Hence X t , t e 0,T , is a solution of 8.24 if and only if If a and 3 are F-adapted, the aforementioned condition holds. But nothing can be said in general. If 3 is deterministic, we can solve 8.24 by using white noise analysis, see Problem 5.8. In this case we get that the solution is X t xexp0 J a s ds 3 s dW s j, t e 0,T . 8.26 If, in addition, a is F-adapted, then X t X t , t e 0, T . 8.6 Application to Insider Trading Modeling 8.6.1 Markets with No Friction Let us return to the financial...

## Skorohod Integrals

We now use the chaos expansions of Theorem 10.2 to define the Skorohod integral with respect to the compensated Poisson random measure N. The approach will be similar to the approach used in Chap. 2 for Brownian motion. Definition 11.1. Let X X t, z , 0 lt t lt T,z G R0, be a stochastic process more precisely a random field such that X t, z is an Ft-measurable random variable for all t, z G 0, T x Ro and E X2 t,z lt to, t, z G 0, T x Ro. Then for each t, z , the random variable X t, z has an...

## Short Introduction to Levy Processes

In this chapter we present some basic definitions and results about Levy processes. We do not aim at being complete in our presentation, but for a smoother reading, we like to include the notions we constantly refer to in this book. We can address the reader to the recent monographs, for example, 8, 32, 114, 212 for a deeper study of Levy processes. Our presentation follows the survey given in 182, Chap. 1 . Let H, F,P be a complete probability space. Definition 9.1. A one-dimensional Levy...

## Integral Representations and the Clark Ocone Formula

In this chapter we present explicit stochastic integral representations for random variables in terms of the Malliavin derivative. The central result is the celebrated Clark-Ocone formula. See 46, 47, 97, 172 . We also discuss some generalization of this formula that turns to be central in the application to hedging in mathematical finance. Another application of the Clark-Ocone formula appears in the sensitivity analysis. This is also presented in the last section of this chapter. Theorem 4.1....

## The Hida Malliavin Derivative on the Space Q 5R

6.1 A New Definition of the Stochastic Gradient and a Generalized Chain Rule In Chap. 3 we have introduced the Malliavin derivative in terms of chaos expansions. However, the original definition was tailored for the Wiener space where the derivative is given as a differential operator. See Appendix. Although the Wiener space is a natural space to work on when dealing with the Brownian motion, this original approach has the disadvantage that the directional derivative in the direction 7 Y t , t...

## The Donsker Delta Function and Applications

In this chapter we use white noise calculus to define the Donsker delta function of a Brownian motion. The Donsker delta function of a Brownian motion can be considered the time derivative of local time of a Brownian motion on a distribution space. We aim at employing this concept to determine explicit formulae for replicating portfolios in a Black-Scholes market for a class of contingent claims. 7.1 Motivation An Application of the Donsker Delta Function to Hedging As a motivation we start by...

## White Noise the Wick Product and Stochastic Integration

This chapter gives an introduction to the white noise analysis and its relation to the analysis discussed in the first two chapters. This is a useful alternative approach for several reasons. First, it allows us to represent the Malliavin derivative as a natural directional derivative or stochastic gradient, to be more precise . Second, it makes it possible to obtain an extension of the Clark-Ocone formula from Bij2 to L2 P . Moreover, it provides a natural platform for the Wick product, which...

## The Skorohod Integral

The Wiener-Ito chaos expansion is a convenient starting point for the introduction of several important stochastic concepts. In this chapter we focus on the Skorohod integral. This stochastic integral, introduced for the first time by A. Skorohod in 1975 216 , may be regarded as an extension of the Ito integral to integrands that are not necessarily F-adapted, see also, for example, 30, 31 . The Skorohod integral is also connected to the Malliavin derivative, which is introduced with full...

## Preface

There are already several excellent books on Malliavin calculus. However, most of them deal only with the theory of Malliavin calculus for Brownian motion, with 35 as an honorable exception. Moreover, most of them discuss only the application to regularity results for solutions of SDEs, as this was the original motivation when Paul Malliavin introduced the infinite-dimensional calculus in 1978 in 157 . In the recent years, Malliavin calculus has found many applications in stochastic control and...

## Malliavin Derivative via Chaos Expansion

The Malliavin calculus see 157 , see also, for example, 53, 72, 159, 168, 211 was originally created as a tool for studying the regularity of densities of solutions of stochastic differential equations. Subsequently, partly due to the papers 172 and 173 , the significance of Malliavin calculus in finance became clear. This triggered a tremendous interest in the subject, also among economists. Today the range of applications has extended even further to include numerical methods, stochastic...

## The Wiener Ito Chaos Expansion

The celebrated Wiener-Ito chaos expansion is fundamental in stochastic analysis. In particular, it plays a crucial role in the Malliavin calculus as it is presented in the sequel. This result which concerns the representation of square integrable random variables in terms of an infinite orthogonal sum was proved in its first version by Wiener in 1938 226 . Later, in 1951, Ito 119 showed that the expansion could be expressed in terms of iterated Ito integrals in the Wiener space setting. Before...

## Chain Rule Malliavin

1 Part I The Continuous Case Brownian Motion 1 The Wiener Ito Chaos Expansion 1.1 Iterated Ito 1.2 The Wiener-Ito Chaos Expansion 2 The Skorohod 2.1 The Skorohod Integral 2.2 Some Basic Properties of the Skorohod 2.3 The Skorohod Integral as an Extension of the Ito Integral 23 3 Malliavin Derivative via Chaos 3.1 The Malliavin 3.2 Computation and Properties of the Malliavin Derivative 29 3.2.1 Chain Rules for Malliavin 3.2.2 Malliavin Derivative and Conditional Expectation 30 3.3 Malliavin...

## Clark-haussmann-ocone Formula

The mathematical theory now known as Malliavin calculus was first introduced by Paul Malliavin in 157 as an infinite-dimensional integration by parts technique. The purpose of this calculus was to prove the results about the smoothness of densities of solutions of stochastic differential equations driven by Brownian motion. For several years this was the only known application. Therefore, since this theory was considered quite complicated by many, Malliavin calculus remained a relatively...