Malliavin’s calculus and stochastic integral representations of functionals of diffusion processes. (English) Zbl 0542.60055

Ocone, Daniel

Stochastics 12, 161-185 (1984).

Let F be a Fréchet differentiable function on the space C[0,1] with continuous paths, with Fréchet derivative \(dF(b)(u)=\int^{1}_{0}u(s)\lambda^ F(ds;b)\) for b and u in C[0,1]. If \(B=\{B(t)| 0\leq t\leq 1\}\) is a Brownian motion and \(\beta_ t=\sigma \{B(s)| o\leq s\leq t\}\) and \(EF(B)=0\), Clark’s formula states that \(F(b)=\int^{1}_{0}E\{\lambda^ F((s,1],\cdot)| \beta_ s\}dB(s)\). This paper extends Clark’s formula to functionals F which are weakly H-differentiable in the sense of Shigekawa. For such F, \[ E\{F(b)\int^{1}_{0}<\alpha(s),dB(s)>\}=E\{\int^{1}_{0}<DF(b)'(s),\;alpha(s)>ds\} \] for all bounded, \(\beta_ t\)-adapted processes \(\alpha\) (s), where DF(b)(s), \(o\leq s\leq 1\), denotes the weak H-derivative of F. Clark’s formula is a simple consequence of this result. A simple proof of this result is given using Stroock’s development of the Malliavin calculus. Malliavin calculus techniques are also used to derive U. G. Haussmann’s [SIAM J. Control Optimization 16, 252-269 (1978; Zbl 0375.60070)] stochastic integral representation of a functional F(y) of the diffusion \(dy=m(t,y)dt+\sigma(t,y)dB\). In doing this, it is shown that y(t,B) is weakly H-differentiable if m and \(\sigma\) have bounded, continuous first derivatives in y.

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60J65	Brownian motion

Keywords:

Haussmann’s formula; stochastic integral representation; Clark’s formula; Malliavin calculus

Citations:

Zbl 0375.60070

References:

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