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Functional Itō calculus and stochastic integral representation of martingales. (English) Zbl 1272.60031
The authors develop an extended Itō calculus relative to continuous semimartingales, applicable to adapted functionals which may depend on the trajectory (up to the current time). They proceed by specifying some simplified Malliavin calculus. Namely, they consider two pathwise derivatives,
– a horizontal derivative is defined by continuing a $$\mathbb{R}^d$$-valued path $$\omega[0,t]$$ into a path $$\omega[0,t+\varepsilon[$$, taken to be constant on $$[t,t+\varepsilon[$$;
– a vertical derivative is defined by varying the path $$\omega[0,t]$$ according to $$(\omega[0,t]+ 1_{\{t\}}\varepsilon e_j)$$.
Such use of the only very particular case of Malliavin derivatives happens to be effective, when considering continuous functionals of both $$(X_t,\langle X\rangle_t)$$, $$X_t= (X(s),0\leq s\leq t)$$ being a continuous semimartingale, and $$\langle X\rangle_t$$ denoting its quadratic variation process.
Examples of functionals to which this extended calculus can be applied are $\int^t_0 g(s, X_s)\,d\langle X\rangle_s,\quad G(t,X_t,\langle X\rangle_t),\quad \operatorname{E}[G(t, X_t,\langle X\rangle_t)/F_s].$ Under convenient regularity assumptions on such functionals, the authors establish the Itō formula, an integration by parts formula, and the Clark-Ocone martingale representation formula. Of course, in the usual setting all this reduces to the normal notions and formulas.

##### MSC:
 60H05 Stochastic integrals 60H07 Stochastic calculus of variations and the Malliavin calculus 60G44 Martingales with continuous parameter 60H25 Random operators and equations (aspects of stochastic analysis)
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##### References:
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