A weak version of path-dependent functional Itô calculus.(English)Zbl 1451.60054

Path dependent functional calculus has been introduced in order to study the dependence of some non anticipating functional $$F(w)=(F(t,w);0\le t\le T)$$ with respect to perturbations on the path $$(w(t);0\le t\le T)$$ ($$F$$ is assumed to be non anticipating in the sense that $$F(t,w)$$ depends only on the path $$(w(s);0\le s\le t)$$). This can be applied to stochastic analysis by replacing $$w$$ by stochastic paths. The work under review considers the case where $$F$$ is not smooth enough for a direct application of this calculus.
The authors want to study some processes $$X$$ adapted to a multidimensional Brownian motion $$B$$, where $$X$$ is considered as a non anticipating functional of $$B$$. The idea is to introduce a notion of embedded discrete structure $$\mathcal Y$$ which consists in a discretisation of $$B$$ associated to a random time subdivision $$T^k=(T_n^k;n\ge0)$$, and in an adapted step process $$X^k$$ converging to the original process $$X$$ as $$k\to\infty$$. Then the functional calculus is worked out in this discrete setting; it leads to finite-dimensional calculations. If this calculus can be extended to the limit $$X$$, one says that the structure $$\mathcal Y$$ is stable, and one can deduce a derivative $$\mathcal{D^Y}X$$ of $$X$$ with respect to perturbations of $$B$$.
In particular (Theorem 4.1), if $$X=\int H\,dB+V$$ (sum of an Itô integral and of another adapted process $$V$$), then under some assumptions one can prove that $$H=\mathcal{D^Y}X$$. Notice that $$V$$ has not necessarily finite variation, and $$X$$ is not necessarily a semimartingale.
In the end of the article, this calculus is applied to backward stochastic differential equations.

MSC:

 60H07 Stochastic calculus of variations and the Malliavin calculus 60G40 Stopping times; optimal stopping problems; gambling theory 60G48 Generalizations of martingales
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