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.


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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