A generalization of the Bismut-Elworthy formula (Q2708587)

The author considers a stochastic differential equation in a separable Hilbert space \(H\) NEWLINE\[NEWLINE d\xi(t)=b(\xi(t))dt+\sigma(\xi(t))dw(t),\quad\xi(0)=x,NEWLINE\]NEWLINE and one denotes, for \(\varphi\) continuous bounded real function on \(H\), \(P_{t}\varphi(x)= E\varphi(\xi(t,x))\), where \(\xi(t,x)\) is an \(H\)-valued adapted solution starting from \(x\). The author proves that if for some \(C^{1}\) bounded function \(\varphi\) on \(H\) we have NEWLINE\[NEWLINE\langle D(P_{t}\varphi)(x),h\rangle_{H} = E\langle D\varphi(\xi(t,x)),\sigma(\xi(t,x))v(t,x,h)\rangle,\quad t>0, NEWLINE\]NEWLINE for a suitable adapted process \(v(t,x,h)\in L^{2}(0,T,L^{2}(\Omega))\), then the following formula holds NEWLINE\[NEWLINE \langle D(P_{t}\varphi)(x),h\rangle_{H} =\frac{1}{t} E\varphi(\xi(t,x)) \int_{0}^{t}\langle v(t,x,h), dw(s)\rangle_{H}. NEWLINE\]NEWLINE This is a generalization of the Bismut-Elworthy formula [see \textit{J.-M. Bismut}, Z. Wahrscheinlichkeitstheorie Verw. Geb. 56, 469-505 (1981; Zbl 0445.60049) or \textit{K. D. Elworthy} and \textit{X.-M. Li}, J. Funct. Anal. 125, No. 1, 252-286 (1994; Zbl 0813.60049)]. Applications are given for stochastic equations having diffusion term possibly degenerate or for stochastic differential equations in finite or infinite dimension whose solutions are not mean-square differentiable with respect to initial datum.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00037].