Hilbert-valued anticipating stochastic differential equations
The authors study the Hilbert space \(H\)-valued solution \(X_ t(x)\) of the stochastic differential equation with Stratonovich integral \[ X_ t(x)=x+\int^ t_ 0\sigma(X_ s(x))\circ dW_ s+\int^ t_ 0 b(X_ s(x))ds,\quad 0\leq t\leq 1,\quad x\in H,\tag{1} \] where \(W\) is a Hilbert space \(K\)-valued Brownian motion with a nuclear covariance operator. Using a generalization of the Kolmogorov continuity criterion for processes indexed by a metric space, as presented in \textit{X. Fernique's} paper [École d'été de probabilités de Saint-Flour XI -- 1981, Lect. Notes Math. 976, 1-74 (1983; Zbl 0507.60027)], they derive the continuity of \(x\mapsto X_ t(x)\) in the compact sets \(B\) of elements \(x\) whose Fourier coefficients in some fixed basis of \(H\) converge to zero fast enough. This result is then applied to prove the existence of the solution of an anticipating version of (1), where the initial value \(x\in H\) is replaced by a \(B\)-valued random variable \(X_ 0\) which may depend on the whole path of \(W\). This is done by using an infinite- dimensional version of Nualart's and Pardoux's formula for Stratonovich integrals depending on a parameter.
stochastic differential equation with Stratonovich integral
Pardoux's formula for Stratonovich integrals
scientific article; zbMATH DE number 522701