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Distributions of ItĂ´ processes: Estimates for the density and for conditional expectations of integral functionals. (English. Russian original) Zbl 0844.60030
Theory Probab. Appl. 39, No. 4, 662-670 (1994); translation from Teor. Veroyatn. Primen. 39, No. 4, 825-833 (1994).
Let for the finite-dimensional equation \[ dy(t,\omega) = f(y(t,\omega), t,\omega) dt + \beta(y(t,\omega), t,\omega)dw(t) \] \(y^{a,s} (t)\) be the solution with the initial condition \(y(s) = a\), which is independent from \(w(t) - w(s)\), \(t \geq s\). The article is devoted to the investigation of the functionals \[ V(x,s,\omega) = E \Biggl\{ \int^t_s \varphi(y^{x,s}(t,\omega), t, \omega) dt/{\mathcal F}_s \Biggr\}, \] where \({\mathcal F}_s = \sigma\{w(\tau) : \tau \leq s\}\). Estimations for the different functional norms of \(V\) are obtained. The main instrument is a stochastic parabolic equation for the conditional density of \(y^{a,s}(t)\) with respect to \({\mathcal F}_t\).

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)