On the generalized differentiability on the initial data of the flow generated by stochastic equation with reflection.(Ukrainian, English)Zbl 1164.60385

Teor. Jmovirn. Mat. Stat. 75, 127-139 (2006); translation in Theory Probab. Math. Stat. 75, 147-160 (2007).
Let $$\phi_{t}(x)$$, $$x\in\mathbb R_{+}^{d}=\mathbb R^{d-1}\times [0,\infty)$$ be a solution of the following stochastic differential equation in $$\mathbb R^{d}$$ with normal reflection at the boundary: $\begin{cases} d\phi_{t}(x)=a_0(\phi_{t}(x))dt+\sum_{k=1}^{m}a_{k}(\phi_{t}(x))dw_{k}(t)+\bar n\xi(x,dt),\;t\geq0,\\ \phi_{0}(x)=x,\;\xi(x,0)=0,\;x\in\mathbb R_{+}^{d},\end{cases}$ where $$\bar n=(0,\dots,0,1)$$, $$\xi(x,t)$$ is non-decreasing on $$t$$ for fixed $$x$$ and increases only at the points, where $$\phi_{t}(x)\in\mathbb R^{d-1}\times\{0\}$$: $$\xi(x,t)=\int_0^{t}1_{\{\phi_{s}(x)\in\mathbb R^{d-1}\times\{0\}\}}\xi(x,ds)$$. It is proved that the random map $$\phi_{t}(\cdot,\omega)$$ is differentiable in the Sobolev sense for almost all $$\omega$$. For the derivative $$\nabla\phi_{t}$$, the stochastic equation is derived.

MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J25 Continuous-time Markov processes on general state spaces
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