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Stochastic differential equations with reflecting boundary conditions. (English) Zbl 0598.60060
In this paper we solve stochastic differential equations with reflecting boundary conditions by a direct approach based on the Skorokhod problem. Our goal will be two-fold: (i) we give a direct approach to the solution of stochastic differential equations with reflecting boundary conditions (without localization arguments) and (ii) reductions to the case including nonsmooth ones (provided convenient restrictions are assumed). Let us describe now more precisely the results and methods presented: first, in the case of a normal reflection inside, say, a smooth bounded open set in ${\bbfR}\sp d$, we show that for each $w\sb t\in C([0,\infty [;{\bbfR}\sp d)$ with $w(0)\in \bar {\cal O}$ there exists a unique solution $(x\sb t,k\sb t)$ of the Skorokhod problem: $$ \gathered x\sb t\in C([0,\infty[, \bar{\cal O}),\quad k\sb t\in C([0,\infty[; {\bbfR}\sp d), \quad k\sb t\in BV(0,T) \text{ for all } T<\infty, \\ x\sb t+k\sb t=w\sb t \text{ for all } t\ge 0, \\ \ \\ k\sb t=\int\sp{t}\sb{0}n(x\sb s)d\vert k\vert\sb s, \vert k\vert\sb t=\int\sp{t}\sb{0}1\sb{(x\sb s\in \partial{\cal O})}d\vert k\vert\sb s \text{ for }t\ge 0, \endgathered \tag{S} $$ where $k\sb t$ stands for the total variation of k on [0,t] and n(x) is the unit outward normal to $\partial {\cal O}$ at x. Having solved the deterministic problem (S), we are able to solve stochastic differential equations with reflection along the normal as for example: on some given probability space $(\Omega,F,F\sb t,P,B\sb t)$ satisfying the usual assumptions, $B\sb t$ being an $F\sb t$-Brownian motion, find a continuous adapted semimartingale $X\sb t$ such that $$ \gathered X\sb t=x+\int\sp{t}\sb{0}\sigma (X\sb s)dB\sb s+\int\sp{t}\sb{0}b(X\sb s)ds-k\sb t, \\ X\sb t\in \bar {\cal O}\text{ for all }t\ge 0,\text{ a.s., $k\sb t$ is a bounded variation process,} \\ \vert k\vert\sb t=\int\sp{t}\sb{0} 1\sb{(X\sb s\in \partial {\cal O})} d\vert k\vert\sb s,\quad k\sb t=\int\sp{t}\sb{0}n(X\sb s)d\vert k\vert\sb s, \endgathered\tag{SDE} $$ for all $t\ge 0$ a.s. We prove that there exists a unique solution $(x\sb t,k\sb t)$ of (SDE) provided, for example, $\sigma$, b are Lipschitz on $\bar {\cal O}$ and ${\cal O}$ is smooth.

60H10Stochastic ordinary differential equations
60G44Martingales with continuous parameter
60H25Random operators and equations
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