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Solutions for a class of stochastic differential equations. (Chinese) Zbl 0593.60067

Let (\(\Omega\),\({\mathcal F},P)\) be a complete probability space. Let (\({\mathcal F}_ t)_{t\leq T}\) be nondecreasing sub-\(\sigma\)-fields of \({\mathcal F}\) and \((W_ t,{\mathcal F}_ t)_{t\leq T}\) be a Wiener process. Let a(t,x) and b(t,x) be two measurable functions in [0,T]\(\times R\). Then as we know, under suitable conditions on a(t,x), b(t,x) and \(\eta\), the stochastic differential equation: \[ (1)\quad d\xi_ t=a(t,\xi_ t)dt+b(t,\xi_ t)dW_ t,\quad \xi_ 0=\eta \] has an unique strong solution. Particularly, when \(a(t,\xi_ t)=a_ 0(t)+a_ 1(t)\xi_ t\) and \(b(t,\xi_ t)=b_ 0(t)+b_ 1(t)\xi_ t,\) under some conditions on \(a_ 0(t)\), \(a_ 1(t)\), \(b_ 0(t)\) and \(b_ 1(t)\), the author shows that (1) can be solved explicitly by his method if and only if \[ \frac{\partial}{\partial t}\{b(t,z)[\frac{\partial (b(t,z))/\partial t}{b^ 2(t,z)}-\frac{\partial}{\partial z}[\frac{a(t,z)}{b(t,z)}]+\frac{1}{2}\frac{\partial^ 2b(t,z)}{\partial \quad z^ 2}]\}=0. \]
Reviewer: Ching-Sung Chou

MSC:

60H20 Stochastic integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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