Limits of the Wong-Zakai type with a modified drift term. (English) Zbl 0733.60082
Stochastic analysis, Proc. Conf. Honor Moshe Zakai 65th Birthday, Haifa/Isr. 1991, 475-493 (1991).
[For the entire collection see Zbl 0724.00018.]
The author considers the a.s. convergence of solutions of stochastic differential equations \[ dx=f_ 0(x)dt+\sum^{m}_{i=1}f_ i(x)dW^{\nu}_ i \] to a solution of the Stratonovich stochastic differential equation \[ dx=(f_ 0(x)+g(x))dt+\sum^{m}_{i=1}f_ i(x)dW_ i, \] where \(W^{\nu}\) is a “regular” approximation of an m- dimensional Wiener process W (m\(\geq 2)\). It is shown that under suitable choice of the approximation, g can be an arbitrary element of the linear span of all the Lie brackets of the \(f_ i\) for \(i=1,...,m\).
The author considers the a.s. convergence of solutions of stochastic differential equations \[ dx=f_ 0(x)dt+\sum^{m}_{i=1}f_ i(x)dW^{\nu}_ i \] to a solution of the Stratonovich stochastic differential equation \[ dx=(f_ 0(x)+g(x))dt+\sum^{m}_{i=1}f_ i(x)dW_ i, \] where \(W^{\nu}\) is a “regular” approximation of an m- dimensional Wiener process W (m\(\geq 2)\). It is shown that under suitable choice of the approximation, g can be an arbitrary element of the linear span of all the Lie brackets of the \(f_ i\) for \(i=1,...,m\).
Reviewer: H.Pragarauskas (Vilnius)
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
