Stochastic Analysis and Related Topics V pp 203-231 | Cite as
Wick Approximation of Quasilinear Stochastic Differential Equations
Conference paper
Abstract
For a \(
\varepsilon \, >\,0
\) be a smooth approximation to 1-dimensional Brownian motion \(
\left\{ {W_s^\varepsilon } \right\}s \geqslant 0
\). We consider the equation where ◇ denotes the Wick product. It is conjectured that (with reasonable conditions on b and σ) a unique strong solution \(
X_t^\varepsilon
\) exists for all ε and that \(
X_t^\varepsilon \to {X_t}\,as\,\varepsilon \, \to \,0\,(i.e.,\,W_s^\varepsilon \, \to \,{W_s})
\), where X t is the solution of the Itô differential equation We prove the conjecture in the quasilinear case, i.e., where σ(s, x) = σs x, where σs is independent of x.
$$
X_t^\varepsilon = \eta + \int\limits_0^t {\sigma (s,X_s^\varepsilon )\diamondsuit W_s^\varepsilon ds + \int\limits_0^t {b(s,X_s^\varepsilon )} } ds;\,0\, \leqslant \,s\, <\,\infty
$$
(1))
$$
{X_t} = \eta + \int\limits_0^t \sigma (s,{X_s})d{W_s} + \int\limits_0^t {b(s,{X_s})ds;\,0\, \leqslant \,s\, <\,\infty }
$$
(2))
The conjecture should be compared to the Wong-Zakai theorem, which says that if we let \(
Y_t^\varepsilon
\) be the solution of the stochastic equation (with o replaced by ordinary product)
then \(
Y_t^\varepsilon \to {Y_t}
\), where Y t is the solution of the Stratonovitch differential equation
$$
Y_t^\varepsilon = \eta + \int\limits_0^t \sigma (s,Y_s^\varepsilon )W_s^\varepsilon ds + \int\limits_0^t b (s,Y_s^\varepsilon )ds;\,0\, \leqslant \,s\, < \,\infty
$$
(1)’)
$$
{Y_{t}} = \eta + \int_{0}^{t} {\sigma \left( {s,{Y_{s}}} \right)} \circ d{W_{s}} + \int_{0}^{t} {b\left( {s,{Y_{s}}} \right)ds;0 \leqslant s} < \infty
$$
(2)’)
Keywords
Stochastic Differential Equation Wiener Space Unique Strong Solution Wiener Measure Stochastic Differential Equa
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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