Euler-Maruyama approximations for SDEs with non-Lipschitz coefficients and applications
The author considers a \(d\)-dimensional system of Itô stochastic differential equations of the form \[ dX(t)= \sigma(X(t))dW(t) + b(X(t))dt,\quad X(0)=x,\;x\in R^d\tag{1} \] where \(W\) is an \(m\)-dimensional Brownian motion. The coefficient functions \(\sigma\) and \(b\) are supposed to satisfy the following non-Lipschitz conditions: \(\| \sigma(x)-\sigma(y)\| ^2 \leq | x-y| ^2 g(| x-y| )\) and \(| b(x)-b(y) | \leq | x-y| g(| x-y| )\), where \(g(x)=C (\log(1/x) \vee K)^{1/\beta}\) for some \(\beta>1\) and \(C\), \(K > 0\). It is then proved that a version of the Euler-Maruyama method converges under these conditions in the \(p\)th mean to the solution of (1). This proof is used to treat (1) with a random and not necessarily adapted initial condition \(\xi\) instead of \(x\). An appropriate extension of the stochastic integral is defined. At the end a theorem concerning a large deviation principle for the anticipative stochastic differential equation is proved. A rate of convergence of the Euler-Maruyama method is not discussed and no numerical experiments illustrate the theoretical results.
Stochastic differential equation with non-Lipschitz coefficients
Anticipative stochastic differential equation
scientific article; zbMATH DE number 5016193
scientific article