Setwise convergence of solution measures of stochastic differential equations (Q1090002)

The main result of this paper is the following theorem, and two of its variants. Theorem. Consider a system of stochastic differential equations \[ (*)\quad dX_ n(t)=f_ n(t,X_ n(t))dt+dW_ t,\quad X_ n(0)=0,\quad t\in [0,1],\quad n\geq 1, \] where \(\{W_ t,t\in [0,1]\}\) is the standard Brownian motion on \((\Omega,\Sigma,P).\) Suppose that the drift coefficient \(f_ n:[0,1]\times {\mathbb{R}}\to {\mathbb{R}}\) satisfies the usual Lipschitz conditions in x uniformly in t, with the Lipschitz constant independent of n, and that \(f_ n(t,\cdot)\) is a Borel function, \(0\leq t\leq 1\). Then (the standard theory implies the existence of a unique solution) the probability measure \(\mu_ n(\cdot)\) associated with the solution \(X_ n\), \(n\geq 1\), of the system (*) under change of measure given by the Girsanov theorem [cf. \textit{D. W. Stroock} and \textit{S. R. S. Varadhan}, Multidimensional diffusion processes, (1979; Zbl 0426.60069)] converges to a probability measure \(\mu_ 0(\cdot)\) provided \[ E(\int^{1}_{0}[f_ n(t,W_ t)-f_ 0(t,W_ t)]^ 2dt)\to 0\text{ as } n\to \infty. \]