On the Weak Convergence of Interpolated Markov Chains to a Diffusion

Abstract

Let $\{\xi_k^n, k = 0, 1, \cdots\}$ denote a $R^r$ valued discrete parameter Markov process for each $n$. For each real $T > 0$, it is shown that suitable piecewise interpolations in $D^r\lbrack 0, T \rbrack$ converge weakly as $n \rightarrow \infty$, to the diffusion given by \begin{equation*}\tag{*} x(t) = x + \int^t_0 f(x(s), s) ds + \int^t_0 \sigma(x(s), s) dw(s),\end{equation*} under essentially the condition that the solution to $(^\ast)$ is unique in the sense of multivariate distributions, $f(\bullet, \bullet), \sigma(\bullet, \bullet)$ are bounded and continuous, and the scaled "infinitesimal" coefficients of the $\{\xi_k^n\}$ are close to $f(\bullet, \bullet)$ and $\sigma(\bullet, \bullet)$. It is not required that $f(\bullet, \bullet)$ or $\sigma(\bullet, \bullet)$ satisfy a uniform Lipschitz condition, nor that $\sigma(\bullet, \bullet)\sigma'(\bullet, \bullet)$ be positive definite. The result extends the result of Gikhman and Skorokhod (1969). Two examples arising in population genetics are given, where $\sigma(\bullet, \bullet)$ is not uniformly Lipschitz.