An efficient data-driven approximation to the stochastic differential equations with non-global Lipschitz coefficient and multiplicative noise

This paper studied the numerical approximation of the stochastic differential equations driven by non-global Lipschitz drift coefficient and multiplicative noise. An efficient data-driven method, called extended continuous latent process flow, was proposed for the underlying problem. Compared with the piecewise construction of a variational posterior process used in the classical continuous latent process flow developed by Deng et al. ^[13], the principle idea of our method was to derive a variational lower bound by constructing a posterior latent process conditional on all information over the whole time interval to maximize the log-likelihood generated by the observations, which reduces the computational cost and, thus, provides a convenient way to approximate the considered equation. Particularly, our new method showed a better approximation to the underlying equation than the classical drift- $\theta$ discretization scheme through numerical error comparison. Numerical experiments were finally reported to demonstrate the effectiveness and generalization performance of the proposed method.