In this talk, we introduce the dynamic stochastic variational inequality (DSVI). The DSVI is an ODE whose right hand side is defined by the natural mapping of a VI (referred to as the first-stage VI) and is coupled with another stochastic VI (referred to as the second-stage SVI). The DSVI provides a unified modeling framework for various applications involving equilibrium/optimality conditions (VI), dynamics (ODE), and uncertainties (stochasticity). We establish solution existence and uniqueness for two classes of DSVIs: the first class is defined by a strongly monotone SVI in the second stage, and the second class pertains to a box-constrained stochastic linear VI with the P-property in the second stage. Preliminary results on switching dynamics of the DSVI are presented. We develop sample average approximation (SAA) and time-stepping schemes to compute the DSVI. The uniform convergence and exponential convergence are established for the SAA under suitable conditions. A time-stepping EDIIS (energy direct inversion on the iterative subspace) method is proposed to solve the differential VI arising from the SAA of the DSVI. Our results are illustrated by an instantaneous dynamic user equilibrium  problem in transportation engineering. This is a joint work with Dr. Xiaojun Chen of the Hong Kong Polytechnic University.