For realistic predictions of vehicle performance in off-road conditions, it is critical to incorporate in the simulation accurate representations of the variability of the terrain profile. It is not practically feasible to measure the terrain at a sufficiently large number of points, or, if measured, to use such data directly in the simulation. Dedicated modeling techniques and computational methods that realistically and efficiently simulate off-road operating conditions are thus necessary. Many studies have been recently conducted to identify effective and appropriate ways to reduce experimental data in order to preserve only essential information needed to re-create the main terrain characteristics, for future use. This thesis focuses on modeling terrain profiles using the finite difference approach for solving linear second-order stochastic partial differential equations. We currently use this approach to model non-stationary terrain profiles in two dimensions (i.e., surface maps). Certain assumptions are made for the values of the model coefficients to obtain the terrain profile through the fast computational approach described, while preserving the stochastic properties of the original terrain topology. The technique developed is illustrated to recreate the stochastic properties of a sample of terrain profile measured experimentally. To further analyze off-road conditions, stochastic soil properties are incorporated into the terrain topology. Soil models can be developed empirically by measuring soil data at several points, or they can be created by using mathematical relations such as the Bekker's pressure-sinkage equation for homogeneous soils. In this thesis, based on a previously developed stochastic soil model, the polynomial chaos method is incorporated in the soil model. In a virtual proving ground, the wheel and soil interaction has to be simulated in order to analyze vehicle maneuverability over different soil types. Simulations have been created on a surface map for different case studies: stepping with a rigid plate, rigid wheel and flexible wheel, and rolling of a rigid wheel and flexible wheel. These case studies had various combinations of stochastic or deterministic terrain profile, stochastic or deterministic soil model, and an object to run across the surface (e.g., deterministic terrain profile, stochastic soil model, rolling rigid wheel). This thesis develops a comprehensive terrain and soil simulation environment for off-road mobility studies. Moreover, the technique developed to simulate stochastic terrain profile can be employed to simulate other stochastic systems modeled by PDEs. / Master of Science