Book Description
In this thesis, we introduce a random vortex dynamics model described by stochastic differential equations. We first explore the existence and uniqueness of the solution. The nonlinear filtering problem for both continuous and jump noises are studied and an approximation of the associated solutions is discussed. We also study the absolute continuity of the law for the solution of our model using Malliavin calculus. The analysis of nonlinear filtering problem for our model is divided into two parts. For the case of continuous noise, we derive a numerical approximation for the nonlinear filtering equations of vortex dynamics in two dimensions using particle filter method. We prove the convergence of this scheme allowing the observation vector to be unbounded. The SDE driven by both continuous and jump noise is used to model stochastic Lagrangian particle dynamics with jumps for the three dimensional Navier-Stokes flow and the associated nonlinear filtering problem is also studied. We apply results from backward Kolmogorov integro-differential equation problem to prove uniqueness of solution to the Zakai equations of nonlinear filtering.