Nonlinear Filtering In Stochastic Volatility Models PDF Download

Filtering deals with recursive estimation of signals from their noisy measurements. A typical setup consists of a Markov process, which cannot be observed directly and is to be "filtered"from the trajectory of another process, related to it statistically. The general idea is to seek a "best estimate"for the true value of the signal, given only some (potentially noisy) observations of that signal. The optimal estimate is given by the conditional expectation and can be generated by a recursive equation, called the filtering equation, driven by the observation process. If the signal/observation model is linear and Gaussian, the filtering problem is called the Kalman-Bucy filter, otherwise is called a nonlinear filter. Being of considerable practical importance in engineering and economics, the filtering theory poses many interesting mathematical problems and it utilizes areas of mathematics such as stochastic calculus, martingales, etc. This thesis focuses on the mathematical aspects of nonlinear filtering for the case when the signal is a jump-diffusion process, i.e. a stochastic process that involves jumps and diffusion. One important objective of the thesis is to describe the evolution of the conditional distribution characterizing the optimal nonlinear filter using a stochastic differential equation known as the Zakai equation. The main contributions of the research are the moment estimates of the multi-dimensional jump-diffusion process which represent the signal in the nonlinear filtering problem, and a new approach for the uniqueness of the measure-valued solution of the stochastic differential equation that describes the evolution of the optimal filter. Applications of the nonlinear filtering theory to financial economics estimation problems including stochastic volatility models are discussed.