Existence and Uniqueness of Solutions for Stochastic Differential Equations

Stochastic differential equations arise typically in situations where for instance the time evolution of a given quantity has some degree of inherent uncertainty. Dating back to Albert Einstein's work in 1905, stochastic differential equations are widely used in applications such as mathematical physics and financial mathematics. Classical examples include the Black-Scholes model, and Ornstein-Uhlenbeck process as the solution of the Langevin equation. In addition, stochastic differential equations have connections to the theory of deterministic partial differential equations, and the Sobolev space theory of deterministic calculus has its counterpart in the stochastic case as well, leading to the so-called Malliavin calculus, or stochastic calculus of variations. There also exists a considerable research literature of stochastic analysis with respect to other processes than Brownian motion, such as Lévy processes. In this thesis we present an existence and uniqueness theorem for stochastic differential equations with respect to a Brownian motion, under the assumption that the coefficients satisfy Lipschitz and linear growth estimates. The theorem is originally due to Kiyosi Itô. In addition, we present a proof of the continuity of the solution with respect to the initial data, assuming it is deterministic. This theorem was originally proved by Tsukasa Fujiwara and Hiroshi Kunita