Stochastic Differential Equations

Abstract

Stochastic differential equations describe the time evolution of certain continuous n-dimensional Markov processes. In contrast with classical differential equations, in addition to the derivative of the function, there is a term that describes the random fluctuations that are coded as an Itô integral with respect to a Brownian motion. Depending on how seriously we take the concrete Brownian motion as the driving force of the noise, we speak of strong and weak solutions. In the first section, we develop the theory of strong solutions under Lipschitz conditions for the coefficients. In the second section, we develop the so-called (local) martingale problem as a method of establishing weak solutions. In the third section, we present some examples in which the method of duality can be used to prove weak uniqueness.

As stochastic differential equations are a very broad subject, and since things quickly become very technical, we only excursively touch some of the most important results, partly without proofs, and illustrate them with examples.