Anticipating Stochastic Integrals and Related Linear Stochastic Differential Equations

Abstract

Itô’s stochastic calculus revolutionized the field of stochastic analysis and has found numerous applications in a wide variety of disciplines. Itô’s theory, even though quite general, cannot handle anticipating stochastic processes as integrands. There have been considerable efforts within the mathematical community to extend Itô’s calculus to account for anticipation. The Ayed–Kuo integral — introduced in 2008 — is one of the most recent developments. It is arguably the most accessible among the theories extending Itô’s calculus — relying solely on probabilistic methods. In this dissertation, we look at the recent advances in this area, highlighting our contributions. First, we extend the class of linear stochastic differential equations with anticipating initial conditions that have closed-form solutions in this theory. Then we prove an extension of Itô’s isometry for the Ayed–Kuo integral using purely probabilistic tools and exploiting the intrinsic nature of the integral. We also prove an optional stopping theorem for near-martingales — the counterpart of martingales in the Ayed–Kuo theory. We study the behavior of conditioned processes corresponding to the solution of a linear stochastic differential equation with anticipating initial conditions. We analyze a particular class of linear stochastic differential equations with anticipating coefficients in the drift term and derive the solution using two methods: (1) by guessing an ansatz and using the general differential formula in the Ayed–Kuo theory, and (2) by introducing a novel “braiding” technique that allows us to construct the solution by an iterative process. We derive explicit solutions in each theory independently and show that both methods yield the same solution under identical conditions. We establish a Freidlin–Wentzell type large deviations principle for solutions of this specific class of anticipating linear stochastic differential equations. Finally, we highlight a few areas of research in this field.