Doctor of Philosophy (PhD)
In 1942, K. Itô published his pioneering paper on stochastic integration with respect to Brownian motion. This work led to the framework for Itô calculus. Note that, Itô calculus is limited in working with knowledge from the future. There have been many generalizations of the stochastic integral in being able to do so. In 2008, W. Ayed and H.-H. Kuo introduced a new stochastic integral by splitting the integrand into the adaptive part and the counterpart called instantly independent. In this doctoral work, we conduct deeper research into the Ayed–Kuo stochastic integral and corresponding anticipating stochastic calculus.
We provide a new proof for the extension of Itô isometry for the Ayed–Kuo stochastic integral which clearly demonstrates the intrinsic nature of the construction of the general integral. Furthermore, we extend classical It\^o theory results for martingales to their Ayed–Kuo stochastic integral analogue, near-martingale. We show the near-martingale property of Ayed–Kuo stochastic integral and optional stopping theorem for near-martingales with bounded stopping times.
Using the general Itô formula for the Ayed–Kuo stochastic integral, we find explicit solutions for linear stochastic differential equations with anticipation. We show existence of solutions for certain classes linear stochastic differential equations with anticipation coming from initial condition as well as from the drift. We present a Trotter inspired product formula to construct the solution. In the process, we also show the uniqueness of the solution. While we mainly rely on the Ayed–Kuo formalism, other theories are used minimally and out of necessity. Using the explicit solution, we show the relation between a solution of an anticipating stochastic differential equations and its Itô projection. Furthermore, we establish Wentzell–Friedlin type large deviation principle for the solution of a class of linear stochastic differential equation with an anticipating drift and non-adapted initial condition.
Shrestha, Pujan, "General Stochastic Calculus and Applications" (2022). LSU Doctoral Dissertations. 5805.