Doctor of Philosophy (PhD)
In this dissertation, we focus mainly on the further study of the new stochastic integral introduced by Ayed and Kuo in 2008. Several properties of this new stochastic integral are obtained. We first introduce the concept of near-martingale for non-adapted stochastic processes. This concept is a generalization of the martingale property for adapted stochastic processes in the It\^o theory. We prove a special case of It\^o isometry for the stochastic integral of certain instantly independent processes. We obtain some formulas for expressing a new stochastic integral in terms of It\^o integrals and Riemann integrals. Several generalized versions of It\^o's formula for the new stochastic integral obtained by Ayed and Kuo are given. We also provide some examples to illustrate the ideas.
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Sae-Tang, Anuwat, "A new theory of stochastic integration" (2011). LSU Doctoral Dissertations. 893.