Solution of a stochastic integral equation using integral contractors

Study of a Stochastic Integral Equation Using Integral Contractors

“…The theorems in this paper generalize some earlier work of Hardiman and Tsokos [4], Lee and Padgett [5] and Padgett and Rao [6]. We extend also the results concerning equation (1) of Tsokos and Padgett [7].…”

“…y £ C[0,a] to the stochastic integral equation (5), whereas in [6], in order to get uniqueness, the existence of solutions for (5) appears as an additional condition.…”

“…Equation (1) was also considered by Padgett and Rao [6]. Let us note that under less general conditions (they allow only linear functions ip G ~H) we obtain also uniqueness of the solution, whereas in [6] one does not. This difference lies in the fact that we are able to prove that under the hypotheses of Theorem 1, for every x,w € C[0,a] there is a solution…”

On stochastic integrodifferential equations via non-linear integral contractors II

“…Later, this approach was appropriately extended by Kuo [6] to the notion of random integral contractors for stochastic differential equations of the Ito type, and also for special classes of stochastic integral and integrodifferential equations in various functional spaces. In particular, we highlight papers [4,5,6,8,9,13,12,14] and esspecially paper [8] by Mao treating stochastic differential-functional equations with semimartingales. The important fact is that in all these papers, with a partial exception of [9], the considered equations were linear with respect to Lebesgue and Ito integrals, so the regular integral contractor was defined as a solution to any linear functional equation.…”

On stochastic integrodifferential equations via non-linear integral contractors I

“…Later, this approach was appropriately extended by Kuo [6] to the notion of random integral contractors for stochastic differential equations of the Ito type, and also for special classes of stochastic integral and integrodifferential equations in various functional spaces. In particular, we highlight papers [4,5,6,8,9,13,12,14] and esspecially paper [8] by Mao treating stochastic differential-functional equations with semimartingales. The important fact is that in all these papers, with a partial exception of [9], the considered equations were linear with respect to Lebesgue and Ito integrals, so the regular integral contractor was defined as a solution to any linear functional equation.…”