Initial Value Problem for the Nonlinear Stochastic Impulsive Integro-Differential Equations
Keywords:
Impulsive stochastic integro-differential equations, initial value condition, expectation, variance, successive approximations, existence and uniqueness of solutionAbstract
This paper examines the Cauchy problem for a nonlinear system of first-order
stochastic integro-differential equations with impulse effects. The model under consideration
combines Wiener-type stochastic perturbations, nonlocal integral interactions, and impulse state
jumps at predetermined points in time. Such systems arise in the mathematical modeling of
control processes, population dynamics, economic systems, technical objects, and other dynamic
processes subject to random perturbations and instantaneous state changes. The original Cauchy
problem is reduced to an equivalent nonlinear stochastic functional-integral equation. A sequence
of stochastic approximations is constructed for the resulting operator, and the necessary a priori
estimates are obtained in the space of piecewise continuous random processes. Using the principle of contraction mappings, the existence and uniqueness of a solution to the problem under
study are proven. The obtained results establish a constructive framework for finding a solution and expand existing methods for studying nonlinear stochastic impulse systems with integral
interactions.
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