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Stability of the Stochastic Differential Equation using Stratonovich - Formula | ||
| College Of Basic Education Research Journal | ||
| Article 30, Volume 20, Issue 4, December 2024, Pages 693-708 PDF (821.82 K) | ||
| Document Type: Research Paper | ||
| DOI: 10.33899/berj.2024.185734 | ||
| Authors | ||
| Ibrahim Mohamed albdrany* 1; Abdulghafoor Jasim Salim* 2 | ||
| 1طالب دراسات عليا | ||
| 2Department of Mathematics, College of Computer Science and Mathematics, University of Mosul. Mosul – Iraq | ||
| Abstract | ||
| In this paper we applied and explain the stability of the solution by using Stratonovich - formula to some non-linear stochastic differential equations, suppose that Lyapunov functions satisfies the given stochastic differential equation, then we the Lyapunov direct second method and Lyapunov theorems by Stratonovich - formula applied to analyze and explain the stability (p-stable, mean square stable) and stochastically asymptotically stable in the large for the solution, then we explain the methods by some examples.we know that the trivial solution is said to be stable if the derivative of Lyapunov function is less than or equal to zero, while if it is only negative-definite then it is asymptotically stable. To find the stability of stochastic differential equation we use the function LV(X_t)≤0 which is equivalence with the inequality (V˙) ̇(X_t)≤0 for deterministic equation, we explain the stability condition for some nonlinear stochastic differential equation by using the direct method (lyapunov direct method), also we explain asymptotically stable in the large not almost this condition is satisfies, that is if the trivial solution is asymptotically stable but not asymptotically stable in large by the fact if the limit is not equal to zero. We explain the methods by several examples,As a future studies one can study the stability (direct method) for some non-linear (harmonic or exponential) stochastic differential equation by using Stratonovich formula for their solution compare it with Ito formula | ||
| Keywords | ||
| stability; ،stochastic differential equation; ؛; ،the Lyapunov function | ||
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