Abstract
In this chapter, we address stochastic impulsive systems with time delay, where the impulse times are state-dependent. Using Itô calculus, we develop the essential foundation of the theory of the mentioned system, namely local and global existence, forward continuation and uniqueness of strong solutions.
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Notes
- 1.
This part of the proof is inspired by that of Theorem 4.2.1 in [1] except the dynamics there are delay-free. We reproduced it here for self-contained proof reading.
- 2.
In fact, if a subsequence is taken, the convergence holds with probability one.
References
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Liu XZ, Ballinger GH (2000) Existence, uniqueness, and boundedness results for impulsive delay differential equations. Appl Anal 74(1–2):71–93
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© 2018 Springer Nature Singapore Pte Ltd. and Higher Education Press, Beijing
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Alwan, M.S., Liu, X. (2018). Fundamental Properties of Stochastic Impulsive Systems with Time Delay. In: Theory of Hybrid Systems: Deterministic and Stochastic. Nonlinear Physical Science. Springer, Singapore. https://doi.org/10.1007/978-981-10-8046-3_3
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DOI: https://doi.org/10.1007/978-981-10-8046-3_3
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