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Computable Bounds of Exponential Moments of Simultaneous Hitting Time for Two Time-Inhomogeneous Atomic Markov Chains

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Stochastic Processes, Statistical Methods, and Engineering Mathematics (SPAS 2019)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 408))

Abstract

In this paper, we study the first simultaneous hitting of the atom by two discrete-time, inhomogeneous Markov chains with values in general phase space. We establish conditions for the existence and find computable bounds for the hitting time’s exponential moment using a geometric drift condition adapted for time-inhomogeneous Markov chains.

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Correspondence to Vitaliy Golomoziy .

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Appendix

Appendix

We state here the Comparison Theorem, it is proved in [7], Theorem 4.3.1.

Theorem 5

Let \(\{\mathcal {V}_n,\ n\ge 0\}\), \(\{\mathcal {Y}_n,\ n\ge 0\}\), and \(\{Z_n,\ n\ge 0\}\) be three \(\{\mathcal {F}_n,\ n\ge 0\}\)-adapted nonnegative processes such that for all \(n\ge 0\),

$$\begin{aligned}\mathbb {E}\left[ \mathcal {V}_{n+1}|\mathcal {F}_n\right] + \mathcal {Z}_n \le \mathcal {V}_n + \mathcal {Y}_n,\ \mathbb {P}\text{-- } \text{ a.s. }.\end{aligned}$$

Then for every \(\{\mathcal {F}_n,\ n\ge 0\}\)-stopping time \(\tau \),

$$\begin{aligned} \mathbb {E}\left[ \mathcal {V}_{\tau }{1}_{\tau < \infty }\right] + \mathbb {E}\left[ \sum \limits _{k=0}^{\tau -1} \mathcal {Z}_k\right] \le \mathbb {E}[\mathcal {V}_0] + \mathbb {E}\left[ \sum \limits _{k=0}^{\tau -1} \mathcal {Y}_{k}\right] .\end{aligned}$$

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Golomoziy, V. (2022). Computable Bounds of Exponential Moments of Simultaneous Hitting Time for Two Time-Inhomogeneous Atomic Markov Chains. In: Malyarenko, A., Ni, Y., Rančić, M., Silvestrov, S. (eds) Stochastic Processes, Statistical Methods, and Engineering Mathematics . SPAS 2019. Springer Proceedings in Mathematics & Statistics, vol 408. Springer, Cham. https://doi.org/10.1007/978-3-031-17820-7_5

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