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Martingale problems for large deviations of Markov processes

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  • Feng, Jin

Abstract

The martingale problems provide a powerful tool for characterizing Markov processes, especially in addressing convergence issues. For each n, let metric space E-valued process Xn be a solution of the martingale problems (i.e.is a martingale), the convergence of in some sense usually implies the weak convergence of Xn=>X, where X is some process characterized by . Our goal here is to establish similar results for another type of limit theorem - large deviations: defining , thenis a martingale. We prove that the convergence of nonlinear operators implies the large deviation principle for the Xns, where the rate function is characterized by a nonlinear transformation of . Furthermore, a 'running cost' interpretation from control theory can be given to this function. The main assumption is a regularity condition on in the sense that for each , bounded viscosity solution ofis unique. This paper considers processes in CE[0,T].

Suggested Citation

  • Feng, Jin, 1999. "Martingale problems for large deviations of Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 81(2), pages 165-216, June.
    • Handle: RePEc:eee:spapps:v:81:y:1999:i:2:p:165-216
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    Cited by:

    1. Swie[combining cedilla]ch, Andrzej, 2009. "A PDE approach to large deviations in Hilbert spaces," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1081-1123, April.
    2. Jin Feng, 2002. "A Stochastic Filtering Approach To Survival Analysis," Statistical Inference for Stochastic Processes, Springer, vol. 5(1), pages 23-53, January.
    3. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.
    4. Popovic, Lea, 2019. "Large deviations of Markov chains with multiple time-scales," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3319-3359.
    5. Kontoyiannis, I. & Meyn, S.P., 2017. "Approximating a diffusion by a finite-state hidden Markov model," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2482-2507.

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