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Markov bridges: SDE representation

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  • Çetin, Umut
  • Danilova, Albina

Abstract

Let X be a Markov process taking values in E with continuous paths and transition function (Ps,t). Given a measure μ on (E,E), a Markov bridge starting at (s,εx) and ending at (T∗,μ) for T∗<∞ has the law of the original process starting at x at time s and conditioned to have law μ at time T∗. We will consider two types of conditioning: (a) weak conditioning when μ is absolutely continuous with respect to Ps,t(x,⋅) and (b) strong conditioning when μ=εz for some z∈E. The main result of this paper is the representation of a Markov bridge as a solution to a stochastic differential equation (SDE) driven by a Brownian motion in a diffusion setting. Under mild conditions on the transition density of the underlying diffusion process we establish the existence and uniqueness of weak and strong solutions of this SDE.

Suggested Citation

  • Çetin, Umut & Danilova, Albina, 2016. "Markov bridges: SDE representation," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 651-679.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:3:p:651-679
    DOI: 10.1016/j.spa.2015.09.015
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    Cited by:

    1. Umut c{C}etin & Alaina Danilova, 2022. "Order routing and market quality: Who benefits from internalisation?," Papers 2212.07827, arXiv.org.
    2. Abel Azze & Bernardo D'Auria & Eduardo Garc'ia-Portugu'es, 2022. "Optimal stopping of Gauss-Markov bridges," Papers 2211.05835, arXiv.org, revised Dec 2023.
    3. Umut c{C}etin, 2016. "Financial equilibrium with asymmetric information and random horizon," Papers 1603.08828, arXiv.org, revised Sep 2017.
    4. Umut Çetin, 2018. "Financial equilibrium with asymmetric information and random horizon," Finance and Stochastics, Springer, vol. 22(1), pages 97-126, January.
    5. Umut c{C}et{i}n, 2018. "Mathematics of Market Microstructure under Asymmetric Information," Papers 1809.03885, arXiv.org.

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