Markov bridges: SDE representation

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• 作者：Umut Ç ; etin^a ; ^{u.cetin@lse.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; Albina Danilova^b ; ^{a.danilova@lse.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Markov bridge ; hh-transform ; Martingale problem ; Weak convergence
• 刊名：Stochastic Processes and their Applications
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：126
• 期：3
• 页码：651-679
• 全文大小：333 K

文摘

Let X$X$ be a Markov process taking values in $E$ with continuous paths and transition function (P_s,t)$\left(P_{s,t}\right)$. Given a measure μ$\mu$ on $(E,\mathcal{E})$, a Markov bridge starting at (s,ε_x)$(s,{\epsilon }_x)$ and ending at (T^∗,μ)$(T^{\ast },\mu )$ for T^∗<∞$T^{\ast }<\infty$ has the law of the original process starting at x$x$ at time s$s$ and conditioned to have law μ$\mu$ at time T^∗$T^{\ast }$. We will consider two types of conditioning: (a) weak conditioning   when μ$\mu$ is absolutely continuous with respect to P_s,t(x,⋅)$P_{s,t}(x,\cdot )$ and (b) strong conditioning   when μ=ε_z$\mu ={\epsilon }_z$ for some $z\in E$. 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.