Bernoulli

• Bernoulli
• Volume 21, Number 2 (2015), 1067-1088.

Finite, integrable and bounded time embeddings for diffusions

Abstract

We solve the Skorokhod embedding problem (SEP) for a general time-homogeneous diffusion $X$: given a distribution $\rho$, we construct a stopping time $\tau$ such that the stopped process $X_{\tau}$ has the distribution $\rho$. Our solution method makes use of martingale representations (in a similar way to Bass (In Seminar on Probability XVII. Lecture Notes in Math. 784 (1983) 221–224 Springer) who solves the SEP for Brownian motion) and draws on law uniqueness of weak solutions of SDEs.

Then we ask if there exist solutions of the SEP which are respectively finite almost surely, integrable or bounded, and when does our proposed construction have these properties. We provide conditions that guarantee existence of finite time solutions. Then, we fully characterize the distributions that can be embedded with integrable stopping times. Finally, we derive necessary, respectively sufficient, conditions under which there exists a bounded embedding.

Article information

Source
Bernoulli Volume 21, Number 2 (2015), 1067-1088.

Dates
First available in Project Euclid: 21 April 2015

http://projecteuclid.org/euclid.bj/1429624971

Digital Object Identifier
doi:10.3150/14-BEJ598

Mathematical Reviews number (MathSciNet)
MR3338657

Zentralblatt MATH identifier
1328.60101

Citation

Ankirchner, Stefan; Hobson, David; Strack, Philipp. Finite, integrable and bounded time embeddings for diffusions. Bernoulli 21 (2015), no. 2, 1067--1088. doi:10.3150/14-BEJ598. http://projecteuclid.org/euclid.bj/1429624971.

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