Bernoulli

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

Finite, integrable and bounded time embeddings for diffusions

Stefan Ankirchner, David Hobson, and Philipp Strack

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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

Permanent link to this document
https://projecteuclid.org/euclid.bj/1429624971

Digital Object Identifier
doi:10.3150/14-BEJ598

Mathematical Reviews number (MathSciNet)
MR3338657

Zentralblatt MATH identifier
1328.60101

Keywords
bounded time embedding Skorokhod’s embedding theorem

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


Export citation

References

  • [1] Ankirchner, S. and Strack, P. (2011). Skorokhod embeddings in bounded time. Stoch. Dyn. 11 215–226.
    Mathematical Reviews (MathSciNet): MR2836522
    Digital Object Identifier: doi:10.1142/S0219493711003255
  • [2] Azéma, J., Gundy, R.F. and Yor, M. (1980). Sur l’intégrabilité uniforme des martingales continues. In Seminar on Probability XIV (Paris, 1978/1979) (French). Lecture Notes in Math. 784 53–61. Berlin: Springer.
  • [3] Bass, R.F. (1983). Skorokhod imbedding via stochastic integrals. In Seminar on Probability XVII. Lecture Notes in Math. 986 221–224. Berlin: Springer.
    Mathematical Reviews (MathSciNet): MR770414
  • [4] Carr, P. and Lee, R. (2010). Hedging variance options on continuous semimartingales. Finance Stoch. 14 179–207.
    Mathematical Reviews (MathSciNet): MR2607762
    Digital Object Identifier: doi:10.1007/s00780-009-0110-3
  • [5] Cox, A.M.G. and Hobson, D.G. (2004). An optimal Skorokhod embedding for diffusions. Stochastic Process. Appl. 111 17–39.
    Mathematical Reviews (MathSciNet): MR2049567
    Digital Object Identifier: doi:10.1016/j.spa.2004.01.003
  • [6] Cox, A.M.G. and Hobson, D.G. (2006). Skorokhod embeddings, minimality and non-centred target distributions. Probab. Theory Related Fields 135 395–414.
    Mathematical Reviews (MathSciNet): MR2240692
    Digital Object Identifier: doi:10.1007/s00440-005-0467-y
  • [7] Cox, A.M.G. and Wang, J. (2013). Root’s barrier: Construction, optimality and applications to variance options. Ann. Appl. Probab. 23 859–894.
    Mathematical Reviews (MathSciNet): MR3076672
    Digital Object Identifier: doi:10.1214/12-AAP857
    Project Euclid: euclid.aoap/1362684848
  • [8] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II., 2nd ed. New York: Wiley.
    Mathematical Reviews (MathSciNet): MR270403
  • [9] Hobson, D. (2011). The Skorokhod embedding problem and model-independent bounds for option prices. In Paris–Princeton Lectures on Mathematical Finance 2010. Lecture Notes in Math. 2003 267–318. Berlin: Springer.
    Mathematical Reviews (MathSciNet): MR2762363
    Digital Object Identifier: doi:10.1007/978-3-642-14660-2_4
  • [10] Karatzas, I. and Shreve, S.E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. New York: Springer.
    Mathematical Reviews (MathSciNet): MR1121940
  • [11] Kotani, S. (2006). On a condition that one-dimensional diffusion processes are martingales. In In Memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX. Lecture Notes in Math. 1874 149–156. Berlin: Springer.
    Mathematical Reviews (MathSciNet): MR2276894
    Digital Object Identifier: doi:10.1007/978-3-540-35513-7_12
  • [12] Monroe, I. (1972). On embedding right continuous martingales in Brownian motion. Ann. Math. Statist. 43 1293–1311.
    Mathematical Reviews (MathSciNet): MR343354
    Digital Object Identifier: doi:10.1214/aoms/1177692480
    Project Euclid: euclid.aoms/1177692480
  • [13] Oberhauser, H. and dos Reis, G. (2013). Root’s barrier, viscosity solutions of obstacle problems and reflected FBSDEs. Available at arXiv:1301.3798.
    arXiv: arXiv:1301.3798
  • [14] Pedersen, J.L. and Peskir, G. (2001). The Azéma–Yor embedding in non-singular diffusions. Stochastic Process. Appl. 96 305–312.
    Mathematical Reviews (MathSciNet): MR1865760
    Digital Object Identifier: doi:10.1016/S0304-4149(01)00120-X
  • [15] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Berlin: Springer.
    Mathematical Reviews (MathSciNet): MR1725357
  • [16] Rogers, L.C.G. and Williams, D. (1987). Diffusions, Markov Processes, and Martingales. Vol. 2. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: Wiley.
    Mathematical Reviews (MathSciNet): MR921238
  • [17] Rogozin, B.A. (1966). Distribution of certain functionals related to boundary value problems for processes with independent increments. Teor. Veroyatn. Primen. 11 656–670.
    Mathematical Reviews (MathSciNet): MR208682
  • [18] Root, D.H. (1969). The existence of certain stopping times on Brownian motion. Ann. Math. Statist. 40 715–718.
    Mathematical Reviews (MathSciNet): MR238394
    Digital Object Identifier: doi:10.1214/aoms/1177697749
    Project Euclid: euclid.aoms/1177697749
  • [19] Seel, C. and Strack, P. (2013). Gambling in contests. J. Econom. Theory 148 2033–2048.
    Mathematical Reviews (MathSciNet): MR3146917
    Digital Object Identifier: doi:10.1016/j.jet.2013.07.005

Bernoulli Society for Mathematical Statistics and Probability

Bernoulli

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more