## Electronic Journal of Probability

### Viscosity methods giving uniqueness for martingale problems

#### Abstract

Let $E$ be a complete, separable metric space and $A$ be an operator on $C_b(E)$. We give an abstract definition of viscosity sub/supersolution of the resolvent equation $\lambda u-Au=h$ and show that, if the comparison principle holds, then the martingale problem for $A$ has a unique solution. Our proofs work also under two alternative definitions of viscosity sub/supersolution which might be useful, in particular, in infinite dimensional spaces, for instance to study measure-valued processes.

We prove the analogous result for stochastic processes that must satisfy boundary conditions, modeled as solutions of constrained martingale problems. In the case of reflecting diffusions in $D\subset {\bf R}^d$, our assumptions allow $D$ to be nonsmooth and the direction of reflection to be degenerate.

Two examples are presented: A diffusion with degenerate oblique direction of reflection and a class of jump diffusion processes with infinite variation jump component and possibly degenerate diffusion matrix.

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 67, 27 pp.

Dates
Accepted: 17 June 2015
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465067173

Digital Object Identifier
doi:10.1214/EJP.v20-3624

Mathematical Reviews number (MathSciNet)
MR3361255

Zentralblatt MATH identifier
1341.60030

Rights

#### Citation

Costantini, Cristina; Kurtz, Thomas. Viscosity methods giving uniqueness for martingale problems. Electron. J. Probab. 20 (2015), paper no. 67, 27 pp. doi:10.1214/EJP.v20-3624. https://projecteuclid.org/euclid.ejp/1465067173

#### References

• Bardi, Martino; Cesaroni, Annalisa; Manca, Luigi. Convergence by viscosity methods in multiscale financial models with stochastic volatility. SIAM J. Financial Math. 1 (2010), no. 1, 230–265. http://dx.doi.org/10.1137/090748147
• Bayraktar, Erhan; Sirbu, Mihai. Stochastic Perron's method and verification without smoothness using viscosity comparison: the linear case. Proc. Amer. Math. Soc. 140 (2012), no. 10, 3645–3654. http://dx.doi.org/10.1090/S0002-9939-2012-11336-X
• Benth, Fred Espen; Karlsen, Kenneth Hvistendahl; Reikvam, Kristin. Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: a viscosity solution approach. Finance Stoch. 5 (2001), no. 3, 275–303. http://dx.doi.org/10.1007/PL00013538
• Costantini, Cristina; Kurtz, Thomas G. Diffusion approximation for transport processes with general reflection boundary conditions. Math. Models Methods Appl. Sci. 16 (2006), no. 5, 717–762. http://dx.doi.org/10.1142/S0218202506001339
• Costantini, Cristina; Papi, Marco; D'Ippoliti, Fernanda. Singular risk-neutral valuation equations. Finance Stoch. 16 (2012), no. 2, 249–274. http://dx.doi.org/10.1007/s00780-011-0166-8
• Crandall, Michael G.; Ishii, Hitoshi; Lions, Pierre-Louis. User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. http://dx.doi.org/10.1090/S0273-0979-1992-00266-5
• Crandall, Michael G.; Lions, Pierre-Louis. Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983), no. 1, 1–42.
• Dynkin, E. B. Markov processes. Vols. I, II. Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. Die Grundlehren der Mathematischen Wissenschaften, Bande 121, 122 Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-Gottingen-Heidelberg 1965 Vol. I: xii+365 pp.; Vol. II: viii+274 pp.
• Ekren, Ibrahim; Keller, Christian; Touzi, Nizar; Zhang, Jianfeng. On viscosity solutions of path dependent PDEs. Ann. Probab. 42 (2014), no. 1, 204–236. http://dx.doi.org/10.1214/12-AOP788
• Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8.
• Feng, Jin; Kurtz, Thomas G. Large deviations for stochastic processes. Mathematical Surveys and Monographs, 131. American Mathematical Society, Providence, RI, 2006. xii+410 pp. ISBN: 978-0-8218-4145-7; 0-8218-4145-9.
• Fleming, Wendell H.; Soner, H. Mete. Controlled Markov processes and viscosity solutions. Second edition. Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006. xviii+429 pp. ISBN: 978-0387-260457; 0-387-26045-5.
• Graham, Carl. McKean-Vlasov Ito-Skorohod equations, and nonlinear diffusions with discrete jump sets. Stochastic Process. Appl. 40 (1992), no. 1, 69–82.
• Lawrence Gray and David Griffeath. Unpublished manuscript, 1977.
• Ikeda, Nobuyuki; Watanabe, Shinzo. Stochastic differential equations and diffusion processes. Second edition. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. xvi+555 pp. ISBN: 0-444-87378-3.
• Jakobsen, Espen R.; Karlsen, Kenneth H. A "maximum principle for semicontinuous functions” applicable to integro-partial differential equations. NoDEA Nonlinear Differential Equations Appl. 13 (2006), no. 2, 137–165. http://dx.doi.org/10.1007/s00030-005-0031-6
• Jakubowski, Adam. A non-Skorohod topology on the Skorohod space. Electron. J. Probab. 2 (1997), no. 4, 21 pp. (electronic). http://dx.doi.org/10.1214/EJP.v2-18
• Kabanov, Yuri; Kluppelberg, Claudia. A geometric approach to portfolio optimization in models with transaction costs. Finance Stoch. 8 (2004), no. 2, 207–227. http://dx.doi.org/10.1007/s00780-003-0114-3
• Kurtz, Thomas G. Martingale problems for constrained Markov problems. Recent advances in stochastic calculus (College Park, MD, 1987), 151–168, Progr. Automat. Info. Systems, Springer, New York, 1990.
• Kurtz, Thomas G. A control formulation for constrained Markov processes. Mathematics of random media (Blacksburg, VA, 1989), 139–150, Lectures in Appl. Math., 27, Amer. Math. Soc., Providence, RI, 1991.
• Kurtz, Thomas G.; Protter, Philip E. Weak convergence of stochastic integrals and differential equations. II. Infinite-dimensional case. Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995), 197–285, Lecture Notes in Math., 1627, Springer, Berlin, 1996.
• Kurtz, Thomas G.; Stockbridge, Richard H. Stationary solutions and forward equations for controlled and singular martingale problems. Electron. J. Probab. 6 (2001), no. 17, 52 pp. (electronic).
• Ma, Jin; Zhang, Jianfeng. On weak solutions of forward-backward SDEs. Probab. Theory Related Fields 151 (2011), no. 3-4, 475–507. http://dx.doi.org/10.1007/s00440-010-0305-8
• Pham, Huyen. Optimal stopping of controlled jump diffusion processes: a viscosity solution approach. J. Math. Systems Estim. Control 8 (1998), no. 1, 27 pp. (electronic).
• Popivanov, P.; Kutev, N. Viscosity solutions to the degenerate oblique derivative problem for fully nonlinear elliptic equations. Math. Nachr. 278 (2005), no. 7-8, 888–903. http://dx.doi.org/10.1002/mana.200310280
• Soner, H. Mete; Touzi, Nizar. Superreplication under gamma constraints. SIAM J. Control Optim. 39 (2000), no. 1, 73–96. http://dx.doi.org/10.1137/S0363012998348991
• Stroock, D.; Varadhan, S. R. S. On degenerate elliptic-parabolic operators of second order and their associated diffusions. Comm. Pure Appl. Math. 25 (1972), 651–713.