Abstract
In the paper we apply methods of the theory of backward stochastic differential equations to prove existence, uniqueness and stochastic representation of solutions of the Cauchy problem for semilinear parabolic equation in divergence form with two time-dependent obstacles. We consider two quite different cases: problems with distinct quasi-continuous obstacles and with irregular obstacles satisfying the so called Mokobodzki condition. As an application we also generalize the Lewy-Stampacchia inequality to non-Radon measures and give new existence result for the Dynkin game problem.
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Andreianov, B., Sbihi, K., Wittbold, P.: On uniqueness and existence of entropy solutions for nonlinear parabolic problems with absorption. J. Evol. Equ. 8, 449–490 (2008)
Aronson, D.G.: Non-negative solutions of linear parabolic equations. Ann. Sc. Norm. Super. Pisa 22, 607–693 (1968)
Bally, V., Matoussi, A.: Weak solutions for SPDEs and backward doubly stochastic differential equations. J. Theoret. Probab. 14, 125–164 (2001)
Bénilan, P., Boccardo, L., Gallouet, T., Gariepy, R., Pierre, M. and Vazquez, J.L.: An L1 theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 22, 240–273 (1995)
Bensoussan, A., Lions J.-L.: Applications of Variational Inequalities in Stochastic Control. North-Holland, Amsterdam (1982)
Blumenthal, M.R., Getoor, R.K.: Markov Processes and Potential Theory. Dover Publications, New York (2007)
Briand, Ph., Delyon, B., Hu, Y., Pardoux, E., Stoica, L.: L p solutions of backward stochastic differential equations. Stoch. Process. Their Appl. 108, 109–129 (2003)
Buckdahn, R., Li, J.: Probabilistic interpretation for systems of Isaacs equations with two reflecting barriers. Nonlinear Differ. Equ. Appl. 16, 381–420 (2009)
Chen, Q.: Optimal obstacle control problem for semilinear evolutionary bilateral variational inequalities. J. Math. Anal. Appl. 307, 677–690 (2005)
Cinlar, E., Jacod, J., Protter, P., Sharpe, M.J.: Semimartingales and Markov processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 54, 161–219 (1980)
Droniou, J., Poretta, A., Prignet, A.: Parabolic capacity and soft measures for nonlinear equations. Potential Anal. 19, 99–161 (2003)
El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S., Quenez, M.C.: Reflected solutions of backward SDEs, and related obstacle problems for PDE’s. Ann. Probab. 25, 702–737 (1997)
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. De Gruyter Studies in Mathematics 19. Walter de Gruyter, New York (1994)
Getoor, R.K., Sharpe, M.J.: Naturality, standardness, and weak duality for Markov processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 67, 1–62 (1984)
Hamadène, S., Hassani, M.: BSDEs with two reflecting barriers: the general result. Probab. Theory Relat. Fields 132, 237–264 (2005)
Klimsiak, T.: On time-dependent functionals of diffusions corresponding to divergence form operators. J. Theor. Probab. doi:10.1007/s10959-011-0381-4 (2011)
Klimsiak, T.: Reflected BSDEs and the obstacle problem for semilinear PDEs in divergence form. Stoch. Process. Their Appl. 122, 134–169 (2012)
Kubo, M.: Variational inequalities with time-dependent constraints in L p. Nonlinear Anal. 73, 390–398 (2010)
Kubo, M., Yamzaki, N.: Periodic solutions of elliptic-parabolic variational inequalities with time-dependent constraints. J. Evol. Equ. 6, 71–93 (2006)
Ladyzenskaya, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and quasi-linear equations of parabolic type. Transl. Math. Monographs 23, Am. Math. Soc., Providence, R.I. (1968)
Lejay, A.: A probabilistic representation of the solution of some quasi-linear PDE with a divergence form operator. Application to existence of weak solutions of FBSDE. Stoch. Process. Their Appl. 110, 145–176 (2004)
Lions, J.-L.: Quelques Méthodes de Résolutions des Problèmes aux Limites Non Linéaires. Dunod, Gauthier Villars, Paris (1969)
Lyons, T.J., Zheng, W.A.: On conditional diffusion processes Proc. Roy. Soc, Edinburgh 115, 243–255 (1990)
Mignot, F., Puel, J.P.: Inéquations d’évolution paraboliques avec convexes dépendant du temps. Applications aux inéquations quasi-variationnelles d’évolution. Arch. Ration. Mech. Anal. 64, 59–91 (1977)
Mokrane, A., Murat, F.: The Lewy-Stampacchia inequality for bilateral problems. Ric. Mat. 53, 139–182 (2004)
Oshima, Y.: Time-dependent Dirichlet forms and related stochastic calculus. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7, 281–316 (2004)
Ouknine, Y., Ndiaye, D.: Weak solutions of semilinear PDEs with obstacle(s) in Sobolev spaces and their probabilistic interpretation via the RFBSDEs and DRFBSDEs. Stoch. Dyn. 2, 247–269 (2008)
Palmeri, M.C.: Homographic approximation for some nonlinear parabolic unilateral problems. J. Convex Anal. 7, 353–373 (2000)
Peng, S.: Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyers type. Probab. Theory Relat. Fields 113, 473–499 (1999)
Peng, S., Xu, M.: The smallest g-supermartingale and reflected BSDE with single and double L2 obstacles. Ann. I.H. Poincare 41, 605–630 (2005)
Pierre, M.: Problemes d’evolution avec contraintes unilatérales et potentiel paraboliques. Comm. Partial Differential Equations 4, 1149–1197 (1979)
Pierre, M.: Parabolic capacity and Sobolev spaces. J. Math. Anal. 14, 522–533 (1983)
Ren, Y., Xia, N.: Generalized reflected BSDE and an obstacle problem for PDEs with nonlinear neumann boundary condition. Stoch. Anal. Appl. 24, 1013–1033 (2006)
Revuz, D.: Mesures associees aux fonctionnelles additives de Markov I. Trans. Am. Math. Soc. 148, 501–531 (1970)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1991)
Rozkosz, A.: Weak convergence of diffusions corresponding to divergence form operators. Stoch. Stoch. Rep. 57, 129–157 (1996)
Rozkosz, A.: On Dirichlet processes associated with second order divergence form operators. Potential Anal. 14, 123–149 (2001)
Rozkosz, A.: Time-inhomogeneous diffusions corresponding to symmetric divergence form operators. Probab. Math. Stat. 22, 231–252 (2002)
Rozkosz, A.: Backward SDEs and Cauchy problem for semilinear equations in divergence form. Probab. Theory Relat. Fields 125, 393–401 (2003)
Rozkosz, A.: On the Feynman-Kac representation for solutions of the Cauchy problem for parabolic equations in divergence form. Stochastics 77, 297–313 (2005)
Stannat, W.: Dirichlet forms and Markov processes: a generalized framework including both elliptic and parabolic cases. Potential Anal. 8, 2–60 (1998)
Stannat, W.: The theory of generalized Dirichlet forms and its application in analysis and stochastics. Mem. Am. Math. Soc. 142, viii+101 (1999)
Stoica, I.L.: A Probabilistic interpretation of the divergence and BSDE’s. Stoch. Process. Their Appl. 103, 31–55 (2003)
Stroock, D.W.: Diffusion semigroups corresponding to uniformly eliptic divergence form operators. In: Seminaire de Probabilities XXII. Lecture Notes in Math., vol. 1321, pp. 316–347 (1988)
Shiqiu, Z., Shengwu, Z.: A generalized existence theorem of reflected BSDEs with double obstacles. Stat. Probab. Lett. 78, 528–536 (2008)
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Research supported by the Polish Minister of Science and Higher Education under Grant N N201 372 436.
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Klimsiak, T. Cauchy Problem for Semilinear Parabolic Equation with Time-Dependent Obstacles: A BSDEs Approach. Potential Anal 39, 99–140 (2013). https://doi.org/10.1007/s11118-012-9323-8
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DOI: https://doi.org/10.1007/s11118-012-9323-8
Keywords
- Semilinear parabolic equation
- Divergence form operator
- Obstacle problem
- Backward stochastic differential equation