Backward Stochastic Differential Equations Associated to a Symmetric Markov Process
Tóm tắt
We consider a second order semi-elliptic differential operator L with measurable coefficients, in divergence form, and the semilinear parabolic system of PDE’s
$$\begin{gathered} \left( {\partial _t + L} \right)u\left( {t,x} \right) + f\left( {t,x,u,\nabla u\sigma } \right) = \forall 0 \leqslant t \leqslant T, \hfill \\ u\left( {T,x} \right) = \Phi \left( x \right). \hfill \\ \end{gathered} $$
We solve this system in the framework of Dirichlet spaces and employ the symmetric Markov process of infinitesimal operator L in order to obtain a precised version of the solution u by solving the corresponding system of backward stochastic differential equations. This precised version verifies pointwise the so called “mild equation”, which is equivalent to the above PDE. As a technical ingrediend we prove a representation theorem for arbitrary martingales which generalises a result of Fukushima for martingale additive functionals. The nonlinear term f satisfies a monotonicity condition with respect to u and a Lipschitz condition with respect to ∇u.
Tài liệu tham khảo
Bally, V. and Matoussi, A.: ‘Stochastic PDE’s and doubly stochastic backward differential equations’, J. Theoret. Probab. 14 (2001), 125–164.
Barles, G. and Lesigne, E.: ‘SDE, BSDE and PDE’, in N. El Karoui and L. Mazliak (eds), Backward Stochastic Differential Equations, Pitman Res. Notes in Math. 364, Longman, 1997.
Blumenthal, R.M. and Getoor, R.K.: Markov Processes and Potential Theory, Academic Press, New York, 1966.
Briand, Ph., Delyon, B., Hu, Y., Pardoux, E. and Stoica, L.: ‘L p solutions of backward stochastic differential equations’, Stochastic Process. Appl. 108 (2003), 109–129.
Fukushima, M., Oshima, Y. and Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, Berlin, 1994.
El Karoui, N.: ‘Backward stochastic differential equations a general introduction’, in Backward Stochastic Differential Equations, Pitman Res. Notes in Math. 364, Longman, 1997.
El Karoui, N., Peng, S. and Quenez, M.C.: ‘Backward stochastic differential equations in finance’, Math. Finance 7 (1997), 1–71.
Kato, T.: ‘Schrödinger operators with singular potentials’, Israel J. Math. 13 (1972).
Ladyzenskaya, O.A., Solonikov, N.N. and Uraltseva, V.A.: Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967.
Lions, J.L.: Equations Différentielles Opérationnelles, Springer, Berlin, 1961.
Pardoux, E.: ‘BSDEs, weak convergence and homogenization of semilinear PDEs’, in F.H. Clarke and R.J. Stern (eds), Nonlinear Analysis, Differential Equations and Control, Kluwer Academic Publishers, 1999, pp. 505–549.
Pardoux, E. and Peng, S.: ‘Adapted solution of a backward stochastic differential equation’, Systems and Control Letters 14 (1990), 55–61.
Pardoux, E. and Peng, S.: ‘Backward stochastic differential equations and quasilinear parabolic partial differential equations and their applications’, in B.L. Rozovskii and R.B. Sowers (eds), Stochastic Partial Differential Equations and Their Applications, Lecture Notes in Control Inform. Sci. 176, Springer, Berlin, 1992, pp. 200–217.
Sharpe, M.: General Theory of Markov Processes, Academic Press, Boston, 1988.