×

Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I. (English) Zbl 1053.60065

A nonlinear stochastic partial differential equation in the Stratonovich form \[ du = \bigl \{ Au + f(t,x,u,\sigma ^ *(x)Du)\bigr \}\,dt + \sum ^ {d}_ {i=1} g_ {i}(t,x,u)\circ\,dB^ {i}_ {t}, \quad (t,x)\in (0,T)\times \mathbb R^ {m}, \tag{1} \] with initial condition \(u(0,\cdot ) = u_ 0\) on \(\mathbb R^ {m}\) is studied. In (1), \(B\) denotes the standard \(d\)-dimensional Wiener process and \(A\) is a second-order differential operator \[ A = \sum ^ {m}_ {i,j=1} \sum ^ {k}_ {l=1}\sigma _ {il}(x) \sigma _ {jl}(x)\partial ^ 2_ {x_ {i}x_ {j}} + \sum ^ {m}_ {i=1} \beta _ {i}(x)\partial _ {x_ {i}}. \] It is supposed that \(u_ 0\) is a continuous function of a polynomial growth, \(\sigma \), \(\beta \) are Lipschitz continuous functions, \(g\) is a continuous function sufficiently smooth in both the second and the third variables, and \(f\) is a progressively measurable random field, almost surely uniformly Lipschitz continuous in all variables.
By extending the well known deterministic concept, the authors propose a new definition of a stochastic viscosity solution to (1). A transformation converting (1) into a partial differential equation with random coefficient (but with no martingale term) is found such that stochastic viscosity solutions to (1) are mapped in a one-to-one way to pathwise viscosity solutions of this random PDE. By means of this transformation and the theory of backward doubly stochastic differential equations, existence of stochastic viscosity solutions of (1) is established. [For part II see below, Zbl 1053.60064.]

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)

Citations:

Zbl 1053.60064
Full Text: DOI

References:

[1] Bensoussan, A., Stochastic Control of Partially Observable Systems (1992), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0776.93094
[2] Buckdahn, R., Ma, J., 2001. Stochastic viscosity solution for nonlinear partial differential equations (Part II), Stochastic Process. Appl. 93 (2001) 205-228.; Buckdahn, R., Ma, J., 2001. Stochastic viscosity solution for nonlinear partial differential equations (Part II), Stochastic Process. Appl. 93 (2001) 205-228. · Zbl 1053.60066
[3] Bardi, M.; Crandall, M. G.; Evans, L. C.; Soner, H. M.; Souganidis, P. E., Viscosity solutions and applications. Viscosity solutions and applications, Lecture Notes in Math., vol. 1660 (1997), Springer: Springer Berlin
[4] Crandall, M. G.; Lions, P. L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277, 1-42 (1983) · Zbl 0599.35024
[5] Crandall, M. G.; Ishii, H.; Lions, P. L., User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (NS), 27, 1-67 (1992) · Zbl 0755.35015
[6] Doss, H., Lien entre équations différentielles stochastiques et ordinaires, Ann. Inst. H. Poincaré, 13, 99-125 (1977) · Zbl 0359.60087
[7] Fleming, W. H.; Soner, H. M., Controlled Markov Processes and Viscosity Solutions (1992), Springer: Springer Berlin, New York
[8] Fujiwara, T.; Kunita, H., Stochastic differential equations of jump type and Lévy processes in diffeomorphism group, J. Math. Kyoto Univ., 25, 1, 71-106 (1989) · Zbl 0575.60065
[9] Kobylansky, M., Résultats d’existence et d’unicité pour des équations différentielles stochastiques rétrogrades avec des générateurs à croissance quadratique, C. R. Acad. Sci. Paris Sér. I Math., 324, 1, 81-86 (1997) · Zbl 0880.60061
[10] Karatzas, I.; Shreve, S., Brownian Motion and Stochastic Calculus (1988), Springer: Springer Berlin · Zbl 0638.60065
[11] Kunita, H., Stochastic Flows and Stochastic Differential Equations. Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Math., vol. 24 (1990), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0743.60052
[12] Lions, P-L.; Souganidis, P. E., Fully nonlinear stochastic partial differential equations, C.R. Acad. Sci. Paris, t., 326, 1, 1085-1092 (1998) · Zbl 1002.60552
[13] Lions, P-L.; Souganidis, P. E., Fully nonlinear stochastic partial differential equations: non-smooth equations and applications, C. R. Acad. Sci. Paris, t., 327, 1, 735-741 (1998) · Zbl 0924.35203
[14] Nualart, D.; Pardoux, E., Stochastic Calculus with Anticipating Integrands, Probab. Theory Related Fields, 78, 535-581 (1988) · Zbl 0629.60061
[15] Ocone, D.; Pardoux, E., A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations, Ann. Inst. H. Poincaré, 25, 1, 39-71 (1989) · Zbl 0674.60057
[16] Pardoux, E., Peng, S., 1992. Backward Stochastic Differential Equations and Quasilinear Parabolic Partial Differential Equations, Lecture Notes in CIS. Springer, Berlin, 176, 200-217.; Pardoux, E., Peng, S., 1992. Backward Stochastic Differential Equations and Quasilinear Parabolic Partial Differential Equations, Lecture Notes in CIS. Springer, Berlin, 176, 200-217. · Zbl 0766.60079
[17] Pardoux, E.; Peng, S., Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Related Fields, 98, 209-227 (1994) · Zbl 0792.60050
[18] Protter, P., Stochastic Integration and Differential equations, A New Approach. (1990), Springer: Springer Berlin · Zbl 0694.60047
[19] Sussmann, H., On the gap between deterministic and stochastic differential equations, Ann. Probab., 6, 19-41 (1978) · Zbl 0391.60056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.