# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
  Bensoussan, A., Stochastic control of partially observable systems, (1992), Cambridge University Press Cambridge · Zbl 0776.93094  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  Bardi, M.; Crandall, M.G.; Evans, L.C.; Soner, H.M.; Souganidis, P.E., Viscosity solutions and applications, Lecture notes in math., vol. 1660, (1997), Springer Berlin  Crandall, M.G.; Lions, P.L., Viscosity solutions of hamilton – jacobi equations, Trans. amer. math. soc., 277, 1-42, (1983) · Zbl 0599.35024  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  Doss, H., Lien entre équations différentielles stochastiques et ordinaires, Ann. inst. H. Poincaré, 13, 99-125, (1977) · Zbl 0359.60087  Fleming, W.H.; Soner, H.M., Controlled Markov processes and viscosity solutions, (1992), Springer Berlin, New York  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)  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)  Karatzas, I.; Shreve, S., Brownian motion and stochastic calculus, (1988), Springer Berlin · Zbl 0638.60065  Kunita, H., Stochastic flows and stochastic differential equations, Cambridge studies in advanced math., vol. 24, (1990), Cambridge University Press Cambridge  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  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  Nualart, D.; Pardoux, E., Stochastic calculus with anticipating integrands, Probab. theory related fields, 78, 535-581, (1988) · Zbl 0629.60061  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  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  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  Protter, P., Stochastic integration and differential equations, A new approach., (1990), Springer Berlin · Zbl 0694.60047  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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.