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Viscosity solutions of nonlinear integro-differential equations. (English) Zbl 0870.45002
The paper is concerned with existence and uniqueness of solutions to the following backward nonlinear integro-differential equation of the form $-\partial_t u(t,x)+F(t,x,u(t,x),Du(t,x),D^2 u(t,x)) -\int M(u(t,x+z),u(t,x))m_{t,x}(dz)=0$ with $$u(T,x)$$ given, where $$D$$, $$D^2$$ are space derivatives. The proof of existence is based on finding sub- and supersolutions and applying Perron’s method. The results find application in some problems of mathematical finance involving stochastic utility model.

##### MSC:
 45K05 Integro-partial differential equations 45G10 Other nonlinear integral equations 91B16 Utility theory 91B28 Finance etc. (MSC2000)
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##### References:
 [1] Barles, G.; Perthame, B., Comparison principle for Dirichlet-type Hamilton-Jacobi equations and singular perturbation of degenerated elliptic equations, Appl. Math. Optim., Vol. 21, 21-44, (1990) · Zbl 0691.49028 [2] Bensoussan, A.; Lions, J. L., Contrôle impulsionnel et inéquations quasi variationnelles., (1982), Dunod Paris · Zbl 0491.93002 [3] Crandall, M. G.; Ishii, H.; Lions, P. L., User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., Vol. 27, 1-67, (1992) · Zbl 0755.35015 [4] Duffie, D.; Epstein, L., Stochastic differential utility, Econometrica, Vol. 60, 353-394, (1992) · Zbl 0763.90005 [5] Duffie, D.; Lions, P. L., PDE solutions of stochastic differential utility, J. Math. Econ., Vol. 21, 577-606, (1992) · Zbl 0768.90006 [6] Gihman, I. I.; Skorohod, A. V., The theory of stochastic processes, Vol. 3, (1979), Springer Berlin · Zbl 0404.60061 [7] Ishii, H., Perron’s method for Hamilton-Jacobi equations, Duke Math. J., Vol. 55, 369-384, (1987) · Zbl 0697.35030 [8] С. Мa, Intertemporal recursive utility in the presence of mixed Poisson-Brownian uncertainty, 1993, Mc Gill University, Montreal, Canada. Working paper series 14. [9] Pardoux, E.; Peng, S. G., Adapted solutions of a backward stochastic differential equation, Systems Control Let., Vol. 14, 55-61, (1990) · Zbl 0692.93064 [10] Sayah, A., Equations d’hamilton-Jacobi du premier ordre avec termes intégro-différentiels, parties I & II, Comm P.D.E., Vol. 16, 1057-1093, (1991) · Zbl 0742.45005 [11] Soner, H. M., Optimal control with state-space constraint II, SIAM J. Control Optim., Vol. 24, 1110-1122, (1986) · Zbl 0619.49013 [12] Soner, H. M., Optimal control of jump-Markov processes and viscosity solutions, in stochastic differential systems, stochastic control theory and applications, (Fleming, W. H.; Lions, P. L., IMA Math. Appl., Vol. 10, (1988), Springer Berlin), 501-511 · Zbl 0850.93889
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