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Non-linear degenerate integro-partial differential evolution equations related to geometric Lévy processes and applications to backward stochastic differential equations. (English) Zbl 1049.60050
The authors are interested in viscosity solutions of nonlinear degenerate parabolic integro-partial differential equations with a given terminal condition. The equations involve an integro-differential operator with respect to a given Radon measure on $$E=\mathbb R^2\setminus\{0\}$$ (the so-called Lévy measure) which may possess a second order singularity at the origin. It is proved that one can have a comparison principle for an unbounded semicontinuous viscosity sub- and supersolution of the main equation. It is possible because of a special structure of the problem. This structure arises naturally in finance applications with geometric Brownian motions, or, more generally, geometric Lévy processes, as the underlying stochastic processes for the asset dynamics. Multi-dimensional (semilinear and linear) integro-partial differential equations occur in applications to backward stochastic differential equations (BSDEs). It is proved (under certain conditions) that the solution to the BSDE provides a unique viscosity solution of a semilinear integro-partial differential equation set on $$(0,\infty)^n$$. The new results on pricing European and American option in general Lévy markets are provided.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 91G80 Financial applications of other theories 45K05 Integro-partial differential equations 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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