Non-linear degenerate integro-partial differential evolution equations related to geometric Lévy processes and applications to backward stochastic differential equations

Abstract

We prove a comparison principle for unbounded semicontinuous viscosity sub- and supersolutions of non-linear degenerate parabolic integro-partial differential equations coming from applications in mathematical finance in which geometric Lévy processes act as the underlying stochastic processes for the assets dynamics. As a consequence of the “geometric form” of these processes, the comparison principle holds without assigning spatial boundary data. We present applications of our result to (i) backward stochastic differential equations (BSDEs) and (ii) pricing of European and American derivatives via BSDEs. Regarding (i), we extend previous results on BSDEs in a Lévy setting and the connection to semilinear integro-partial differential equations.

Keywords:

• Integro-partial differential equations
• Viscosity solutions
• Comparison principles
• Lévy processes
• Backward stochastic differential equations
• Option pricing
• Primary: 45K55
• Secondary: 49L20, 60H10

Acknowledgements

K.H. Karlsen is supported in part by the BeMatA program of the Research Council of Norway and the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282. We are grateful to an anonymous referee for very careful reading of the first version of this paper.

Notes

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