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

Abstract

Abstract. We prove a comparison principle for unbounded semicontinuous viscosity sub-and supersolutions of nonlinear 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 and (ii) pricing of European and American derivatives via backward stochastic differential equations. Regarding (i), we extend previous results on backward stochastic differential equations in a Lévy setting and the connection to semilinear integro-partial differential equations.

Error estimates for approximate solutions to Bellman equations associated with controlled jump-diffusions

“…With the use of this equivalent definition and the maximum principle for semiconinuous functions adapted to integro-PDEs in [38], it is standard to prove existence, uniqueness, and regularity results, see, e.g., [4,7,38,37,49]. Here we state such results without proofs.…”

“…Such equations may not possess smooth solutions (not even in the linear case), and the appropriate notion of solutions is that of viscosity solutions. The theory of viscosity solutions for (pure) partial differential equations is highly developed [24,31], and there has been an interest in recent years to apply this theory to integro-partial differential equations [1,2,3,4,5,7,14,15,16,32,37,38,47,48,49,51,52,54,55]. Of particular relevance to the present paper are the works [37,38].…”

The Minimal Entropy Martingale Measure and Numerical Option Pricing for the Barndorff–Nielsen–Shephard Stochastic Volatility Model

“…We end this Section with a few words on convergence of our numerical procedure. In the present paper we have not considered this question from a theoretical point of view, but refer the reader to the works by Amadori [1], Amadori, Karlsen and La Chioma [2] and Jakobsen and Karlsen [12], where convergence is analyzed for integroPDEs similar to ours. In order to justify that our numerical solution of Λ indeed converges, we have tested the algorithm with step-wise refining of the grid.…”

Asymptotics for large time of solutions to nonlinear system associated with the penetration of a magnetic field into a substance

“…(see, for example, [1]- [4], [6]- [8], [10]- [12], [19], [26], [27], [29]). One kind of integro-differential systems arises in mathematical modelling of the process of penetrating of a magnetic field into a substance.…”