Nonlinear integro-differential evolution problems arising in option pricing: a viscosity solutions approach

By Annalisa L. Amadori. Differential and Integral Equations, 2003

67 citations recorded by the source corpus.

A class of nonlinear integro-differential Cauchy problems is studied by means of the viscosity solutions approach. In view of financial applications, we are interested in continuous initial data with exponential growth at infinity. Existence and uniqueness of solution is obtained through Perron’s method, via a comparison principle; besides, a first order regularity result is given. This extension of the standard theory of viscosity solutions allows to price derivatives in jump–diffusion markets with correlated assets, even in the presence of a large investor, by means of the PDE’s approach. In particular, derivatives may be perfectly hedged in a completed market.

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/9b37053d8e0dda784c22b45c4b4ff821e2da02d4

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Citations in the corpus

Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes (2010, methodology): “In a separate piece of work [15], we adapt this approach to the nonlocal Bellman equation of controlled jump-diffusion processes, which is drawing a lot of interests these days due to its applications in mathematical finance (see for example [2, 3, 12, 13, 20] and the references therein).”
Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations (2017, unknown): “sqrt(shape [1]+ output_size ))) 78 hidden = tf.”
Holder estimates for solutions of integra-differential equations like the fractional laplace (2006, background): “Then for a small α, u ∈ C(Br) and sup x,y∈Br |u(x)− u(y)| |x− y| ≤ C 1 rα ‖u‖∞ Moreover, if u is merely continuous, the conditions Mu ≥ 0 and M−u ≤ 0 in B2r could be taken in the viscosity sense (for more on viscosity solutions to nonlocal equations see [1], [2], [5], [15], [3] and [4]).”
CONTINUOUS DEPENDENCE ESTIMATES FOR VISCOSITY SOLUTIONS OF INTEGRO-PDES (2005, unknown): “In recent years we have witnessed an interest in extending viscosity solution theory to integro-PDEs [1, 2, 3, 6, 11, 12, 13, 35, 41, 43, 45, 46, 48, 49].”
A “maximum principle for semicontinuous functions” applicable to integro-partial differential equations (2006, background): “comparison/uniqueness results for (semicontinuous and unbounded) viscosity solutions are given in [1, 2, 3].”
About the Pricing Equations in Finance (2011, background): “[58], Amadori [2, 3] and Ma and Cvitanic [76], among others.”