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Nonlinear integro-differential evolution problems arising in option pricing: a viscosity solutions approach. (English) Zbl 1052.35083
The class of nonlinear integro-differential Cauchy problems \[ \begin{gathered} \partial_tu+ F(x,t,u,l{\mathcal I},Du, D^2u)= 0,\quad (x,t)\in\mathbb{R}^n\times (0, T],\\ u(x,0)= u_0(x),\quad x\in\mathbb{R}^N,\end{gathered} \] where the integral term \({\mathcal I}u\) is given by \[ {\mathcal I}u(x, t)= \int_{\mathbb{R}^N} M(u(x+z,t), u(x,t))\,d\mu_{x,t}(z) \] is studied by means of the viscosity solutions approach. In view of financial applications, the author is 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. The 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 PDEs approach. In particular, derivatives may be perfectly hedged in a completed market.

MSC:
35K55 Nonlinear parabolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35R10 Partial functional-differential equations
60J75 Jump processes (MSC2010)
91G80 Financial applications of other theories
45K05 Integro-partial differential equations
35K15 Initial value problems for second-order parabolic equations
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