A convergent difference scheme for a class of partial integro-differential equations modeling pricing under uncertainty (Q2796853)

In this paper, the authors present a finite difference scheme to approximate viscosity solutions of a class of partial integro-differential equations of the form NEWLINENEWLINE\[NEWLINE \partial_{t}u-\sup_{(b,q,\upsilon)\in\Theta}A_{(b,q,\upsilon)}u=0, \quad u(0)=u_{0},\tag{*} NEWLINE\]NEWLINE NEWLINEwhere the operator \( A \) is given by NEWLINENEWLINE\[NEWLINE A_{(b,q,\upsilon)}u=b\partial_{x}u+\frac{1}{2}q^{2}\partial_{xx}u+\int_{R}(u(x+z)-u(x)-z\partial_{x}u(x))\upsilon(dz) NEWLINE\]NEWLINE NEWLINEwith some set \( \Theta \subseteq \mathbb{R}\times \mathbb{R}\times \Im \), where \( \Im \) denotes the set of Levy measures, that is, NEWLINENEWLINE\[NEWLINE \Im =\left\{\upsilon\text{ measures on } \mathbb{R}: \int_{R}1\wedge|z|^{2}\upsilon(dz)<\infty\right\}. NEWLINE\]NEWLINE NEWLINEIt may be noted that (*) describes a pricing model under uncertainty. NEWLINEIn this article, the authors focus on the special case of uncertainty in the Levy measure. It is shown that the approximations converge to the unique viscosity solution as the discretization tends to zero. An asymptotic rate of the convergence is discussed. Several numerical examples are given to illustrate the results obtained.