title: 'On solutions of a partial integro-differential equation in Bessel potential spaces with applications in option pricing models'
---

Partial integro-differential equation, sectorial operator, analytic semigroup, Bessel potential space, option pricing under Lévy stochastic process, Lévy measure

Introduction
============

In this paper, we analyze solutions to the semilinear parabolic partial integro-differential equation (PIDE) of the form: $$\begin{aligned}
\frac{\partial u}{\partial \tau}(\tau,x) &=& \frac{\sigma^2}{2}\frac{\partial^2 u}{\partial x^2}(\tau,x)  + \omega \frac{\partial u}{\partial x}(\tau,x) + g(\tau, u(\tau,x))
\nonumber
\\
&&  + \int_{\mathbb{R}}\left[ u(\tau, x+z)-u(\tau, x)- (e^z-1) \frac{\partial u}{\partial x}(\tau, x)\, \right] \nu({\mathrm{d}}z),
\label{PDE-u}
\\
&&u(0,x)=u_0(x),
\nonumber\end{aligned}$$ $x\in \mathbb{R}, \tau\in(0,T)$, where $g$ is Hölder continuous in the $\tau$ variable and it is Lipschitz continuous in the $u$ variable. Here $\nu $ is a positive Radon measure on $\mathbb{R}$ such that $\int_{\mathbb{R}} \min(z^2,1) \nu ({\mathrm{d}}z)<\infty$.

Our purpose is to prove existence and uniqueness of a solution to (\[PDE-u\]) in the framework of Bessel potential spaces. These functional spaces represent a nested scale $\{X^\gamma\}_{\gamma\ge0}$ of Banach spaces such that $$X^1\equiv D(A) \hookrightarrow X^{\gamma_1} \hookrightarrow X^{\gamma_2} \hookrightarrow X^0\equiv X,$$ for any $0\le \gamma_2\le \gamma_1\le 1$ where $A$ is a sectorial operator in the Banach space $X$ with a dense domain $D(A)\subset X$. For example, if $A=-\Delta$ is the Laplacian 