in pricing American style of options.

In order to prove existence, continuation and uniqueness of a solution to the problem (\[problem\_transformed\]) we follow the qualitative theory of semilinear abstract parabolic equations developed by Henry in [@Henry1981]. First, we recall the concept of an analytic semigroup of linear operators and a sectorial operator in a Banach space.

[@Henry1981] A family of bounded linear operators $\left\{S(t), t\geq 0\right\}$ in a Banach space $X$ is called an analytic semigroup if it satisfies the following conditions:

-   $S(0)=I, S(t)S(s)=S(s)S(t)=S(t+s)$, for all $t,s\geq 0$;

-   $S(t) u\rightarrow u$ when $t\rightarrow 0^{+}$ for all $u\in X$;

-   $t\rightarrow S(t) u$ is a real analytic function on $0< t< \infty$ for each $u\in X$.

The associated infinitesimal generator $A$ is defined as follows: $A u = \lim_{t\rightarrow 0^{+}} \frac{1}{t} (S(t)u-u)$ and its domain $D(A)\subseteq X$ consists of those elements $u\in X$ for which the limit exists in the space $X$.

[@Henry1981] Let $S_{a,\phi}=\left\{\lambda \in \mathbb{C}: \phi\leq \arg(\lambda-a)\leq 2\pi-\phi \right\}$ be a sector of complex numbers. A closed densely defined linear operator $A: D(A)\subset X \rightarrow X$ is called a sectorial operator if there exists a constant $M\geq 0$ such that $\Vert(A-\lambda )^{-1}\Vert\leq M/|\lambda -a|$ for all $\lambda\in 