  The purpose of this review paper is to present our recent results on
nonlinear and nonlocal mathematical models arising from modern financial
mathematics. It is based on our four papers written jointly by J. Cruz, M.
Grossinho, D. Sevcovic, and C. Udeani, as well as parts of PhD thesis by J.
Cruz. We investigated linear and nonlinear partial integro-differential
equations (PIDEs) arising from option pricing and portfolio selection problems
and studied the systematic relationships between the PIDEs with option pricing
theory and Black--Scholes models. First, we relax the liquid and complete
market assumptions and extend the models that study the market's illiquidity to
the case where the underlying asset price follows a L\'evy stochastic process
with jumps. Then, we establish the corresponding PIDE for option pricing under
suitable assumptions. The qualitative properties of solutions to nonlocal
linear and nonlinear PIDE are presented using the theory of abstract semilinear
parabolic equation in the scale of Bessel potential spaces. The existence and
uniqueness of solutions to the PIDE for a general class of the so-called
admissible L\'evy measures satisfying suitable growth conditions at infinity
and origin are also established in the multidimensional space.
