• Stochastic processes
• Volterra process

The main purpose of this thesis is the study of some classes of Volterra processes with jumps, and in particular we will focus on equations of the form

dXt = K * b(Xt)dt + K * \sigma (Xt)dW_t + K * \gamma(Xt)dN_t
X0 = x0

where b, \sigma, and \gamma are continuous coefficients with conditions to be discussed later, K is a convolution kernel with regularity condition to be discussed in Chapter 1, W_t is a Brownian motion and N_t is a compound Poisson process with jumps of positive heights.
Our interest comes from application in financial modeling. Classical affine models constitute the most popular framework for building tractable models in finance. However a growing body of empirical research indicates that volatility fluctuates more rapidly than Brownian motion, which is inconsistent with standard semimartingales models. For this purpose fractional adaptations of models such as the one above have emerged.
So far mainly diffusion-type models have been considered. We focus on the introduction of jumps. This will allow us to gain more flexibility in modeling.
Our starting point, in Chapter 1, is the article Affine Volterra Processes. Here the authors consider an equation of the form dXt = K * b(Xt)dt + K *\sigma(Xt)dWt
with b an affine continuous coefficient and \sigma continuous. In this first chapter we will study the conditions under which there exists a unique solution. Being interested in considering the solution of the first equation as the volatility of a process, we will then focus on the question of when this solution is R+-valued and in turn also \mathbb{R}_d^+ valued. At the end of this Chapter we will also give some examples of affine Volterra processes and study some properties of those models.
In Chapter 2 we will consider a class of Volterra processes with jumps. We will first introduce Hawkes processes, and then we will prove the existence of a solution of an equation of the form
dXt = K * b(Xt)dt + K * \gamma(Xt)d_Nt
Where e N_t is a compensated compound Poisson process. To this end we will
first follow the work of Protter. Noticing that the condition in this
article are not satisfied by our convolution Kernel K we will then move to
the idea presented in another article. Exploiting stochastic calculus for Poisson processes we will prove the existence of a strong solution to the previous equation under Lipschitz conditions.
In Chapter 3 we will finally study the first Equation. We will prove the existence of a solution as a Corollary of what we have proved in the previous chapters. We will then discuss when the solution are \mathbb{R}_d^+ valued, finding a condition on the coefficients.