Comparison Principles for Stochastic Volterra Equations

Abstract:

Motivated by rough volatility models in mathematical finance, stochastic
Volterra equations (SVEs)
\[
X_t = g(t) + \int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}b(X_s)\,d s
+ \int_0^t \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\sigma(X^{i}_s)\,d B_s
\]
with $\alpha\in(1/2,1)$ and $\sigma$ H\"older continuous of order
$\eta\in[1/2,1]$ have received a great deal of study in recent years.
However, they fall neither into the class of semimartingales nor into
the class of Markov processes. Therefore, the classical framework as the
It\^o formula and classical semigroup theory is not applicable. In this
talk, we are concerned with Comparison Principles for stochastic
Volterra equations. That is, suppose
\[
X^{i}_t = g_i(t) + \int_0^t K(t-s)b_i(X^{i}_s)\,d s + \int_0^t
K(t-s)\sigma(X^{i}_s)\,d B_s,\quad i=1,2,
\]
with $g_1\leq g_2$ then $\mathbb{P}(X^1_t\leq X^2_t,\,t\geq0)=1.$
However, in general, SVEs with non-Lipschitz diffusion coefficients
suffer from the lack of pathwise uniqueness and a desired comparison
principle is out of reach (since comparison principles imply pathwise
uniqueness). To address this problem, we construct a coupling satisfying
a comparison principle. We prove this under different assumptions on the
initial data $(g_i,K,b_i,\sigma)$ and discuss an example.
This is joint work with Martin Friesen (Dublin City University).