Comparison Principles for Stochastic Volterra Equations

Periodic seminar of the Department of mathematics
8 November 2023
Start time 
5:00 pm
PovoZero - Via Sommarive 14, Povo (Trento)
Seminar Room "-1" (Povo 0) and Zoom (please contact to get the code)
Dipartimento di Matematica
Target audience: 
University community
UniTrento students
Contact person: 
Prof. Luigi Amedeo Bianchi, Prof. Stefano Bonaccorsi, Prof. Michele Coghi
Contact details: 
Università degli Studi Trento 38123 Povo (TN) - Staff Dipartimento di Matematica
+39 04 61/281508-1625-1701-3898-1980-1511
Ole Cañadas (Dublin City University)


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).