Functional Itô versus Banach space stochastic calculus and strict solutions of semilinear path-dependent equations

2016
Publication type:
Paper in peer-reviewed journals
Journal:
Infinite Dimensional Analysis, Quantum Probability and Related Topics, vol. 19 (4)
Keywords :
Functional Itô calculus; Banach space valued stochastic calculus; Path-dependent partial differential equation; strict solutions; calculus via regularization.
Abstract:
Functional Itô calculus was introduced in order to expand a functional $F(t, X_{\cdot+t}, X_t)$ depending on time $t$, past and present values of the process $X$. Another possibility to expand $F(t, X_{\cdot+t}, X_t)$ consists in considering the path $X_{\cdot+t}=\{X_{x+t},\,x\in[-T,0]\}$ as an element of the Banach space of continuous functions on $C([-T,0])$ and to use Banach space stochastic calculus. The aim of this paper is threefold. 1) To reformulate functional Itô calculus, separating time and past, making use of the regularization procedures which matches more naturally the notion of horizontal derivative which is one of the tools of that calculus. 2) To exploit this reformulation in order to discuss the (not obvious) relation between the functional and the Banach space approaches. 3) To study existence and uniqueness of smooth solutions to path-dependent partial differential equations which naturally arise in the study of functional Itô calculus. More precisely, we study a path-dependent equation of Kolmogorov type which is related to the window process of the solution to an It\^o stochastic differential equation with path-dependent coefficients. We also study a semilinear version of that equation.
BibTeX:
@article{Cos-Rus-2016,
    author={Andrea Cosso and Francesco Russo },
    title={Functional Itô versus Banach space stochastic calculus and 
           strict solutions of semilinear path-dependent equations },
    doi={10.1142/S0219025716500247 },
    journal={Infinite Dimensional Analysis, Quantum Probability and Related 
           Topics },
    year={2016 },
    volume={19 (4) },
}