Functional Itô calculus was introduced in order to expand a functional F(t,X.+t,Xt) depending on time t, past and present values of the process X. Another possibility to expand F(t,X.+t,Xt) consists in considering the path X.+t = {Xx+t, x ∈ [-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 match 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 Itô stochastic differential equation with path-dependent coefficients. We also study a semilinear version of that equation.

Functional Itô versus Banach space stochastic calculus and strict solutions of semilinear path-dependent equations / A. Cosso, F. Russo. - In: INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS. - ISSN 0219-0257. - 19:4(2016 Dec), pp. 1650024.1-1650024.44. [10.1142/S0219025716500247]

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

A. Cosso
Primo
;
2016-12

Abstract

Functional Itô calculus was introduced in order to expand a functional F(t,X.+t,Xt) depending on time t, past and present values of the process X. Another possibility to expand F(t,X.+t,Xt) consists in considering the path X.+t = {Xx+t, x ∈ [-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 match 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 Itô stochastic differential equation with path-dependent coefficients. We also study a semilinear version of that equation.
Banach space valued stochastic calculus; calculus via regularization; functional Itô calculus; path-dependent partial differential equation; strict solutions; statistical and nonlinear physics; statistics and probability; mathematical physics; applied mathematics
Settore MAT/06 - Probabilita' e Statistica Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/931968
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