The Annals of Probability

An infinite-dimensional approach to path-dependent Kolmogorov equations

Franco Flandoli and Giovanni Zanco

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Abstract

In this paper, a Banach space framework is introduced in order to deal with finite-dimensional path-dependent stochastic differential equations. A version of Kolmogorov backward equation is formulated and solved both in the space of $L^{p}$ paths and in the space of continuous paths using the associated stochastic differential equation, thus establishing a relation between path-dependent SDEs and PDEs in analogy with the classical case. Finally, it is shown how to establish a connection between such Kolmogorov equation and the analogue finite-dimensional equation that can be formulated in terms of the path-dependent derivatives recently introduced by Dupire, Cont and Fournié.

Article information

Source
Ann. Probab., Volume 44, Number 4 (2016), 2643-2693.

Dates
Received: December 2013
Revised: April 2015
First available in Project Euclid: 2 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1470139151

Digital Object Identifier
doi:10.1214/15-AOP1031

Mathematical Reviews number (MathSciNet)
MR3531677

Zentralblatt MATH identifier
1356.60101

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.) 35C99: None of the above, but in this section 35K99: None of the above, but in this section

Keywords
Path-dependent SDEs path-dependent PDEs delay equations stochastic calculus in Banach spaces Kolmogorov equations

Citation

Flandoli, Franco; Zanco, Giovanni. An infinite-dimensional approach to path-dependent Kolmogorov equations. Ann. Probab. 44 (2016), no. 4, 2643--2693. doi:10.1214/15-AOP1031. https://projecteuclid.org/euclid.aop/1470139151


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