First, we revisit basic theory of functional Itô/path-dependent calculus, using the formulation of calculus via regularization. Relations with the corresponding Banach space valued calculus are explored. The second part of the paper is devoted to the study of the Kolmogorov type equation associated with the so called window Brownian motion, called path-dependent heat equation, for which well-posedness at the level of strict solutions is established. Then, a notion of strong approximating solution, called strong-viscosity solution, is introduced which is supposed to be a substitution tool to the viscosity solution. For that kind of solution, we also prove existence and uniqueness.

Functional and Banach Space Stochastic Calculi: Path-Dependent Kolmogorov Equations Associated with the Frame of a Brownian Motion / A. Cosso, F. Russo (SPRINGER PROCEEDINGS IN MATHEMATICS & STATISTICS). - In: Stochastics of Environmental and Financial Economics / [a cura di] F. Espen Benth, G. Di Nunno. - [s.l] : Springer, 2016. - ISBN 978-3-319-23424-3. - pp. 27-80 (( convegno Conference on Stochastics for Environmental and Financial Economics tenutosi a Oslo nel 2014 [10.1007/978-3-319-23425-0_2].

### Functional and Banach Space Stochastic Calculi: Path-Dependent Kolmogorov Equations Associated with the Frame of a Brownian Motion

#### Abstract

First, we revisit basic theory of functional Itô/path-dependent calculus, using the formulation of calculus via regularization. Relations with the corresponding Banach space valued calculus are explored. The second part of the paper is devoted to the study of the Kolmogorov type equation associated with the so called window Brownian motion, called path-dependent heat equation, for which well-posedness at the level of strict solutions is established. Then, a notion of strong approximating solution, called strong-viscosity solution, is introduced which is supposed to be a substitution tool to the viscosity solution. For that kind of solution, we also prove existence and uniqueness.
##### Scheda breve Scheda completa Scheda completa (DC) Horizontal and vertical derivative; functional Itô/path-dependent calculus; Banach space stochastic calculus; Strong-viscosity solutions; Calculus via regularization
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/2434/931979`
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