Title:
Functional Itô Calculus for Lévy Processes (With a View Towards Mathematical Finance)
Functional Itô Calculus for Lévy Processes (With a View Towards Mathematical Finance)
Author(s)
Viquez Bolanos, Jorge Aurelio Aurelio
Advisor(s)
Houdré, Christian
Editor(s)
Collections
Supplementary to
Permanent Link
Abstract
We examine the relationship between Dupire’s functional derivative and a
variant of the functional derivative developed by Kim for analyzing functionals in systems with delay. Our findings demonstrate that if Dupire’s space derivatives exist, differentiability in any continuous functional direction implies differentiability in any other direction, including the constant one.
Additionally, we establish that co-invariant differentiable functionals can lead to a functional Itô formula in the Cont and Fournié path-wise setting under the right regularity conditions.
Next, our attention turns to functional extensions of the Meyer-Tanaka formula and the efforts made to characterize the zero-energy term for integral representations of functionals of semimartingales. Using Eisenbaum’s idea for
reversible semimartingales, we obtain an optimal integration formula for Lévy
processes, which avoids imposing additional regularity requirements on the functional’s space derivative and extends other approaches using the stationary and martingale properties of Lévy processes.
Finally, we address the topic of integral representations for the Delta of
a path-dependent pay-off, which generalizes Benth, Di Nunno, and Khedher’s framework for the approximation of functionals of jump-diffusions to cases where they may be driven by a process satisfying a path-dependent differential equation. Our results extend Jazaerli and Saporito’s formula for the Delta of functionals to the jump-diffusion case. We propose an adjoint formula for the horizontal derivative, hoping to obtain more tractable formulas for the Delta of value options with strongly path-dependent pay-offs.
Sponsor
Date Issued
2023-07-24
Extent
Resource Type
Text
Resource Subtype
Dissertation