- Series
- Dissertation Defense
- Time
- Thursday, June 22, 2023 - 3:00pm for 1 hour (actually 50 minutes)
- Location
- Skiles 006/Zoom
- Speaker
- Jorge Aurelio Víquez Bolaños – Georgia Tech – jviquez6@gatech.edu – https://jviquez6.math.gatech.edu/
- Organizer
- Jorge Viquez
Zoom link. Meeting ID: 914 2801 6313, Passcode: 501018
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 Ito 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 strongly path-dependent functionals.
Committee
- Prof. Christian Houdré - School of Mathematics, Georgia Tech (advisor)
- Prof. Michael Damron - School of Mathematics, Georgia Tech
- Prof. Rachel Kuske - School of Mathematics, Georgia Tech
- Prof. Andrzej Święch - School of Mathematics, Georgia Tech
- Prof. José Figueroa-López - Department of Mathematics and Statistics, Washington University in St. Louis
- Prof. Bruno Dupire - Department of Mathematics, New York University