(Georgia Institute of Technology, 2023-07-24)
Viquez Bolanos, Jorge Aurelio Aurelio
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.
(Georgia Institute of Technology, 2021-02-15)
Li, Hangfan
Gap data problems are very popular recently, since scientists are more curious about what occurs during a period where information might be missing or unrecorded. Here, a nonparametric method called Imputed Empirical estimating (IEE) method will be illustrated. Moreover, using IEE into the medical field to estimate T_1, which is the first recovery time after an acute myocardial infraction will be discussed as well. Simulation studies are shown to assess the accuracy of the IEE estimate and demonstrate that the IEE method outperformed all other algorithms. An IEE estimate of the survival function based on the real-life data will also be provided to show the real-world application. Mathematical proofs will be provided if applicable.