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Gerardo Hernandez-del-Valle

Publications and source records attributed to Gerardo Hernandez-del-Valle.

7 recordsLinked to original sources
arXiv2026

A Recursive Representation of Compatibility Matrices for Transported Differential Operators

We develop a recursive algebraic framework for the compatibility matrices arising in moving-boundary problems for the heat equation. Such matrices naturally appear in the analysis of boundary trace operators, including Dirichlet-to-Neumann maps and first-passage problems for Brownian motion. The construction is based on two recursive hierarchies: transported differential operators in the interior and boundary operators generated by successive differentiation of the boundary condition. Their interaction produces a recursive family of compatibility matrices whose size grows linearly with the order of the transported operator. The main contribution of the paper is a decomposition of every compatibility matrix into a universal reference matrix and a sequence of transport perturbations. This representation isolates the boundary geometry from the recursive transport corrections and shows that only the transport rows change at each recursive level. Consequently, the determinant of the compatibility matrix may be interpreted as the determinant of a reference matrix together with successive low-rank perturbations. The recursive construction is illustrated through explicit examples, which suggest additional algebraic structure in the reference determinants. The resulting recursive representation provides a systematic organization of compatibility matrices and establishes an algebraic foundation for future investigations of recursive determinant formulas, boundary trace operators, and Dirichlet-to-Neumann maps for moving-boundary problems.

math.AP↗
arXiv2016

A solution to the heat equation with a cubic moving boundary

In this work we find a solution to problem of the heat equation which is annihiliated at a cubic boundary $f$. The solution turns out to be the convolution between the fundamental solution of the heat equation and a function $ϕ$ which solves a third order ODE. However we believe that the main contribution is the procedure itself, which links in a rather straightforward way, solutions of the heat equation $v$ with moving boundaries $f$ through the convolution of the heat kernel with $\mathbb{C}^p$ funtions $ϕ$.

math.GM↗
arXiv2012

On the first time that an Ito process hits a barrier

This work deals with first hitting time densities of Ito processes whose local drift can be modeled in terms of a solution to Burgers equation. In particular, we derive the densities of the first time that these processes reach a moving boundary. We distinguish two cases: (a) the case in which the process has unbounded state space before absorption, and (b) the case in which the process has bounded state space before absorption. The reason as to why this distinction has to be made will be clarified. Next, we classify processes whose local drift can be expressed as a linear combination to solutions of Burgers equation. For example the local drift of a Bessel process of order 5 can be modeled as the sum of two solutions to Burgers equation and thus will be classified as of class $\mathcal{B}^2$. Alternatively, the Bessel process of order 3 has a local drift that can be modeled as a solution to Burgers equation and thus will be classified as of class $\mathcal{B}^1$. Examples of diffusions within class $\mathcal{B}^1$, and hence those to which the results described within apply, are: Brownian motion with linear drfit, the 3D Bessel process, the 3D Bessel bridge, and the Brownian bridge.

math.PR↗
arXiv2009

Optimal execution of Portfolio transactions with geometric price process

In this paper we derive the optimal execution trajectory for a trader who wishes to buy or sell a large position of shares which evolve as a geometric Brownian process in contrast to the arithmetic model which prevails in the existing literature, and with a general temporary impact $h$. We provide a couple of examples which illustrate the results. We would like to stress the fact that in this paper we use understandable user-friendly techniques.

q-fin.PM↗
arXiv2009

On Schroedinger's equation, 3-dimensional bessel bridges, and passage time problems

We obtain explicit solutions for the density $φ_T$ of the first-time $T$ that a one-dimensional Brownian process $B$ reaches the twice, continuously differentiable moving boundary $f$ and such that $f''(t)\geq 0$ for all $t\in \mathbb{R}^+$. We do so by finding the expected value of some functionals of a 3-dimensional Bessel bridge $\tilde{X}$ and exploiting its relationship with first-passage time problems as pointed out by Kardaras (2007). It turns out that this problem is related to Schrödinger's equation with time-dependent linear potential, see Feng (2001).

math.PR↗
arXiv2009

On the first passage time density of a continuous Martingale over a moving boundary

In this paper we derive the density $φ$ of the first time $T$ that a continuous martingale $M$ with non-random quadratic variation $ _\cdot:=\int_0^\cdot h^2(u)du$ hits a moving boundary $f$ which is twice continuously differentiable, and $f'/h\in\mathbb{C}^2[0,\infty)$. Thus, this work is an extension to case in which $M$ is in fact a one-dimensional standard Brownian motion $B$, as studied in Hernandez-del-Valle (2007).

math.PR↗