arXiv subjects

J. M. Pedraza

Publications and source records attributed to J. M. Pedraza.

3 recordsLinked to original sources

arXiv2025

On the last zero process with an application in corporate bankruptcy

For a spectrally negative Lévy process $X$, consider $g_t$, the last time $X$ is below the level zero before time $t\geq 0$. We use a perturbation method for Lévy processes to derive an Itô formula for the three-dimensional process $\{(g_t,t, X_t), t\geq 0 \}$ and its infinitesimal generator. Moreover, with $U_t:=t-g_t$, the length of a current positive excursion, we derive a general formula that allows us to calculate a functional of the whole path of $ (U, X)=\{(U_t, X_t),t\geq 0\}$ in terms of the positive and negative excursions of the process $X$. As a corollary, we find the joint Laplace transform of $(U_{\mathbf{e}_q}, X_{\mathbf{e}_q})$, where $\mathbf{e}_q$ is an independent exponential time, and the q-potential measure of the process $(U, X)$. Furthermore, using the results mentioned above, we find a solution to a general optimal stopping problem depending on $(U, X)$ with an application in corporate bankruptcy. Lastly, we establish a link between the optimal prediction of $g_{\infty}$ and optimal stopping problems in terms of $(U, X)$ as per Baurdoux and Pedraza (2024).

math.PR↗

arXiv2023

$L_p$ optimal prediction of the last zero of a spectrally negative Lévy process

Given a spectrally negative Lévy process $X$ drifting to infinity, (inspired on the early ideas of Shiryaev (2002)) we are interested in finding a stopping time that minimises the $L^p$ distance ($p>1$) with $g$, the last time $X$ is negative. The solution is substantially more difficult compared to the case $p=1$, for which it was shown in Baurdoux and Pedraza (2020b) that it is optimal to stop as soon as $X$ exceeds a constant barrier. In the case of $p>1$ treated here, we prove that solving this optimal prediction problem is equivalent to solving an optimal stopping problem in terms of a two-dimensional strong Markov process that incorporates the length of the current excursion away from $0$. We show that an optimal stopping time is now given by the first time that $X$ exceeds a non-increasing and non-negative curve depending on the length of the current excursion away from $0$. We further characterise the optimal boundary and the value function as the unique solution of a non-linear system of integral equations within a subclass of functions. As examples, the case of a Brownian motion with drift and a Brownian motion with drift perturbed by a Poisson process with exponential jumps are considered.

math.PR↗

arXiv2018

Predicting the last zero of a spectrally negative Lévy process

Last passage times arise in a number of areas of applied probability, including risk theory and degradation models. Such times are obviously not stopping times since they depend on the whole path of the underlying process. We consider the problem of finding a stopping time that minimises the $L^1$-distance to the last time a spectrally negative Lévy process $X$ is below zero. Examples of related problems in a finite horizon setting for processes with continuous paths are Du Toit et al. (2008) and Glover and Hulley (2014), where the last zero is predicted for a Brownian motion with drift, and for a transient diffusion, respectively. As we consider the infinite horizon setting, the problem is interesting only when the Lévy process drifts to $\infty$ which we will assume throughout. Existing results allow us to rewrite the problem as a classic optimal stopping problem, i.e. with an adapted payoff process. We use a direct method to show that an optimal stopping time is given by the first passage time above a level defined in terms of the median of the convolution with itself of the distribution function of $-\inf_{t\geq 0}X_t$. We also characterise when continuous and/or smooth fit holds.

math.PR↗