arXiv subjects

Erik J. Baurdoux

Publications and source records attributed to Erik J. Baurdoux.

8 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↗

arXiv2021

Predicting the Last Zero before an exponential time of a Spectrally Negative Lévy Process

Given a spectrally negative Lévy process, we predict, in a $L_1$ sense, the last passage time of the process below zero before an independent exponential time. This optimal prediction problem generalises Baurdoux and Pedraza (2020) where the infinite horizon problem is solved. Using a similar argument as that in Urusov (2005), we show that this optimal prediction problem is equivalent to solving an optimal prediction problem in a finite horizon setting. Surprisingly (unlike the infinite horizon problem) an optimal stopping time is based on a curve that is killed at the moment the mean of the exponential time is reached. That is, an optimal stopping time is the first time the process crosses above a non-negative, continuous and non-increasing curve depending on time. This curve and the value function are characterised as a solution of a system of non-linear integral equations which can be understood as a generalisation of the free boundary equations (see e.g. Peskir and Shiryaev (2006) Chapter IV.14.1) in the presence of jumps. As an example, we calculate numerically such curve in the Brownian motion case and a compound Poisson process with exponential sized jumps perturbed by a Brownian motion.

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↗

arXiv2015

Optimality of doubly reflected Levy processes in singular control

We consider a class of two-sided singular control problems. A controller either increases or decreases a given spectrally negative Levy process so as to minimize the total costs comprising of the running and control costs where the latter is proportional to the size of control. We provide a sufficient condition for the optimality of a double barrier strategy, and in particular show that it holds when the running cost function is convex. Using the fluctuation theory of doubly reflected Levy processes, we express concisely the optimal strategy as well as the value function using the scale function. Numerical examples are provided to confirm the analytical results.

math.OC↗

arXiv2014

Optimal double stopping of a Brownian bridge

We study optimal double stopping problems driven by a Brownian bridge. The objective is to maximize the expected spread between the payoffs achieved at the two stopping times. We study several cases where the solutions can be solved explicitly by strategies of threshold type.

math.OC↗

arXiv2013

A direct method for solving optimal stopping problems for Lévy processes

We propose an alternative approach for solving a number of well-studied optimal stopping problems for Lévy processes. Instead of the usual method of guess-and-verify based on martingale properties of the value function, we suggest a more direct method by showing that the general theory of optimal stopping for strong Markov processes together with some elementary observations imply that the stopping set must be of a certain form for the optimal stopping problems we consider. The independence of increments and the strong Markov property of Lévy processes then allow us to use straightforward optimisation over a real-valued parameter to determine this stopping set. We illustrate this approach by applying it to the McKean optimal stopping problem (American put), the Novikov--Shiryaev optimal stopping problem and the Shepp--Shiryaev optimal stopping problem (Russian option).

math.PR↗

arXiv2010

Further calculations for the McKean stochastic game for a spectrally negative Levy process: from a point to an interval

Following Baurdoux and Kyprianou [2] we consider the McKean stochastic game, a game version of the McKean optimal stopping problem (American put), driven by a spectrally negative Levy process. We improve their characterisation of a saddle point for this game when the driving process has a Gaussian component and negative jumps. In particular we show that the exercise region of the minimiser consists of a singleton when the penalty parameter is larger than some threshold and 'thickens' to a full interval when the penalty parameter drops below this threshold. Expressions in terms of scale functions for the general case and in terms of polynomials for a specific jump-diffusion case are provided.

math.PR↗