arXiv subjects

Mohammed Louriki

Publications and source records attributed to Mohammed Louriki.

10 recordsLinked to original sources

arXiv2026

Diffusion bridge with randomized initial and terminal times and its application to fish migration

We mathematically model the dynamics of the number of migratory fish observed at a fixed location along a river in a random environment. Particularly, as a new approach, we construct a stochastic differential equation that incorporates the influence of environmental factors on the fluctuations in the start and end of migration. The model is a diffusion bridge with a non-Lipschitz diffusion coefficient, called the Cox-Ingersoll-Ross bridge, and has random initial and terminal times arising from time-change, so that the influences of environmental factors can be efficiently incorporated. The well-posedness of the model is first established, which is considered novel and significant in applied mathematics. Second, we estimate the parameters of the model based on the latest multiyear daily data set for the upstream migration of Plecoglossus altivelis altivelis (Ayu) by relying on the hypothesis that water temperature affects the migration of the fish, which has been suggested in existing studies. We also explore the application of the proposed model to the challenging task of analyzing environmental DNA data. This study advances the development of a theory of fish migration that is simple yet can take environmental factors into account.

q-bio.PE↗

arXiv2026

Stopping Times in the Filtration of a Brownian Motion Stopped at its Last Passage Time

We investigate the structural properties of the last passage time $σ_z^λ$ at level $z > 0$ of a Brownian motion with positive drift $λ> 0$, denoted $B^λ = (B_t + λt)_{t \geq 0}$, in the filtration generated by the process $ξ^{λ,z} = (B^λ_{t \wedge σ_z^λ})_{t \geq 0}$. We compute the compensator of $σ_z^λ$ and establish that it is the unique totally inaccessible stopping time in the filtration of $ξ^{λ,z}$. Moreover, we provide a canonical decomposition of arbitrary stopping times: for any stopping time $T$, the restriction of $T$ to the set $\{T = σ_z^λ\}$ is totally inaccessible, while its restriction to $\{T \neq σ_z^λ\}$ is predictable. Although the paths of $ξ^{λ,z}$ are continuous, the process fails to satisfy the Feller property and is not strong Markov. Nevertheless, we show that its natural filtration is quasi-left-continuous. To overcome these limitations, we consider the extended process $ζ^{λ,z} = (\mathbb{I}_{\{t < σ_z^λ\}}, ξ_t^{λ,z})_{t \geq 0}$, and prove that it is a Feller process. We compute its infinitesimal generator, which allows us to characterize the associated class of martingales and identify the solutions to certain partial differential equations.

math.PR↗

arXiv2025

McKean-Vlasov processes of bridge type

In this paper, we introduce and study McKean-Vlasov processes of bridge type. Specifically, we examine a stochastic differential equation (SDE) of the form: $$\mathrm{d} ξ_t=-μ(t,\mathbb{E}[φ_1(ξ_t)]) \frac{ξ_t}{T-t} \mathrm{d} t+σ(t,\mathbb{E}[φ_2(ξ_t)]) \mathrm{d} W_t,\,\, t<T,$$ where $μ$ and $σ$ are deterministic functions that depend on time $t$ and the expectation of given functions $φ_1$ and $φ_2$ of the process, and $W$ is a Brownian motion. We establish the existence and uniqueness of solutions to this equation and analyze the behavior of the process as $t$ approaches $T$. Furthermore, we provide conditions ensuring the pinned property of the process $ξ$. Finally, we explore explicit solutions in specific cases of interest, including power-weighted expectations and second moments in the drift.

math.PR↗

arXiv2024

The Impact of Pinning Points on Memorylessness in Lévy Random Bridges

Random Bridges have gained significant attention in recent years due to their potential applications in various areas, particularly in information-based asset pricing models. This paper aims to explore the potential influence of the pinning point's distribution on the memorylessness and stochastic dynamics of the bridge process. We introduce Lévy bridges with random length and random pinning points and analyze their Markov property. Our study demonstrates that the Markov property of Lévy bridges depends on the nature of the distribution of their pinning points. The law of any random variables can be decomposed into singular continuous, discrete, and absolutely continuous parts with respect to the Lebesgue measure (Lebesgue's decomposition theorem). We show that the Markov property holds when the pinning points' law does not have an absolutely continuous part. Conversely, the Lévy bridge fails to exhibit Markovian behavior when the pinning point has an absolutely continuous part.

math.PR↗

arXiv2024

Information-Based Approach: Pricing of a Credit Risky Asset in the Presence of Default Time

We extend the information-based asset-pricing framework by Brody, Hughston \& Macrina to incorporate a stochastic bankruptcy time for the writer of the asset. Our model introduces a non-defaultable cash flow $Z_T$ to be made at time $T$, alongside the time $τ$ of a possible bankruptcy of the writer of the asset are in line with the filtration generated by a Brownian random bridge with length $ν=τ\wedge T$ and pinning point $σZ_T$, where $σ$ is a constant. Quantities $Z_T$ and $τ$ are not necessarily independent. The model does not depend crucially on the interpretation of $τ$ as a bankruptcy time. We derived the price process of the asset and compute the prices of associated options. The dynamics of the price process satisfy a diffusion equation. Employing the approach of P.-A.~ Meyer, we provide the explicit computation of the compensator of $ν$. Leveraging special properties of the bridge process, we also provide the explicit expression of the compensator of $Z_T\,\mathbb{I}_{[ν,+\infty)}$. The resulting conclusion highlights the totally inaccessible property of the stopping time $ν$. This characteristic is particularly suitable for financial markets where the time of default of a writer cannot be predictable from any other signal in the system until default happens.

math.PR↗

arXiv2021

Brownian bridge with random length and pinning point for modelling of financial information

In this paper, we introduce an extension of a Brownian bridge with a random length by including uncertainty also in the pinning level of the bridge. The main result of this work is that unlike for deterministic pinning point, the bridge process fails to be Markovian if the pining point distribution is absolutely continuous with respect to the Lebesgue measure. Further results include the derivation of formulae to calculate the conditional expectation of various functions of the random pinning time, the random pinning location, and the future value of the Brownian bridge, given an observation of the underlying process. For the specific case that the pining point has a two-point distribution, we state further properties of the Brownian bridge, e.g., the right continuity of its natural filtration and its semi-martingale decomposition. The newly introduced process can be used to model the flow of information about the behaviour of a gas storage contract holder; concerning whether to inject or withdraw gas at some random future time.

math.PR↗

arXiv2021

Bridge-Type Processes Associated with L\'evy Processes and Their Decompositions

We study a class of stochastic bridge-type processes whose terminal pinning value is random and is generated by an underlying stochastic process. In contrast with classical bridges, the construction depends not only on the terminal value of the driving process but also on its evolution before the terminal time. This dynamic stochastic input breaks some of the classical Markovian structure and requires a separate analysis of the semimartingale decomposition in the natural filtration. We first analyze the Brownian case, which provides a Gaussian reference model, and show that the corresponding process is not Markovian in its natural filtration. We then extend the study to non-Gaussian L\'evy drivers, focusing on finite variation jump processes and on L\'evy processes with both Gaussian and jump components. In each case, we study the Doob--Meyer decomposition in the natural filtration.

math.PR↗

arXiv2019

Lévy bridges with random length

In this paper our first goal is to give precise definition of the Lévy bridges with random length. Our second task is to establish the Markov property of this process with respect to its completed natural filtration and thus with respect to the usual augmentation of this one. This property will be crucial for the right-continuity of completed natural filtration.

math.PR↗

arXiv2018

Bridges with random length: Gamma case

The aim objective of this paper is to show that certain basic properties of gamma bridges with deterministic length stay true also for gamma bridges with random length. Among them the Markov property as well as the canonical decomposition with respect to the usual augmentation of its natural filtration, which leads us to conclude that its completed natural filtration is right continuous.

math.PR↗

arXiv2017

Bridges with random length: Gaussian-Markovian case

Motivated by the Brownian bridge on random interval considered by Bedini et al \cite{BBE}, we introduce and study Gaussian bridges with random length with special emphasis to the Markov property. We prove that if the starting process is Markov then this property was kept by the bridge with respect to the usual augmentation of its natural filtration. This leads us to conclude that the completed natural filtration of the bridge satisfies the usual conditions of right-continuity and completeness.

math.PR↗