arXiv subjects

Antoine-Marie Bogso

Publications and source records attributed to Antoine-Marie Bogso.

10 recordsLinked to original sources

arXiv2026

Stochastic Optimal Control for Jump Diffusion Models with Singular Drifts

We study a stochastic optimal control problem for jump-diffusion systems whose drift coefficient is piecewise Lipschitz continuous and exhibits threshold-induced discontinuities. Such dynamics naturally arise in applications with intervention policies triggered by safety levels, notably in insurance surplus management with dividend payments and capital injections. These features place the problem outside the scope of classical stochastic maximum principle (SMP) results, which rely on global smoothness assumptions. We establish both necessary and sufficient optimality conditions for this class of control problems. Our approach combines a Sobolev-type representation of the first variation process with smooth approximations and Ekeland's variational principle. As application, we study an optimal premium adjustment and reserve management policies for an insurance whose surplus is modelled by threshold-based dividend and capital injection policies.

math.OC↗

arXiv2025

Smoothness of solutions of hyperbolic stochastic partial differential equations with $L^{\infty}$-vector fields

In this paper we are interested in a quasi-linear hyperbolic stochastic differential equation (HSPDE) when the vector field is merely bounded and measurable. Although the deterministic counterpart of such equation may be ill-posed (in the sense that uniqueness or even existence might not be valid), we show for the first time that the corresponding HSPDE has a unique (Malliavin differentiable) strong solution. Our approach for proving this result rests on: 1) tools from Malliavin calculus and 2) variational techniques introduced in [Davie, Int. Math. Res. Not., Vol. 2007] non trivially extended to the case of SDEs in the plane by using an algorithm for the selection of certain rectangles. As a by product, we also obtain the Sobolev differentiability of the solution with respect to its initial value. The results derived here constitute a significant improvement of those in the current literature on SDEs on the plane and can be regarded as an analogous equivalent of the pioneering works by [Zvonkin, Math. URSS Sbornik, 22:129-149] and [Veretennikov, Theory Probab. Appl., 24:354-366] in the case of one-parameter SDEs with singular drift.

math.PR↗

arXiv2025

Strong solutions of fractional Brownian sheet driven SDEs with integrable drift

We prove the existence of a unique Malliavin differentiable strong solution to a stochastic differential equation on the plane with merely integrable coefficients driven by the fractional Brownian sheet with Hurst parameters less than 1/2. The proof of this result relies on a compactness criterion for square integrable Wiener functionals from Malliavin calculus ([Da Prato, Malliavin and Nualart, 1992]), variational techniques developed in the case of fractional Brownian motion ([Baños, Nielssen, and Proske, 2020]) and the concept of sectorial local nondeterminism (introduced in [Khoshnevisan and Xiao, 2007]). The latter concept enable us to improve the bound of the Hurst parameter (compare with [Baños, Nielssen, and Proske, 2020]).

math.PR↗

arXiv2022

Malliavin differentiability of solutions of hyperbolic stochastic partial differential equations with irregular drifts

We prove path-by-path uniqueness of solution to hyperbolic stochastic partial differential equations when the drift coefficient is the difference of two componentwise monotone Borel measurable functions of spatial linear growth. The Yamada-Watanabe principle for SDE driven by Brownian sheet then allows to derive strong uniqueness for such equation and thus extending the results in [Bogso, Dieye and Menoukeu Pamen, Elect. J. Probab., 27:1-26, 2022] and [Nualart and Tindel, Potential Anal., 7(3):661--680, 1997]. Assuming that the drift is globally bounded, we show that the unique strong solution is Malliavin differentiable. The case of spatial linear growth drift coefficient is also studied.

math.PR↗

arXiv2022

Stochastic integration with respect to local time of the Brownian sheet and regularising properties of Brownian sheet paths

In this work, we generalise the stochastic local time space integration introduced in \cite{Ei00} to the case of Brownian sheet. %We develop a stochastic local time-space calculus with respect to the Brownian sheet. This allows us to prove a generalised two-parameter Itô formula and derive Davie type inequalities for the Brownian sheet. Such estimates are useful to obtain regularity bounds for some averaging type operators along Brownian sheet curves.

math.PR↗

arXiv2022

Path-by-path uniqueness of multidimensional SDE's on the plane with nondecreasing coefficients

In this paper we study path-by-path uniqueness for multidimensional stochastic differential equations driven by the Brownian sheet. We assume that the drift coefficient is unbounded, verifies a spatial linear growth condition and is componentwise nondeacreasing. Our approach consists of showing the result for bounded and componentwise nondecreasing drift using both a local time-space representation and a law of iterated logarithm for Brownian sheets. The desired result follows using a Gronwall type lemma on the plane. As a by product, we obtain the existence of a unique strong solution of multidimensional SDEs driven by the Brownian sheet when the drift is non-decreasing and satisfies a spatial linear growth condition.

math.PR↗

arXiv2019

European Option Pricing of electricity under exponential functional of Lévy processes with Price-Cap principle

We propose a new model for electricity pricing based on the price cap principle. The particularity of the model is that the asset price is an exponential functional of a jump Lévy process. This model can capture both mean reversion and jumps which are observed in electricity market. It is shown that the value of an European option of this asset is the unique viscosity solution of a partial integro-differential equation (PIDE). A numerical approximation of this solution by the finite differences method is provided. The consistency, stability and convergence results of the scheme are given. Numerical simulations are performed under a smooth initial condition.

q-fin.PR↗

arXiv2019

Self-similar martingales derived from Root embedding

Given a family $(μ_λ,λ\geq0)$ of integrable mean-zero probability measures such that, for every $λ\geq0$, $μ_λ$ is the image of $μ_1$ under the homothety $y\longmapsto\sqrtλy$, we provide a necessary and sufficient condition on $μ_1$ under which the Root embedding algorithm yields a self-similar martingale with one-dimensional marginals $(μ_λ,λ\geq0)$. Precisely, if $τ_λ$ and $R_λ$ denote the Root solution to the Skorokhod embedding problem (SEP) and the Root regular barrier for $μ_λ$ respectively, then this condition is equivalent to the property that $(R_λ,λ\geq0)$ is non-increasing in the sense of inclusion, which in turn is equivalent to the assertion that $(τ_λ,λ\geq0)$ is non-decreasing a.s. We show that there are many examples for which this result applies and we provide some numerical simulations to illustrate the monotonicity property of regular barriers $(R_λ,λ\geq0)$ in this case.

math.PR↗

arXiv2018

Mean residual life processes and associated submartingales

We use Madan-Yor's argument to construct associated submartingales to a class of two-parameter processes that are ordered by the increasing convex dominance. This class includes processes which have MTP$_2$ integrated survival functions. We prove that the integrated survival function of an integrable two-parameter process is MTP$_2$ if and only if it is TP$_2$ in each pair of arguments when the remaining argument is fixed. This result can not be deduced from known results since there are several two-parameter processes whose integrated survival functions do not have interval support. The MTP$_2$ property of certain MRL processes is useful to exhibit numerous other processes having the same property.

math.PR↗

arXiv2017

Weak decreasing stochastic order

We introduce the notion of weak decreasing stochastic (WDS) ordering for real-valued processes with negative means, which, to our knowledge, has not been studied before. Thanks to Madan-Yor's argument, it follows that the WDS ordering is a necessary and sufficient condition for a process with negative mean to be embeddable in a standard Brownian motion by the Cox and Hobson extension of the Azéma-Yor algorithm. Since the decreasing stochastic order is stronger than the WDS order, then, for every stochastically non-decreasing family of probability measures with densities, the Cox-Hobson stopping times provide an associated Markov process. The quantile process associated to a stochastically non-decreasing process is not necessarily Markovian.

math.PR↗