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Alexandre Popier

Publications and source records attributed to Alexandre Popier.

At least 19 recordsLinked to original sources
arXiv2026

Multi-state model with temporal-consistent survival analysis for homogeneous Markov chains

In this study, we consider sequences drawn from time-homogeneous Markov chains and introduce a novel approach for estimating first hitting-time distributions to specified terminal states. Our method- ology is based on the temporal-consistent survival analysis that facilitates the construction of consistent estimators of the distributions from any estimates of the transition rate and transition probabilities. In this line of work, we also discuss the issue of cured individuals with chains that never reach a termi- nal state, and propose an estimator of the cure rate. Furthermore, we derive non-asymptotic theoretical guarantees for our approach and apply our methodology with kernel type estimators. The latter approach is illustrated in a simulation study using generic data and a real-life application involving patients un- dergoing bone marrow transplants.

math.ST↗
arXiv2026

Time discretization of BSDEs with singular terminal condition using asymptotic expansion

We consider a class of backward stochastic differential equations (BSDEs) with singular terminal condition and develop a numerical scheme to approximate their solution. To this end, we extend an asymptotic development of the BSDE solution known from the power case, which arises from optimal liquidation problems, to more general generators. This expansion allows to obtain a suitable approximation of the BSDE solution close to the terminal time. Using this as a terminal condition, we analyze the error of a backward Euler implicit scheme and detail its dependence on the terminal condition.

math.OC↗
arXiv2026

Homogenization of nonlocal equations in randomly evolving media. Diffusion approximation

The paper deals with homogenization and higher order approximations of solutions to nonlocal evolution equations of convolution type whose coefficients are periodic in the spatial variables and random stationary in time. We assume that the convolution kernel has finite moments up to order three. Under proper mixing assumptions, we study the limit behavior of the normalized difference between solutions of the original and the homogenized problems and show that this difference converges to the solution of a linear stochastic partial differential equation.

math.AP↗
arXiv2026

Matrix Riccati BSDEs with singular terminal condition and stochastic LQ control with linear terminal constraint

We analyze a class of multidimensional linear-quadratic stochastic control problems with random coefficients, motivated by multi-asset optimal trade execution. The problems feature non-diffusive controlled state dynamics and a terminal constraint that restricts the terminal state to a prescribed random linear subspace. We derive the associated Riccati backward stochastic differential equation (BSDE) and identify a suitable formalization of its singular terminal condition. Via a penalization approach, we establish existence of a minimal supersolution of the Riccati BSDE and use it to characterize both the value function and the optimal control. We analyze the asymptotic behavior of the supersolution near terminal time and discuss special cases where closed-form solutions can be obtained.

math.OC↗
arXiv2025

Ratio limit theorem for renewal processes

We consider a renewal process which models a cumulative shock model that fails when the accumulation of shocks up-crosses a certain threshold. The ratio limit properties of the probabilities of non-failure after n cumulative shocks are studied. We establish that the ratio of survival probabilities converges to the probability that the renewal epoch equals zero. This limit holds for any renewal process, subject only to mild regularity conditions on the individual shock random variable. Precisions on the rates of convergence are provided depending on the support structure and the regularity of the distribution. Arguments are provided to highlight the coherence between this new results and the pre-existing results on the behavior of summands of i.i.d. real random variables.

math.PR↗
arXiv2025

Malliavin calculcus for a Hawkes process

We develop a Malliavin calculus for nonlinear Hawkes processes in the sense of Carlen and Pardoux. This approach, based on perturbations of the jump times of the process, enables the construction of a local Dirichlet form. As an application, we establish criteria for the absolute continuity of solutions to stochastic differential equations driven by Hawkes processes. We also derive sensitivity formulas for the valuation of financial derivatives with respect to model parameters.

math.PR↗
arXiv2025

Malliavin derivative and sensitivity for optimal liquidation

We prove that the solution of the backward stochastic differential equation with terminal singularity has a Malliavin derivative, which is the limit of the derivative of the approximating sequence. We also provide the asymptotic behavior of this derivative close to the terminal time. We apply this result to the regularity of the related partial differential equation and to the sensitivity of the liquidation problem.

math.PR↗
arXiv2025

A dependent and censored first hitting-time model with compound Poisson processes

We consider a bivariate first hitting-time model in which durations are the crossing times of dependent compound Poisson processes with fixed thresholds. The identifiability of the model is discussed, and likelihood estimators of the model parameters are proposed. We obtain the asymptotic properties of the estimators and underline their finite sample performance with a simulation study on synthetic data. The practical applicability of our approach is demonstrated by an application using data from patients suffering from mushroom poisoning.

stat.ME↗
arXiv2023

Optimal Liquidation with Conditions on Minimum Price

The classical optimal trading problem is the closure of a position in an asset over a time interval; the trader maximizes an expected utility under the constraint that the position be fully closed by terminal time. Since the asset price is stochastic, the liquidation constraint may be too restrictive; the trader may want to relax it or slow down/stop trading depending on price behavior. We consider two additional parameters that serve these purposes within the Almgren-Chriss framework: a binary valued process $I$ that prescribes when trading takes place and a measurable set $S$ that prescribes when full liquidation is required. We give four examples for $S$ and $I$ which are defined in terms of a lower bound for the price process. The terminal cost of the control problem is $\infty$ over $S$ representing the liquidation constraint. The permanent price impact parameter enters the problem as the negative part of the terminal cost over $S^c$. $I$ modifies the running cost. A terminal cost that can take negative values implies 1) the backward stochastic differential equation (BSDE) associated with the value function of the control problem can explode to $-\infty$ backward in time and 2) existence results on minimal supersolutions of BSDE with singular terminal values and monotone drivers are not directly applicable. A key part of the solution is an assumption that balances market volume and the permanent price impact parameter and a lower bound on the BSDE based on this assumption. When liquidation costs are quadratic, the problem is convex and, under a general filtration, the minimal supersolution of the BSDE gives the value function and the optimal control. For the non-quadratic case, we assume a stochastic volatility model and focus on choices of $I$ and $S$ that are Markovian or can be broken into Markovian pieces. These give PDE/PDE-system representations for the value functions.

math.PR↗
arXiv2020

Continuity problem for singular BSDE with random terminal time

We study a class of nonlinear BSDEs with a superlinear driver process f adapted to a filtration F and over a random time interval [[0, S]] where S is a stopping time of F. The terminal condition $\xi$ is allowed to take the value +$\infty$, i.e., singular. Our goal is to show existence of solutions to the BSDE in this setting. We will do so by proving that the minimal supersolution to the BSDE is a solution, i.e., attains the terminal values with probability 1. We consider three types of terminal values: 1) Markovian: i.e., $\xi$ is of the form $\xi$ = g($\Xi$ S) where $\Xi$ is a continuous Markovian diffusion process and S is a hitting time of $\Xi$ and g is a deterministic function 2) terminal conditions of the form $\xi$ = $\infty$ $\times$ 1 {$\tau$ $\le$S} and 3) $\xi$ 2 = $\infty$ $\times$ 1 {$\tau$ >S} where $\tau$ is another stopping time. For general $\xi$ we prove the minimal supersolution is continuous at time S provided that F is left continuous at time S. We call a stopping time S solvable with respect to a given BSDE and filtration if the BSDE has a minimal supersolution with terminal value $\infty$ at terminal time S. The concept of solvability plays a key role in many of the arguments. Finally, we discuss implications of our results on the Markovian terminal conditions to solution of nonlinear elliptic PDE with singular boundary conditions.

math.AP↗
arXiv2020

Asymptotic decomposition of solutions to parabolic equations with a random microstructure

We consider a Cauchy problem for a divergence form second order parabolic operator with rapidly oscillating coefficients that are periodic in spatial variables and random stationary ergodic in time. As was already proved, in this case the homogenized operator is deterministic. We obtain the leading terms of the asymptotic expansion of the solution, these terms being deterministic functions, and show that a properly renormalized difference between the solution and the said leading terms converges to a solution of some SPDE.

math.AP↗
arXiv2020

Backward stochastic Volterra integral equations with jumps in a general filtration

In this paper, we study backward stochastic Volterra integral equations introduced in [26, 45] and extend the existence, uniqueness or comparison results for general filtration as in [31] (not only Brownian-Poisson setting). We also consider Lp-data and explore the time regularity of the solution in the It{\^o} setting, which is also new in this jump setting.

math.PR↗
arXiv2019

Backward Stochastic Differential Equations with Non-Markovian Singular Terminal Conditions with General Driver and Filtration

We consider a class of Backward Stochastic Differential Equations with superlinear driver process $f$ adapted to a filtration supporting at least a $d$ dimensional Brownian motion and a Poisson random measure on ${\mathbb R}^m- \{0\}.$ We consider the following class of terminal conditions $\xi_1 = \infty \cdot 1_{\{\tau_1 \le T\}}$ where $\tau_1$ is any stopping time with a bounded density in a neighborhood of $T$ and $\xi_2 = \infty \cdot 1_{A_T}$ where $A_t$, $t \in [0,T]$ is a decreasing sequence of events adapted to the filtration ${\mathcal F}_t$ that is continuous in probability at $T$. A special case for $\xi_2$ is $A_T = \{\tau_2 > T\}$ where $\tau_2$ is any stopping time such that $P(\tau_2 =T) =0.$ In this setting we prove that the minimal supersolutions of the BSDE are in fact solutions, i.e., they attain almost surely their terminal values. We further show that the first exit time from a time varying domain of a $d$-dimensional diffusion process driven by the Brownian motion with strongly elliptic covariance matrix does have a continuous density; therefore such exit times can be used as $\tau_1$ and $\tau_2$ to define the terminal conditions $\xi_1$ and $\xi_2.$ The proof of existence of the density is based on the classical Green's functions for the associated PDE.

math.PR↗
arXiv2019

On the fundamental solution of heat and stochastic heat equations

We consider the generic divergence form second order parabolic equation with coefficients that are regular in the spatial variables and just measurable in time. We show that the spatial derivatives of its fundamental solution admit upper bounds that agree with the Aronson type estimate and only depend on the ellipticity constants of the equation and the L $\infty$ norm of the spatial derivatives of its coefficients. We also study the corresponding stochastic partial differential equations and prove that under natural assumptions on the noise the equation admits a mild solution, given by anticipating stochastic integration.

math.AP↗
arXiv2018

A Mean Field Game of Optimal Portfolio Liquidation

We consider a mean field game (MFG) of optimal portfolio liquidation under asymmetric information. We prove that the solution to the MFG can be characterized in terms of a FBSDE with possibly singular terminal condition on the backward component or, equivalently, in terms of a FBSDE with finite terminal value, yet singular driver. Extending the method of continuation to linear-quadratic FBSDE with singular driver we prove that the MFG has a unique solution. Our existence and uniqueness result allows to prove that the MFG with possibly singular terminal condition can be approximated by a sequence of MFGs with finite terminal values.

math.OC↗