arXiv subjects

Astrid Hilbert

Publications and source records attributed to Astrid Hilbert.

17 recordsLinked to original sources

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: $$\mathrm{d} \xi_t=-\mu(t,\mathbb{E}[\varphi_1(\xi_t)]) \frac{\xi_t}{T-t} \mathrm{d} t+\sigma(t,\mathbb{E}[\varphi_2(\xi_t)]) \mathrm{d} W_t,\,\, t<T,$$ where $\mu$ and $\sigma$ are deterministic functions that depend on time $t$ and the expectation of given functions $\varphi_1$ and $\varphi_2$ 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 $\xi$. Finally, we explore explicit solutions in specific cases of interest, including power-weighted expectations and second moments in the drift.

math.PR↗

arXiv2023

Optimal Stopping Under Model Uncertainty in a General Setting

We consider the optimal stopping time problem under model uncertainty $R(v)= {\text{ess}\sup\limits}_{ \mathbb{P} \in \mathcal{P}} {\text{ess}\sup\limits}_{\tau \in \mathcal{S}_v} E^\mathbb{P}[Y(\tau) \vert \mathcal{F}_v]$, for every stopping time $v$, set in the framework of families of random variables indexed by stopping times. This setting is more general than the classical setup of stochastic processes, and particularly allows for general payoff processes that are not necessarily right-continuous. Under weaker integrability, and regularity assumptions on the reward family $Y=(Y(v), v\in \mathcal{S})$, we show the existence of an optimal stopping time. We then proceed to find sufficient conditions for the existence of an optimal model. For this purpose, we present a universal Doob-Meyer-Mertens's decomposition for the Snell envelope family associated with $Y$ in the sense that it holds simultaneously for all $\mathbb{P} \in \mathcal{P}$. This decomposition is then employed to prove the existence of an optimal probability model and study its properties.

math.PR↗

arXiv2022

Stochastic differential equations with respect to optional semimartingales and two reflecting regulated barriers

In this work, we introduce a new Skorokhod problem with two reflecting barriers when the trajectories of the driven process and the barriers are right and left limited. We show that this problem has an explicit unique solution in a deterministic case. Then, we apply our result to study the existence and uniqueness of solutions of reflected stochastic differential equations with respect to optional semimartingales. The study is carried out on a probability space that does not necessarily satisfy the usual conditions.

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↗

arXiv2021

Reflected backward stochastic differential equations with optional barriers: monotone approximation

In this short note we consider RBSDE with Lipschitz drivers and barrier processes that are optional and right upper semicontinuous. We treat the case when the barrier can be represented as a decreasing limit of cadlag barriers. We combine well known existence results for cadlag barriers with comparison arguments for the control process to construct solutions. Finally, we highlight the connection of such RBSDEs with usual cadlag BSDEs.

math.PR↗

arXiv2021

SPDEs with space interactions and application to population modelling

We consider optimal control of a new type of non-local stochastic partial differential equations (SPDEs). The SPDEs have space interactions, in the sense that the dynamics of the system at time $t$ and position in space x also depend on the space-mean of values at neighbouring points. This is a model with many applications, e.g. to population growth studies and epidemiology. We prove the existence and uniqueness of solutions of a class of SPDEs with space interactions, and we show that, under some conditions, the solutions are positive for all times if the initial values are. Sufficient and necessary maximum principles for the optimal control of such systems are derived. Finally, we apply the results to study an optimal vaccine strategy problem for an epidemic by modelling the population density as a space-mean stochastic reaction-diffusion equation.

math.OC↗

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↗

arXiv2019

Singular control of SPDEs with space-mean dynamics

We consider the problem of optimal singular control of a stochastic partial differential equation (SPDE) with space-mean dependence. Such systems are proposed as models for population growth in a random environment. We obtain sufficient and necessary maximum principles for such control problems. The corresponding adjoint equation is a reflected backward stochastic partial differential equation (BSPDE) with space-mean dependence. We prove existence and uniqueness results for such equations. As an application we study optimal harvesting from a population modelled as an SPDE with space-mean dependence.

math.OC↗

arXiv2018

Mean-field optimal control problem of SDDEs driven by fractional Brownian motion

We consider a mean-field optimal control problem for stochastic differential equations with delay driven by fractional Brownian motion with Hurst parameter greater than one half. Stochastic optimal control problems driven by fractional Brownian motion can not be studied using classical methods, because the fractional Brownian motion is neither a Markov process nor a semi-martingale. However, using the fractional White noise calculus combined with some special tools related to the differentiation for functions of measures, we establish and prove necessary and sufficient stochastic maximum principles. To illustrate our study, we consider two applications: we solve a problem of optimal consumption from a cash flow with delay and a linear-quadratique (LQ) problem with delay.

math.OC↗

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

On the collapse of trial solutions for a damped-driven non-linear Schrödinger equation

We consider the focusing 2D non-linear Schrödinger equation, perturbed by a damping term, and driven by multiplicative noise. We show that a physically motivated trial solution does not collapse for any admissible initial condition although the exponent of the non-linearity is critical. Our method is based on the construction of a global solution to a singular stochastic Hamiltonian system used to connect trial solution and Schrödinger equation.

math-ph↗

arXiv2017

An Approximate Nash Equilibrium for Pure Jump Markov Games of Mean-field-type on Continuous State Space

We investigate mean-field games from the point of view of a large number of indistinguishable players which eventually converges to infinity. The players are weakly coupled via their empirical measure. The dynamics of the states of the individual players is governed by a non-autonomous pure jump type semi group in a Euclidean space, which is not necessarily smoothing. Investigations are conducted in the framework of non-linear Markov processes. We show that the individual optimal strategy results from a consistent coupling of an optimal control problem with a forward non-autonomous dynamics. In the limit as the number $N$ of players goes to infinity this leads to a jump-type analog of the well-known non-linear McKean-Vlasov dynamics. The case where one player has an individual preference different from the ones of the remaining players is also covered. The two results combined reveal an epsilon-Nash Equilibrium for the $N$-player games.

math.OC↗

arXiv2015

A 1/n Nash equilibrium for non-linear Markov games of mean-field-type on finite state space

We investigate mean field games for players, who are weakly coupled via their empirical measure. To this end we investigate time-dependent pure jump type propagators over a finite space in the framework of non-linear Markov processes. We show that the individual optimal strategy results from a consistent coupling of an optimal control problem with a forward non-autonomous dynamics which leads to the well-known Mckean-Vlasov dynamics in the limit as the number N of players goes to infinity. The case where one player has an individual preference different to the ones of the remaining players is also covered. The limiting system represents a 1/N-Nash Equilibrium for the approximating system of N players.

math.OC↗

arXiv2013

On the functional Hodrick-Prescott Filter with non-compact operators

We study a version of the functional Hodrick-Prescott filter where the associated operator is not necessarily compact, but merely closed and densely defined with closed range. We show that the associated optimal smoothing operator preserves the structure obtained in the compact case, when the underlying distribution of the data is Gaussian.

math.ST↗

arXiv2013

Smoluchowski-Kramers Limit for a System Subject to a Mean-Field Drift

We establish a scaling limit for autonomous stochastic Newton equations, the solutions are often called nonlinear stochastic oscillators, where the nonlinear drift includes a mean field term of McKean type and the driving noise is Gaussian. Uniform convergence in L^2 sense is achieved by applying L^2-type estimates and the Gronwall Theorem. The approximation is also called Smoluchowski-Kramers limit and is a particular averaging technique studied by Papanicolaou. It reveals an approximation of diffusions with a mean-field contribution in the drift by stochastic nonlinear oscillators with differentiable trajectories

math.PR↗

arXiv2009

Asymptotic Expansions for the Heat Kernel and the Trace of a Stochastic Geodesic Flow

We analyze the asymptotic behaviour of the heat kernel defined by a stochastically perturbed geodesic flow on the cotangent bundle of a Riemannian manifold for small time and small diffusion parameter. This extends WKB-type methods to a particular case of a degenerate Hamiltonian. We derive uniform bounds for the solution of the degenerate Hamiltonian boundary value problem for small time. From this equivalence of solutions of the Hamiltonian equations and the corresponding Hamilton Jacobi equation follows. The results are exploited to derive two sided estimates and multiplicative asymptotics for the heat kernel and the trace.

math.FA↗

arXiv2008

Complete account of randomness in the EPR-Bohm-Bell experiment

We show that paradoxical consequences of violations of Bell's inequality are induced by the use of an unsuitable probabilistic description for the EPR-Bohm-Bell experiment. The conventional description (due to Bell) is based on a combination of statistical data collected for different settings of polarization beam splitters (PBSs). In fact, such data consists of some conditional probabilities which only partially define a probability space. Ignoring this conditioning leads to apparent contradictions in the classical probabilistic model (due to Kolmogorov). We show how to make a completely consistent probabilistic model by taking into account the probabilities of selecting the settings of the PBSs. Our model matches both the experimental data and is consistent with classical probability theory.

quant-ph↗