Pinned Diffusions and Markov Bridges

Hildebrandt, Florian; Roelly, Sylvie

doi:10.1007/s10959-019-00954-5

Cited by 3 publications

(4 citation statements)

References 9 publications

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“…is Gaussian for any t ∈ T − and i ∈ I. Following the arguments of [29,38], we apply Itô's integration-by-parts formula to get…”

Section: And (Ii)mentioning

confidence: 99%

“…one of the conditions we shall ask from f will nullify the situation where f can be a constant. Accordingly, if we choose ρ = 0, the decoupling property of mean-field models is still maintained, but where the individual eigenvalues tend to mutually independent pinned diffusions (see [29,38]) as n → ∞: essentially re-coupling at time T, unlike Ornstein-Uhlenbeck processes. Therefore, we will encounter examples where we get mutually independent α-Wiener bridges (see [9,10]) in the mean-field limit, where the Brownian bridge is the archetypal subclass.…”

Section: Introductionmentioning

confidence: 99%

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Time-convergent random matrices from mean-field pinned interacting eigenvalues

Mengütürk

2022

J. Appl. Probab.

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We study a multivariate system over a finite lifespan represented by a Hermitian-valued random matrix process whose eigenvalues (i) interact in a mean-field way and (ii) converge to their weighted ensemble average at their terminal time. We prove that such a system is guaranteed to converge in time to the identity matrix that is scaled by a Gaussian random variable whose variance is inversely proportional to the dimension of the matrix. As the size of the system grows asymptotically, the eigenvalues tend to mutually independent diffusions that converge to zero at their terminal time, a Brownian bridge being the archetypal example. Unlike commonly studied random matrices that have non-colliding eigenvalues, the proposed eigenvalues of the given system here may collide. We provide the dynamics of the eigenvalue gap matrix, which is a random skew-symmetric matrix that converges in time to the $\textbf{0}$ matrix. Our framework can be applied in producing mean-field interacting counterparts of stochastic quantum reduction models for which the convergence points are determined with respect to the average state of the entire composite system.

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“…is Gaussian for any t ∈ T − and i ∈ I. Following the arguments of [29,38], we apply Itô's integration-by-parts formula to get…”

Section: And (Ii)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Time-convergent random matrices from mean-field pinned interacting eigenvalues

Mengütürk

2022

J. Appl. Probab.

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“…When one considers a system with ρ = 0, the covariance of the system persists to be non-zero in the double-limit. For i = j, trajectories from (23) are shown below for various α values and number of particles. On the left side, n = 100 is fixed, and on the right-hand side α = 1 is fixed (figure 4).…”

Section: A Numerical Study On Mean-field Convergencementioning

confidence: 99%

“…The objective of this paper is to propose a mathematical link between interacting particle diffusions where each element is influenced by the rest of the system in some mean-field way (see [1,2,7,8,14,15,17,18,26]) and pinned diffusions defined over finite time horizons (see [4,12,13,21,23]). We shall note that pinned diffusions focus on convergence behaviour with respect to time and do not display any interaction through an ensemble-average.…”

Section: Introductionmentioning

confidence: 99%

A family of interacting particle systems pinned to their ensemble average

Mengütürk¹

2021

J. Phys. A: Math. Theor.

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We study a family of interacting particle diffusions that converge to their random ensemble-average at some fixed future time for any given number of elements in the system. We impose certain conditions on a class of differential equations that model particle dynamics which are influenced by the average statistic of the whole system. As such, these processes achieve mean-field limits when the number of particles increase asymptotically, whereby the iterated limits over time vs the number of units are commutative. In a specific case, we show that the mean-field limit of the system consists of mutually independent α-Wiener bridges, instead of Ornstein–Uhlenbeck processes typically seen in the classical setting. We provide examples with simulations for demonstration.

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Optimal stopping of Gauss–Markov bridges

Azze,

D’Auria,

García-Portugués

2024

Adv. Appl. Probab.

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We solve the non-discounted, finite-horizon optimal stopping problem of a Gauss–Markov bridge by using a time-space transformation approach. The associated optimal stopping boundary is proved to be Lipschitz continuous on any closed interval that excludes the horizon, and it is characterized by the unique solution of an integral equation. A Picard iteration algorithm is discussed and implemented to exemplify the numerical computation and geometry of the optimal stopping boundary for some illustrative cases.

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Pinned Diffusions and Markov Bridges

Cited by 3 publications

References 9 publications

Time-convergent random matrices from mean-field pinned interacting eigenvalues

Time-convergent random matrices from mean-field pinned interacting eigenvalues

A family of interacting particle systems pinned to their ensemble average

Optimal stopping of Gauss–Markov bridges

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