Exact and Efficient Sampling of Conditioned Walks

Adorisio, Matteo; Pezzotta, Alberto; Mulatier, Clélia de; Micheletti, Cristian; Celani, Antonio

doi:10.1007/s10955-017-1911-y

Cited by 17 publications

(21 citation statements)

References 39 publications

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“…With such a drift, the boundary now corresponds to an entrance boundary [1], which means that the boundary cannot be reached from the interior of the state space (here the interval ] − ∞, a[). Originally introduced in the mathematical literature, and since then widely studied in this field [10,109,110] for both semi-infinite and finite domains, the taboo process and its later generalizations have recently found applications in physics [111] where they are relevant for studying confined polymers [112]. For a physicist-oriented survey, we refer to the recent article [9].…”

Section: Application To the Brownian Motion With Drift µ For Finite H...mentioning

confidence: 99%

Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon

Monthus¹,

Mazzolo²

2022

Preprint

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When the unconditioned process is a diffusion living on the half-line x ∈] − ∞, a[ in the presence of an absorbing boundary condition at position x = a, we construct various conditioned processes corresponding to finite or infinite horizon. When the time horizon is finite T < +∞, the conditioning consists in imposing the probability P * (y, T ) to be surviving at time T and at the position y ∈ ] − ∞, a[, as well as the probability γ * (Ta) to have been absorbed at the previous time Ta ∈ [0, T ]. When the time horizon is infinite T = +∞, the conditioning consists in imposing the probability γ * (Ta) to have been absorbed at the time Ta ∈ [0, +∞[, whose normalization [1−S * (∞)] determines the conditioned probability S * (∞) ∈ [0, 1] of forever-survival. This case of infinite horizon T = +∞ can be thus reformulated as the conditioning of diffusion processes with respect to their first-passagetime properties at position a. This general framework is applied to the explicit case where the unconditioned process is the Brownian motion with uniform drift µ in order to generate stochastic trajectories satisfying various types of conditioning constraints. Finally, we describe the links with the dynamical large deviations at Level 2.5 and the stochastic control theory.

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Section: Application To the Brownian Motion With Drift µ For Finite H...mentioning

confidence: 99%

Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon

Monthus¹,

Mazzolo²

2022

Preprint

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“…We now show how the lazy resampling methods introduced in Section 4.2 can help speed up dSMC significantly, while at the same time retaining the same variance as the original method. In order to do so, similarly to Deligiannidis et al (2020, Section 4.1), we consider a constrained random walk model studied, for example, in Del Moral and and Adorisio et al (2018). While, contrarily to these works, we are not concerned with exact simulation, this model is helpful in understanding the impact of the weights variance on the total runtime and variance of dSMC with lazy resampling.…”

Section: Speed-up and Variance Reduction Via Lazy Resamplingmentioning

confidence: 99%

De-Sequentialized Monte Carlo: a parallel-in-time particle smoother

Corenflos¹,

Chopin²,

Särkkä³

2022

Preprint

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Particle smoothers are SMC (Sequential Monte Carlo) algorithms designed to approximate the joint distribution of the states given observations from a state-space model. We propose dSMC (de-Sequentialized Monte Carlo), a new particle smoother that is able to process T observations in O(log T ) time on parallel architecture. This compares favourably with standard particle smoothers, the complexity of which is linear in T . We derive Lp convergence results for dSMC, with an explicit upper bound, polynomial in T . We then discuss how to reduce the variance of the smoothing estimates computed by dSMC by (i) designing good proposal distributions for sampling the particles at the initialization of the algorithm, as well as by (ii) using lazy resampling to increase the number of particles used in dSMC. Finally, we design a particle Gibbs sampler based on dSMC, which is able to perform parameter inference in a state-space model at a O(log(T )) cost on parallel hardware.

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“…Taboo processes find applications in mathematical finance where market information is often modeled by conditioning [14,15,16]. It is only very recently that these processes have attracted, at last, the physicists' attention [17,18,19]. Now, let us be more specific about what we mean by taboo processes.…”

Section: General Settingmentioning

confidence: 99%

“…In this paragraph, we consider taboo processes in three-dimensional bounded domains. In particular, we study the behavior of the taboo process evolving inside a spherical shell, a non-convex region between two concentric spheres of radius a and b, D A = r ∈ R 3 : a < |r| < b. Taboo processes in other 3-dimensional geometries, like a cylinder, have been recently reported in [19] under the name of "Brownian motion conditioned to stay inside a cylinder". For the spherical shell geometry, as in the two-dimensional annulus case, we start with a standard Brownian motion (here a three-dimensional one) and we seek the lowest eigenfunction of −Lϕ(r) = −1/2∆ϕ(r) = λϕ(r) (∆ being the 3-dimensional Laplacian) for the spherical shell with Dirichlet boundary conditions.…”

Section: Three Dimensional Taboo Processmentioning

confidence: 99%

Sweetest taboo processes

Mazzolo¹

2018

J. Stat. Mech.

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Brownian dynamics play a key role in understanding the diffusive transport of micro particles in a bounded environment. In geometries containing confining walls, physical laws determine the behavior of the random trajectories at the boundaries. For impenetrable walls, imposing reflecting boundary conditions to the Brownian particles leads to dynamics described by reflecting stochastic differential equations. In practice, these stochastic differential equations as well as their refinements are quite challenging to handle, and more importantly, many physical processes are better modeled by processes conditioned to stay in a prescribed bounded region. In the mathematical literature, these processes are known as taboo processes, and despite their simplicity, at least compared to the reflecting stochastic differential equations approach, are surprisingly not much exploited in physics. This paper explores some aspect of taboo processes and other constrained processes in simple geometries: Interval in one dimension, circular annulus in two dimensions, hollow sphere in three dimensions, and more. In particular, for the two-dimensional taboo process in a circular annulus, the Gaussian behavior of the stochastic angle is established.

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Exact and Efficient Sampling of Conditioned Walks

Cited by 17 publications

References 39 publications

Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon

Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon

De-Sequentialized Monte Carlo: a parallel-in-time particle smoother

Sweetest taboo processes

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