Generalized Gaussian bridges

Sottinen, Tommi; Yazigi, Adil

doi:10.1016/j.spa.2014.04.002

Cited by 34 publications

(53 citation statements)

References 21 publications

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“…In solving this problem, he introduced a transformation of the Wiener process, now referred to as Doob's h-transform, which was later adapted under the same name to deal with other conditionings of stochastic processes, including the Brownian bridge [3], Gaussian bridges [4][5][6], and the Schrödinger bridge [7][8][9][10][11], obtained by conditioning a process on reaching a certain target distribution in time as opposed to a target point. Doob's transform also appears prominently in the theory of quasi-stationary distributions [12][13][14][15][16], which describes in the simplest case the conditioning of a process never to reach an absorbing state.…”

Section: Introductionmentioning

confidence: 99%

Nonequilibrium Markov Processes Conditioned on Large Deviations

Chétrite

Touchette

2014

Ann. Henri Poincaré

327

762

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International audienceWe consider the problem of conditioning a Markov process on a rare event and of representing this conditioned process by a conditioning-free process, called the effective or driven process. The basic assumption is that the rare event used in the conditioning is a large-deviation-type event, characterized by a convex rate function. Under this assumption, we construct the driven process via a generalization of Doob's h-transform, used in the context of bridge processes, and show that this process is equivalent to the conditioned process in the long-time limit. The notion of equivalence that we consider is based on the logarithmic equivalence of path measures, and implies that the two processes have the same typical states. In constructing the driven process, we also prove equivalence with the so-called exponential tilting of the Markov process, often used with importance sampling to simulate rare events and giving rise, from the point of view of statistical mechanics, to a nonequilibrium version of the canonical ensemble. Other links between our results and the topics of bridge processes, quasi-stationary distributions, stochastic control, and conditional limit theorems are mentioned

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Section: Introductionmentioning

confidence: 99%

Nonequilibrium Markov Processes Conditioned on Large Deviations

Chétrite

Touchette

2014

Ann. Henri Poincaré

327

762

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“…What can be said about the spectrum of a bridge, given the spectrum of its base process? We show how this question can be answered asymptotically for a family of processes, including the fractional Brownian motion.Various aspects of general Gaussian bridges are discussed in [11], [25]. Aside of being interesting mathematical objects, they are important ingredients in applications, such as statistical hypothesis testing [15], exact sampling of diffusions [3], etc.The covariance operator of the bridge with kernel (1.5) is a rank one perturbation of the covariance operator of its base process.…”

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confidence: 99%

On the eigenproblem for Gaussian bridges

Chigansky¹,

Kleptsyna²,

Marushkevych³

2020

Bernoulli

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Spectral decomposition of the covariance operator is one of the main building blocks in the theory and applications of Gaussian processes. Unfortunately it is notoriously hard to derive in a closed form. In this paper we consider the eigenproblem for Gaussian bridges. Given a base process, its bridge is obtained by conditioning the trajectories to start and terminate at the given points. What can be said about the spectrum of a bridge, given the spectrum of its base process? We show how this question can be answered asymptotically for a family of processes, including the fractional Brownian motion.Various aspects of general Gaussian bridges are discussed in [11], [25]. Aside of being interesting mathematical objects, they are important ingredients in applications, such as statistical hypothesis testing [15], exact sampling of diffusions [3], etc.The covariance operator of the bridge with kernel (1.5) is a rank one perturbation of the covariance operator of its base process. This explains similarity between (1.3) and (1.4) and suggests that the spectra of the two processes must be closely related in general. This is indeed the case and one can find an exact expression for the Fredholm determinant of K in terms of the Fredholm determinant of K even for more general finite rank perturbations (see, e.g., [26], Ch. II, 4.6 in [13]). As mentioned above, the precise formulas for the eigenvalues and eigenfunctions of K are rarely known; however, the exact asymptotic approximations can be more tractable. This raises the following question:Can the exact asymptotics of the eigenvalues and the eigenfunctions for the bridge be deduced from that of the base process ?A rough answer to this question is given by the general perturbation theory [14], which implies that the eigenvalues of K and K agree in the leading asymptotic term, as it happens for (1.3) and (1.4) (see, e.g., the proof of Lemma 2 in [4]). More delicate spectral discrepancies are harder to exhibit and seem to be highly dependent on the perturbation structure. This is vividly demonstrated in the paper [21], where the kernels of the following form are considered, cf. (1.5):K Q (s, t) = K(s, t) + Qψ(s)ψ(t).(1.6)Here Q is a scalar real valued parameter and ψ is a function in the range of K. It turns out that for any Q greater than a certain critical value Q * , the spectrum of K Q coincides

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“…The following theorem, similar to Theorem 3.1 in [17], gives mean and covariance function of the generalized conditioned process.…”

Section: Conditional Lawmentioning

confidence: 85%

Pathwise asymptotics for Volterra processes conditioned to a noisy version of the Brownian motion

Pacchiarotti

2020

Modern Stochastics: Theory and Applications

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In this paper we investigate a problem of large deviations for continuous Volterra processes under the influence of model disturbances. More precisely, we study the behavior, in the near future after T , of a Volterra process driven by a Brownian motion in a case where the Brownian motion is not directly observable, but only a noisy version is observed or some linear functionals of the noisy version are observed. Some examples are discussed in both cases.

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Generalized Gaussian bridges

Cited by 34 publications

References 21 publications

Nonequilibrium Markov Processes Conditioned on Large Deviations

Nonequilibrium Markov Processes Conditioned on Large Deviations

On the eigenproblem for Gaussian bridges

Pathwise asymptotics for Volterra processes conditioned to a noisy version of the Brownian motion

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