Representations of multidimensional linear process bridges

Barczy, Matyas; Kern, Peter

doi:10.1515/rose-2013-0009

Cited by 7 publications

(25 citation statements)

References 28 publications

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“…where we have introduced the additional notations, partly borrowed from [22], As in the two preceding paragraphs, it remains to evaluate the state transition matrix Φ(t, τ ) of Eq. (35).…”

Section: Conditioned Linear Processmentioning

confidence: 99%

“…In a recent article, Barczy and Kern [22] have studied a linear process Z t given by the stochastic differential equation,…”

Section: A Linear Bridges: Link With the Formulation Of Barczy And Kernmentioning

confidence: 99%

“…Note that in the case of a linear bridge, the solution of the stochastic differential equation Eq. (62) is known [22],…”

Section: A Linear Bridges: Link With the Formulation Of Barczy And Kernmentioning

confidence: 99%

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Constraint Ornstein-Uhlenbeck bridges

Mazzolo

2017

Journal of Mathematical Physics

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In this paper, we study the Ornstein-Uhlenbeck bridge process (i.e. the Ornstein-Uhlenbeck process conditioned to start and end at fixed points) constraints to have a fixed area under its path. We present both anticipative (in this case, we need the knowledge of the future of the path) and non-anticipative versions of the stochastic process. We obtain the anticipative description thanks to the theory of generalized Gaussian bridges while the non-anticipative representation comes from the theory of stochastic control. For this last representation, a stochastic differential equation is derived which leads to an effective Langevin equation. Finally, we extend our theoretical findings to linear bridge processes.Comment: 20 pages, 5 figure

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Section: Conditioned Linear Processmentioning

confidence: 99%

“…In a recent article, Barczy and Kern [22] have studied a linear process Z t given by the stochastic differential equation,…”

Section: A Linear Bridges: Link With the Formulation Of Barczy And Kernmentioning

confidence: 99%

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Constraint Ornstein-Uhlenbeck bridges

Mazzolo

2017

Journal of Mathematical Physics

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“…This is a local linearization of the nonlinear diffusion over a small time window (30,39). First we construct bridge distributions for general multivariate linear diffusions (4). If at time s we have X s = d and at time T , X T = e then the distribution of X t for 0 ≤ s < t ≤ T can be shown to be Gaussian with mean…”

Section: Sampling Of Diffusion Pathsmentioning

confidence: 99%

Systematic physics constrained parameter estimation of stochastic differential equations

Peavoy¹,

Franzke²,

Roberts³

2015

Computational Statistics & Data Analysis

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A systematic Bayesian framework is developed for physics constrained parameter inference of stochastic differential equations (SDE) from partial observations. Physical constraints are derived for stochastic climate models but are applicable for many fluid systems. A condition is derived for global stability of stochastic climate models based on energy conservation. Stochastic climate models are globally stable when a quadratic form, which is related to the cubic nonlinear operator, is negative definite. A new algorithm for the efficient sampling of such negative definite matrices is developed and also for imputing unobserved data which improve the accuracy of the parameter estimates. The performance of this framework is evaluated on two conceptual climate models.

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“…Further, since E(Y 1 ) = 0, in this case we have y 1 = 0. In the present paper, we do not intend to study whether the process (Y t ) t∈[0,1] given by (1.1) can be considered as a bridge in the sense that it can be derived from some appropriate stochastic process (for more information on this procedure, see Barczy and Kern [7]).…”

mentioning

confidence: 99%

Karhunen–Loève expansion for a generalization of Wiener bridge

Barczy

Lovas

2018

Lith Math J

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We derive a Karhunen-Loève expansion of the Gauss process B t − g(t) 1 0 g (u) dB u , t ∈ [0, 1], where (B t ) t∈[0,1] is a standard Wiener process and g : [0, 1] → R is a twice continuously differentiable function with g(0) = 0 and 1 0 (g (u)) 2 du = 1. This process is an important limit process in the theory of goodness-of-fit tests. We formulate two special cases with the function g(t) = √ 2 π sin(πt), t ∈ [0, 1], and g(t) = t, t ∈ [0, 1], respectively. The latter one corresponds to the Wiener bridge over [0, 1] from 0 to 0. IntroductionIn this note we present a new class of Gauss processes, generalizing the Wiener bridge, for which Karhunen-Loève (KL) expansion can be given explicitly. We point out that there are only few Gauss processes for which the KL expansion is explicitly known. To give some examples, we mention the Wiener process (see, e.g., Ash and Gardner [5, Example 1.4.4]), the Ornstein-Uhlenbeck process (see, e.g., Papoulis [24, Problem 12.7] or Corlay and Pagès [10, Section 5.4 B]), the Wiener bridge (see, e.g., Deheuvels [12, Remark 1.1]), Kac-Kiefer-Wolfowitz process (see, Kac, Kiefer and Wolfowitz [16] and Nazarov and Petrova [23]), weighted Wiener processes and bridges (Deheuvels and Martynov [13]), Jandhyala-MacNeill process (Jandhyala and MacNeill [15, Section 4]), a generalization of Wiener bridge (Pycke [25]), generalized Anderson-Darling process (Pycke [26]), Rodríguez-Viollaz process (Pycke [ 27]), scaled Wiener bridges or also called α-Wiener bridges (Barczy and Iglói [6]), limit processes related to Cramér-von Mises goodness-of-fit tests for hypotheses that an observed diffusion process has a sign-type trend coefficient (Gassem [14]), detrended Wiener processes (Ai, Li and Liu [3]), additive Wiener processes and bridges (Liu [18]), additive Slepian processes (Liu, Huang and Mao [19]), Spartan spatial random fields (Tsantili and Hristopulos [28]), the demeaned stationary Ornstein-Uhlenbeck process (Ai [2]), and the additive two-sided Brownian motion (Ai and Sun [4]). We also mention that KL expansions of Gauss processes have found several applications in small deviation theory, for a complete bibliography, see Lifshits [17]. Here we only mention two papers of Nazarov and Nikitin [20], [22].

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Representations of multidimensional linear process bridges

Cited by 7 publications

References 28 publications

Constraint Ornstein-Uhlenbeck bridges

Constraint Ornstein-Uhlenbeck bridges

Systematic physics constrained parameter estimation of stochastic differential equations

Karhunen–Loève expansion for a generalization of Wiener bridge

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