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1

Large deviations for Brownian bridges with prescribed terminal densities

Saize Stefane, Yang Xiangfeng

Probability and Mathematical Statistics

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2023

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Vol. 43, Fasc. 1

141--154

EN

We consider a family of Brownian bridges (depending on some small parameter) over a finite time interval whose initial position is deterministically fixed, and whose terminal position possesses a prescribed density. Large deviations for this family are studied with the help of the Girsanov transformation.

2

Optimal importance sampling for continuous Gaussian fields

Pacchiarotti Barbara

Journal of Applied Analysis

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2020

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Vol. 26, nr 2

161--171

EN

We consider the problem of selecting a change of mean which minimizes the variance of Monte Carlo estimators for the expectation of a functional of a continuous Gaussian field, in particular continuous Gaussian processes. Functionals of Gaussian fields have taken up an important position in many fields including statistical physics of disordered systems and mathematical finance (see, for example, [A. Comtet, C.Monthus and M. Yor, Exponential functionals of Brownian motion and disordered systems, J. Appl. Probab. 35 (1998), no. 2, 255-271], [D. Dufresne, The integral of geometric Brownian motion, Adv. in Appl. Probab. 33 (2001), no. 1, 223-241], [N. Privault and W. I. Uy, Monte Carlo computation of the Laplace transform of exponential Brownian functionals, Methodol. Comput. Appl. Probab. 15 (2013), no. 3, 511-524] and [V. R. Fatalov, On the Laplace method for Gaussian measures in a Banach space, Theory Probab. Appl. 58 (2014), no. 2, 216-241]. Naturally, the problem of computing the expectation of such functionals, for example the Laplace transform, is an important issue in such fields. Some examples are considered, which, for particular Gaussian processes, can be related to option pricing.

3

Large deviations for generalized conditioned Gaussian Processes and their Bridges

Pacchiarotti Barbara

Probability and Mathematical Statistics

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2019

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Vol. 39, Fasc. 1

159--181

EN

We study the asymptotic behavior of a Gaussian process conditioned to n linear functionals of its paths and of the bridge of such a process. In particular, functional large deviation results are stated for small time. Two examples are considered.

4

Cramér type large deviations for trimmed L-statistics

Gribkova N.

Probability and Mathematical Statistics

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2017

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Vol. 37, Fasc. 1

101--118

EN

In this paper, we propose a new approach to the investigation of asymptotic properties of trimmed L-statistics and we apply it to the Cramér type large deviation problem. Our results can be compared with those in Callaert et al. (1982) – the first and, as far as we know, the single article where some results on probabilities of large deviations for the trimmed L-statistics were obtained, but under some strict and unnatural conditions. Our approach is to approximate the trimmed L-statistic by a non-trimmed L-statistic (with smooth weight function) based onWinsorized random variables. Using this method, we establish the Cramér type large deviation results for the trimmed L-statistics under quite mild and natural conditions.

5

The rate of convergence in the precise large deviation theorem

Baltrūnas A.

Probability and Mathematical Statistics

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2002

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Vol. 22, Fasc. 2

343--354

EN

Let X1, X2, . . . be i.i.d. random variables with a common d.f. F. Let Sn = X1 +. . .+ Xn, n ≥1, and Mn = max k≤n Xk, n≥1. In this paper for a large class of subexponential distributions we estimate the rate of convergence. ∆n(t) = P(Sn > t) − P(Mn> t), where n ≥ 1 and t ≥ 0. We close this paper with some examples.

6

Large deviation principle for set-valued union processes

Schreiber T.

Probability and Mathematical Statistics

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2000

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Vol. 20, Fasc. 2

273--285

EN

The purpose of the paper is to establish a large deviation principle for a certain class of increasing set-valued processes obeyingMarkovian dynamics. The obtained result is then applied to investigate the asymptotics of the sequence of successive convex hulls generated by uniform samples on a d-dimensional ball.