The first passage problem for generalized Ornstein-Uhlenbeck processes with non-positive jumps

Hadjiev, D.

doi:10.1007/bfb0075840

Cited by 27 publications

(31 citation statements)

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“…When κ < 0 and l > 0, this is a simple and special case of the model treated by Hadjiev [3] and Novikov [11] where one also finds sufficient conditions that must be imposed on G + for the formula (26) to be valid.…”

Section: Notation and Set-upmentioning

confidence: 99%

“…Hadjiev [3] then obtains the Laplace transform in terms of real valued integrals under a certain condition; in particular it suffices that the Lévy process has a Brownian component. For the same type of process Novikov [11] also finds the Laplace transform, but under more general conditions.…”

Section: Introductionmentioning

confidence: 99%

“…As noted in (3), the PDMP X is an Ornstein-Uhlenbeck process solving a stochastic differential equation driven by a compound Poisson process. It is easy to generalize this to dX t = (α + κ X t ) dt + dU t since if X solves this SDE, thenX solves (3) whereX t = X t + α κ and thus Laplace transforms of exit times and undershoots etc.…”

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confidence: 99%

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Exit times for a class of piecewise exponential Markov processes with two-sided jumps

Jacobsen

Jensen²

2007

Stochastic Processes and their Applications

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We consider first passage times for piecewise exponential Markov processes that may be viewed as Ornstein-Uhlenbeck processes driven by compound Poisson processes. We allow for two-sided jumps and as a main result we derive the joint Laplace transform of the first passage time of a lower level and the resulting undershoot when passage happens as a consequence of a downward (negative) jump. The Laplace transform is determined using complex contour integrals and we illustrate how the choice of contours depends in a crucial manner on the particular form of the negative jump part, which is allowed to belong to a dense class of probabilities. We give extensions of the main result to two-sided exit problems where the negative jumps are as before but now it is also required that the positive jumps have a distribution of the same type. Further, extensions are given for the case where the driving Lévy process is the sum of a compound Poisson process and an independent Brownian motion. Examples are used to illustrate the theoretical results and include the numerical evaluation of some concrete exit probabilities. Also, some of the examples show that for specific values of the model parameters it is possible to obtain closed form expressions for the Laplace transform, as is the case when residue calculus may be used for evaluating the relevant contour integrals.

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Section: Notation and Set-upmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Exit times for a class of piecewise exponential Markov processes with two-sided jumps

Jacobsen

Jensen²

2007

Stochastic Processes and their Applications

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“…In the literature, this process is cited as an important example of a different class of random processes: shot noise processes (see, e.g., [3]), filtered Poisson processes [4], generalized OrnsteinUhlenbeck (OU) processes (see [5], [6], and [7]), etc.…”

Section: Introductionmentioning

confidence: 99%

“…Different approaches were used for studying this problem: integral equations (see, e.g., [1], [10], and [21]); martingale techniques ( [7], [8], and [9]), etc. In this paper, we also apply the martingale technique, namely a special parametric family of martingales (Theorem 1).…”

Section: Introductionmentioning

confidence: 99%

Martingales and First-Passage Times for Ornstein--Uhlenbeck Processes with a Jump Component

Novikov¹

2004

Theory Probab. Appl.

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Fluctuations of Lévy Processes with Applications

2014

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The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.

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The first passage problem for generalized Ornstein-Uhlenbeck processes with non-positive jumps

Cited by 27 publications

References 5 publications

Exit times for a class of piecewise exponential Markov processes with two-sided jumps

Exit times for a class of piecewise exponential Markov processes with two-sided jumps

Martingales and First-Passage Times for Ornstein--Uhlenbeck Processes with a Jump Component

Fluctuations of Lévy Processes with Applications

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