Exact conditions for no ruin for the generalised Ornstein-Uhlenbeck process

Damien Bankovsky; Allan Sly

doi:10.1016/j.spa.2009.01.001

Exact conditions for no ruin for the generalised Ornstein-Uhlenbeck process

Damien Bankovsky, Allan Sly

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

For a bivariate Lévy process (ξ_t, η_t)_{t ≥ 0} the generalised Ornstein-Uhlenbeck (GOU) process is defined as V_t {colon equals} e^ξt (z + ∫₀^t e^{- ξs -} d η_s), t ≥ 0, where z ∈ R. We define necessary and sufficient conditions under which the infinite horizon ruin probability for the process is zero. These conditions are stated in terms of the canonical characteristics of the Lévy process and reveal the effect of the dependence relationship between ξ and η. We also present technical results which explain the structure of the lower bound of the GOU.

Original language	English (US)
Pages (from-to)	2544-2562
Number of pages	19
Journal	Stochastic Processes and their Applications
Volume	119
Issue number	8
DOIs	https://doi.org/10.1016/j.spa.2009.01.001
State	Published - Aug 2009
Externally published	Yes

All Science Journal Classification (ASJC) codes

Statistics and Probability
Modeling and Simulation
Applied Mathematics

Keywords

Exponential functionals of Lévy processes
Generalised Ornstein-Uhlenbeck process
Lévy processes
Ruin probability

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10.1016/j.spa.2009.01.001

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title = "Exact conditions for no ruin for the generalised Ornstein-Uhlenbeck process",

abstract = "For a bivariate L{\'e}vy process (ξt, ηt)t ≥ 0 the generalised Ornstein-Uhlenbeck (GOU) process is defined as Vt {colon equals} eξt (z + ∫0t e- ξs - d ηs), t ≥ 0, where z ∈ R. We define necessary and sufficient conditions under which the infinite horizon ruin probability for the process is zero. These conditions are stated in terms of the canonical characteristics of the L{\'e}vy process and reveal the effect of the dependence relationship between ξ and η. We also present technical results which explain the structure of the lower bound of the GOU.",

keywords = "Exponential functionals of L{\'e}vy processes, Generalised Ornstein-Uhlenbeck process, L{\'e}vy processes, Ruin probability",

author = "Damien Bankovsky and Allan Sly",

note = "Funding Information: We are very grateful to Professor Ross Maller and two anonymous referees. Allan Sly is supported by NSF grants DMS-0528488 and DMS-0548249.",

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doi = "10.1016/j.spa.2009.01.001",

language = "English (US)",

volume = "119",

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journal = "Stochastic Processes and their Applications",

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T1 - Exact conditions for no ruin for the generalised Ornstein-Uhlenbeck process

AU - Bankovsky, Damien

AU - Sly, Allan

N1 - Funding Information: We are very grateful to Professor Ross Maller and two anonymous referees. Allan Sly is supported by NSF grants DMS-0528488 and DMS-0548249.

PY - 2009/8

Y1 - 2009/8

N2 - For a bivariate Lévy process (ξt, ηt)t ≥ 0 the generalised Ornstein-Uhlenbeck (GOU) process is defined as Vt {colon equals} eξt (z + ∫0t e- ξs - d ηs), t ≥ 0, where z ∈ R. We define necessary and sufficient conditions under which the infinite horizon ruin probability for the process is zero. These conditions are stated in terms of the canonical characteristics of the Lévy process and reveal the effect of the dependence relationship between ξ and η. We also present technical results which explain the structure of the lower bound of the GOU.

AB - For a bivariate Lévy process (ξt, ηt)t ≥ 0 the generalised Ornstein-Uhlenbeck (GOU) process is defined as Vt {colon equals} eξt (z + ∫0t e- ξs - d ηs), t ≥ 0, where z ∈ R. We define necessary and sufficient conditions under which the infinite horizon ruin probability for the process is zero. These conditions are stated in terms of the canonical characteristics of the Lévy process and reveal the effect of the dependence relationship between ξ and η. We also present technical results which explain the structure of the lower bound of the GOU.

KW - Exponential functionals of Lévy processes

KW - Generalised Ornstein-Uhlenbeck process

KW - Lévy processes

KW - Ruin probability

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SN - 0304-4149

VL - 119

SP - 2544

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JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 8

ER -

Exact conditions for no ruin for the generalised Ornstein-Uhlenbeck process

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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