Exponential Families of Stochastic Processes

Küchler, Uwe; Sørensen, Michael

doi:10.1007/b98954

Cited by 169 publications

(128 citation statements)

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“…see Proposition 2.1.3 of Kűchler and Sørensen (1997) and Theorem 3.9 of Kyprianou (2006). It turns out that, in the light of this change of measure, it is important to introduce an additional parameter to the scale functions described above.…”

Section: Distributional Identitiesmentioning

confidence: 94%

Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process

Kyprianou

Palmowski

2007

J. Appl. Probab.

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We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muler (2005), andAvram et al. (2007), which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically, we build on recent work in the actuarial literature concerning calculations of the nth moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than the existing literature, in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical Cramér-Lundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integro-differential equations and, for the case of the nth moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators.Keywords: Reflected Lévy process; passage problem; integrated exponential Lévy process; insurance risk process; ruin 2000 Mathematics Subject Classification: Primary 60K05; 60K15; 91B30 Secondary 60G70; 60J55 Lévy insurance risk processesRecall that the Cramér-Lundberg model corresponds to a Lévy process X CL = {X CL t : t ≥ 0} with characteristic exponent given byfor θ ∈ R, such that lim t↑∞ X CL t = ∞. In other words, X CL is a compound Poisson process with arrival rate λ CL > 0 and negative jumps, corresponding to claims, having common distribution function F with finite mean 1/µ CL as well as a drift c CL > 0, corresponding to a steady income due to premiums, which necessarily satisfies the net profit condition c CL − λ CL /µ CL > 0. Suppose instead that we work with a general spectrally negative Lévy process, that is, a Lévy process X = {X t : t ≥ 0} with Lévy measure satisfying ((0, ∞)) = 0. At such a degree of generality, the analogue of the condition c CL − λ CL /µ CL > 0 may be taken to be E(X 1 ) > 0. Such processes have been considered recently by Huzak et al.

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Section: Distributional Identitiesmentioning

confidence: 94%

Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process

Kyprianou

Palmowski

2007

J. Appl. Probab.

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“…The martingale condition retains when we replace t with a locally predictable and continuous time change T t (Küchler and Sørensen (1997)):…”

Section: Constructing Risk-neutral Return Dynamicsmentioning

confidence: 99%

“…The literature has taken different approaches in arriving at the dynamics under a measure change. For measure changes defined by exponential martingales of a Lévy processes X, it is convenient to remember that ϕ P X (s) = ϕ Q X (s + γ) − ϕ Q X (γ) and that the drift adjustment of X is captured by η = ϕ P (1) − ϕ Q (1) (Küchler and Sørensen (1997)). For the simple case in (43) with X = σW , we have ϕ Q σW (1) = 1 2 σ 2 , and…”

Section: Market Price Of Risks and Statistical Dynamicsmentioning

confidence: 99%

Modeling Financial Security Returns Using Levy Processes

2006

SSRN Journal

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“…Sørensen [45] gave a review of sequential maximum likelihood estimation in linearly parametrized diffusion type processes. Sørensen [46] (see also [24]) studied similar properties of SMLE for exponential families of stochastic processes. Musiela [40,41] studied sequential ML estimation in a linear diffusion model.…”

Section: Introductionmentioning

confidence: 96%

Sequential Maximum Likelihood Estimation in Nonlinear Nonmarkov Diffusion Type Processes

Bishwal¹

2018

DSA

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ABSTRACT:We obtain the strong consistency, uniform asymptotic normality and local asymptotic minimaxity (in the Hajek-LeCam sense) of the two stage sequential maximum likelihood estimator of a parameter appearing nonlinearly in the drift coefficient of a stochastic differential equation when the corresponding non-Markov diffusion type process is observed until the observed Fisher information of the process exceeds a predetermined level of precision and the level becomes large. Main results are illustrated by the exponential memory non-Markovian Ornstein-Uhlenbeck process.

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Exponential Families of Stochastic Processes

Cited by 169 publications

References 0 publications

Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process

Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process

Modeling Financial Security Returns Using Levy Processes

Sequential Maximum Likelihood Estimation in Nonlinear Nonmarkov Diffusion Type Processes

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