Exponential Functionals of Lévy Processes

“…We prove the lemma in the reverse order of its statement: we take (11) (10); it is easily checked that (9) holds (see below), and hence also (8) by identification of moments. We now provide some details for the proof of (9) for t = s/α.…”

Section: Proofmentioning

confidence: 99%

On Subordinators, Self-Similar Markov Processes and Some Factorizations of the Exponential Variable

Bertoin

Yor

2001

Electron. Commun. Probab.

Self Cite

103

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“…where B is a standard Brownian motion and Z a squared Bessel process of dimension 0 started from l. Moreover, according to P. Carmona, F. Petit and M. Yor [CPY01], we have the equality in law…”

Section: Proof Of Theorem 13(iii) To Evaluate Pmentioning

confidence: 94%

Penalization of a Positively Recurrent Diffusion by an Exponential Function of its Local Time

Profeta¹

2010

Publ. Res. Inst. Math. Sci.

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Using Krein's theory of strings, we penalize here a large class of positively recurrent diffusions by an exponential function of their local time. After a brief study of the processes so penalized, we show that on this example the principle of penalization can be iterated, and that the family of probabilities we get forms a group. We conclude by an application to Bessel processes of dimension δ ∈ ]0, 2[ which are reflected at 1. Let P x and E x denote, respectively, the probability measure and the expectation associated with X when started from x ≥ 0. We assume that X is defined on the canonical space Ω := C(R + → R + ) (where R + := [0, +∞[), and we denote by (F t , t ≥ 0) its natural filtration, with F ∞ := t≥0 F t .Let us start by giving a definition of penalization (see also Theorem 3.1):Definition 1.1. Let (Γ t , t ≥ 0) be a measurable process taking positive values and such that 0 < E x [Γ t ] < ∞ for every t > 0 and every x ∈ I. We say that the process (Γ t , t ≥ 0) satisfies the penalization principle if there exists a probability measure Q x defined on (Ω, F ∞ ) such that

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“…In order to see the connection, extend X 0 to a process (X t ) t∈R + defined via 6) and note that (Z t , X t ) t∈R + is a stationary process under P. Fix T > 0 and define the process…”

Section: The Distribution Of (Z 0 X 0 ) Via Diffusion Time Reversalmentioning

confidence: 99%

Continuous-time perpetuities and time reversal of diffusions

Kardaras

Robertson

2016

Finance Stoch

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We consider the problem of estimating the joint distribution of a continuous-time perpetuity and the underlying factors which govern the cash flow rate, in an ergodic Markovian model. Two approaches are used to obtain the distribution. The first identifies a partial differential equation for the conditional cumulative distribution function of the perpetuity given the initial factor value, which under certain conditions ensures the existence of a density for the perpetuity. The second (and more general) approach, using techniques of time reversal, identifies the joint law as the stationary distribution of an ergodic multidimensional diffusion. This latter approach allows efficient use of Monte Carlo simulation, as the distribution is obtained by sampling a single path of the reversed process.

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Exponential Functionals of Lévy Processes

Abstract: This text surveys properties and applications of the exponential functional t 0 exp(−ξs)ds of real-valued Lévy processes ξ = (ξt, t ≥ 0).

Cited by 77 publications

References 51 publications

On Subordinators, Self-Similar Markov Processes and Some Factorizations of the Exponential Variable

On Subordinators, Self-Similar Markov Processes and Some Factorizations of the Exponential Variable

Penalization of a Positively Recurrent Diffusion by an Exponential Function of its Local Time

Continuous-time perpetuities and time reversal of diffusions

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