Lévy processes and infinitely divisible distributions

By 佐藤健一. 2013

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Citations in the corpus

Causal interpretation of stochastic differential equations (2013, methodology): “For a treatment of the theory of Markov processes and Lévy processes, and in particular for notions such as Feller processes, Feller semigroups, generators, Lévy processes and so forth, see [14, 3, 34].”
STOCHASTIC INTEGRALS WITH RESPECT TO L´ EVY PROCESSES AND INFINITELY DIVISIBLE DISTRIBUTIONS (2011, background): “Indeed, an exponential distribution is selfdecomposable and hence LT1 has selfdecomposable distribution by the result of Sato [46].”
Pricing and Hedging in Exponential Lévy Models: Review of Recent Results (2011, background): “20 in [48], Xt t → b almost surely as t → 0.”
Error analysis of finite difference and Markov chain approximations for option pricing (2018, unknown): “The Laplace transform of 𝑇 is given by (see, e.g., Sato, 1999) 𝐸[𝑒−𝜆𝑇𝑡] = ∫[0,∞) 𝑒 −𝜆𝑠𝑞𝑡(𝑑𝑠) = 𝑒−𝜙(𝜆)𝑡, 𝜙(𝜆) = 𝛾𝜆 + ∫(0,∞)(1 − 𝑒 −𝜆𝑠)𝜈(𝑑𝑠), (4.1) where 𝑞𝑡 is the distribution of 𝑇𝑡, and 𝛾 ≥ 0 is the drift of the subordinator.”
MCMC ESTIMATION OF LÉVY JUMP MODELS USING STOCK AND OPTION PRICES (2011, methodology): “7 Other studies that estimate Lévy processes using underlying or option prices include Barndorff-Nielsen and Shephard (2004), Belomestny and Reiss (2006), Cont and Tankov (2004a), Griffin and Steel (2006), among others.”
Optimal dividends problem with a terminal value for spectrally positive Levy processes (2013, unknown): “For more details on spectrally positive Lévy processes, the reader is referred to Bertoin (1996), Sato (1999) and Kyprianou (2006).”
Estimates of transition densities and their derivatives for jump Lévy processes (2013, background): “, in Section 26 of [26], where estimates of tails of corresponding distributions are included.”
On Gerber–Shiu functions and optimal dividend distribution for a Lévy risk process in the presence of a penalty function (2011, background): “Provided the value function v∗ were sufficiently regular, it would hold at level x of the reserves that Ex[e−q(t∧T −0 )v∗(Xt∧T −0 )] = v∗(x) + t ( v∗(x) − qv∗(x)) + o(t) for t ↘ 0, where T −0 = inf{t ≥ 0 :Xt 0}, and denotes the infinitesimal generator of the Feller semi-group of X which acts on f ”
Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations (2009, background): “For definition and properties of Lévy processes, consult [1] or [15].”
Asymptotic properties of Brownian motion delayed by inverse subordinators (2013, background): “It is known [53] that S1/2(τ ) is the 1/2-stable subordinator with Laplace transform”
A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations (2011, methodology): “For a general account on Lévy processes we refer the reader to the books by [Ber98] and [Sat99].”
Fluctuations of Lévy Processes with Applications (2014, background): “There can be no doubt, particularly to the more experienced reader, that the current text has been heavily influenced by the outstanding books of Bertoin (1996) and Sato (1999), and especially the former which also takes a predominantly pathwise approach to its content.”
On magnitude, asymptotics and duration of drawdowns for L\'{e}vy models (2015, background): “For any fixed t > 0, by Theorem 27.7 of Sato [33], we know that the law of X̃t is absolute continuous with respect to the Lebesgue measure, so is the law of Xt from the property of convolutions.”
Non-life Insurance Mathematics (2014, background): “Let z1, z2 be two bounded measurable solutions of (7).”