Stochastic Integration and Differential Equations

Protter, Philip

doi:10.1007/978-3-662-10061-5

Cited by 1,888 publications

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“…The strong Markov property follows from the fact that we have a strong solution, the integrators are Lévy processes and the integrand functions doesn't depend on time. (See Theorem V.32 in Protter [35], where the Lipschitz continuity is just to guarantee the existence and uniqueness of the solution.) Finally, from Itô's formula it is easy to show that the infinitesimal generator of Z is given by (6).…”

Section: Proposition 1 Suppose That (B σ G H) Are Admissible Parammentioning

confidence: 99%

Branching Processes in a Lévy Random Environment

Palau

Pardo

2017

Acta Appl Math

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In this paper, we introduce branching processes in a Lévy random environment. In order to define this class of processes, we study a particular class of non-negative stochastic differential equations driven by a white noise and Poisson random measures which are mutually independent. Following similar techniques as in Dawson and Li (Ann. Probab. 40:813-857, 2012) and Li and Pu (Electron. Commun. Probab. 17(33):1-13, 2012), we obtain existence and uniqueness of strong local solutions of such stochastic equations. We use the latter result to construct continuous state branching processes with immigration and competition in a Lévy random environment as a strong solution of a stochastic differential equation. We also study the long term behaviour of two interesting examples: the case with no immigration and no competition and the case with linear growth and logistic competition.

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Section: Proposition 1 Suppose That (B σ G H) Are Admissible Parammentioning

confidence: 99%

Branching Processes in a Lévy Random Environment

Palau

Pardo

2017

Acta Appl Math

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“…Using the same argument as in the proof of Theorem 15 on page 380 of [28], we have that, for any positive test function f (·) on [0, ∞),…”

Section: Lemma 3 For M > Xmentioning

confidence: 99%

Occupation Times, Drawdowns, and Drawups for One-Dimensional Regular Diffusions

Zhang

2015

Adv. Appl. Probab.

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The drawdown process of a one-dimensional regular diffusion process X is given by X reflected at its running maximum. The drawup process is given by X reflected at its running minimum. We calculate the probability that a drawdown precedes a drawup in an exponential time-horizon. We then study the law of the occupation times of the drawdown process and the drawup process. These results are applied to address problems in risk analysis and for option pricing of the drawdown process. Finally, we present examples of Brownian motion with drift and three-dimensional Bessel processes, where we prove an identity in law.

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“…However, as τ > τ m , τ is a predictable stopping time, see Ch. III in Protter [63]. As the jumps of D are totally inaccessible, since they are induced by a (inhomogeneous) Poisson random measure, τ a.s. cannot coincide with a jump time.…”

Section: Then Property (B) Implies That There Exist Constantsmentioning

confidence: 99%

Robust Portfolio Choice and Indifference Valuation

Laeven

Stadje²

2014

Mathematics of OR

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We solve, theoretically and numerically, the problems of optimal portfolio choice and indifference valuation in a general continuous-time setting. The setting features (i) ambiguity and ambiguity averse preferences, (ii) discontinuities in the asset price processes, with a general and possibly infinite activity jump part next to a continuous diffusion part, and (iii) general and possibly non-convex trading constraints. We characterize our solutions as solutions to Backward Stochastic Differential Equations (BSDEs). We prove existence and uniqueness of the solution to the general class of BSDEs with jumps having a drift (or driver) that grows at most quadratically, encompassing the solutions to our portfolio choice and valuation problems as special cases. We provide an explicit decomposition of the excess return on an asset into a risk premium and an ambiguity premium, and a further decomposition into a piece stemming from the diffusion part and a piece stemming from the jump part. We further compute our solutions in a few examples by numerically solving the corresponding BSDEs using regression techniques.

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Stochastic Integration and Differential Equations

Cited by 1,888 publications

References 0 publications

Branching Processes in a Lévy Random Environment

Branching Processes in a Lévy Random Environment

Occupation Times, Drawdowns, and Drawups for One-Dimensional Regular Diffusions

Robust Portfolio Choice and Indifference Valuation

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