scholarly journals Malliavin Differentiability of Solutions of SPDEs with Lévy White Noise

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Raluca M. Balan ◽  
Cheikh B. Ndongo
Keyword(s):  
White Noise ◽  
Wave Equations ◽  
Stochastic Partial Differential Equation ◽  
Stochastic Integral ◽  
Random Measure ◽  
Poisson Random Measure ◽  
Stochastic Integral Equation ◽  
Lévy White Noise ◽  
Differentiability Of Solutions ◽  
Compensated Poisson Random Measure

We consider a stochastic partial differential equation (SPDE) driven by a Lévy white noise, with Lipschitz multiplicative term σ. We prove that, under some conditions, this equation has a unique random field solution. These conditions are verified by the stochastic heat and wave equations. We introduce the basic elements of Malliavin calculus with respect to the compensated Poisson random measure associated with the Lévy white noise. If σ is affine, we prove that the solution is Malliavin differentiable and its Malliavin derivative satisfies a stochastic integral equation.

EXISTENCE, UNIQUENESS AND REGULARITY w.r.t. THE INITIAL CONDITION OF MILD SOLUTIONS OF SPDEs DRIVEN BY POISSON NOISE

2010 ◽  
Vol 13 (01) ◽  
pp. 133-163 ◽  
Author(s):  
CLAUDIA INGRID PRÉVÔT
Keyword(s):  
Mild Solution ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
Separable Hilbert Space ◽  
Mild Solutions ◽  
Random Measure ◽  
Initial Condition ◽  
Poisson Random Measure ◽  
Unique Mild Solution ◽  
Compensated Poisson Random Measure

In this paper we investigate stochastic partial differential equations in a separable Hilbert space driven by a compensated Poisson random measure. Our interest is directed towards the existence and uniqueness of mild solutions and their regularity w.r.t. the inital condition. We show the existence of a unique mild solution and prove the Gâteaux differentiability of the mild solution w.r.t. the initial condition. As a consequence, we obtain a gradient estimate for the Gâteaux derivative of the mild solution and for the resolvent associated to the mild solution.

scholarly journals Interconnection between Wick multiplication and integration on spaces of nonregular generalized functions in the Lévy white noise analysis

2019 ◽  
Vol 11 (1) ◽  
pp. 70-88
Author(s):  
N.A. Kachanovsky ◽  
T.O. Kachanovska
Keyword(s):  
White Noise ◽  
Stochastic Integral ◽  
Noise Analysis ◽  
Stochastic Equations ◽  
Generalized Functions ◽  
Wick Product ◽  
Pettis Integral ◽  
White Noise Analysis ◽  
Lévy White Noise

We deal with spaces of nonregular generalized functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our aim is to describe a relationship between Wick multiplication and integration on these spaces. More exactly, we show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of an extended stochastic integral; establish an analog of this result for a Pettis integral (a weak integral); and prove a theorem about a representation of the extended stochastic integral via the Pettis integral from the Wick product of the original integrand by a Lévy white noise. As examples of an application of our results, we consider some stochastic equations with Wick type nonlinearities.

Global solutions to stochastic Volterra equations driven by Lévy noise

2018 ◽  
Vol 21 (5) ◽  
pp. 1170-1202 ◽  
Author(s):  
Erika Hausenblas ◽  
Mihály Kovács
Keyword(s):  
Memory Function ◽  
Random Measure ◽  
Global Solutions ◽  
Lévy Noise ◽  
Volterra Equations ◽  
Poisson Random Measure ◽  
Intensity Measure ◽  
Nonlinear Maps ◽  
Compensated Poisson Random Measure ◽  
Levy Noise

Abstract In this paper we investigate the existence and uniqueness of semilinear stochastic Volterra equations driven by multiplicative Lévy noise of pure jump type. In particular, we consider the equation $$\begin{array}{} \left\{ \begin{aligned} du(t) & = \left( A\int_0 ^t b(t-s) u(s)\,ds\right) \, dt + F(t,u(t))\,dt \\ & {} + \int_ZG(t,u(t), z) \tilde \eta(dz,dt) + \int_{Z_L}G_L(t,u(t), z) \eta_L(dz,dt),\, t\in (0,T],\\ u(0)&=u_0, \end{aligned} \right. \end{array} $$ where Z and ZL are Banach spaces, η̃ is a time-homogeneous compensated Poisson random measure on Z with intensity measure ν (capturing the small jumps), and ηL is a time-homogeneous Poisson random measure on ZL independent to η̃ with finite intensity measure νL (capturing the large jumps). Here, A is a selfadjoint operator on a Hilbert space H, b is a scalar memory function and F, G and GL are nonlinear mappings. We provide conditions on b, F G and GL under which a unique global solution exists. We also present an example from the theory of linear viscoelasticity where our result is applicable. The specific kernel b(t) = cρtρ−2, 1 < ρ < 2, corresponds to a fractional-in-time stochastic equation and the nonlinear maps F and G can include fractional powers of A.

scholarly journals Operators of stochastic differentiation on spaces of nonregular generalized functions of Levy white noise analysis

2016 ◽  
Vol 8 (1) ◽  
pp. 83-106
Author(s):  
N.A. Kachanovsky
Keyword(s):  
White Noise ◽  
Stochastic Integral ◽  
Noise Analysis ◽  
Gaussian White Noise ◽  
Generalized Functions ◽  
Test Functions ◽  
White Noise Analysis ◽  
Lévy White Noise ◽  
Stochastic Derivative ◽  
Further Development

The operators of stochastic differentiation, which are closely related with the extended Skorohod stochastic integral and with the Hida stochastic derivative, play an important role in the classical (Gaussian) white noise analysis. In particular, these operators can be used in order to study some properties of the extended stochastic integral and of solutions of stochastic equations with Wick-type nonlinearities. During recent years the operators of stochastic differentiation were introduced and studied, in particular, in the framework of the Meixner white noise analysis, in the same way as on spaces of regular test and generalized functions and on spaces of nonregular test functions of the Levy white noise analysis. In the present paper we make the next natural step: introduce and study operators of stochastic differentiation on spaces of nonregular generalized functions of the Levy white noise analysis (i.e., on spaces of generalized functions that belong to the so-called nonregular rigging of the space of square integrable with respect to the measure of a Levy white noise functions). In so doing, we use Lytvynov's generalization of the chaotic representation property. The researches of the present paper can be considered as a contribution in a further development of the Levy white noise analysis. 

scholarly journals On operators of stochastic differentiation on spaces of regular test and generalized functions of Lévy white noise analysis

2014 ◽  
Vol 6 (2) ◽  
pp. 212-229 ◽  
Author(s):  
M.M. Dyriv ◽  
N.A. Kachanovsky
Keyword(s):  
White Noise ◽  
Stochastic Integral ◽  
Noise Analysis ◽  
Gaussian White Noise ◽  
Stochastic Equations ◽  
Generalized Functions ◽  
White Noise Analysis ◽  
Unbounded Operators ◽  
Lévy White Noise ◽  
Stochastic Derivative

The operators of stochastic differentiation, which are closely related with the extended Skorohod stochastic integral and with the Hida stochastic derivative, play an important role in the classical (Gaussian) white noise analysis. In particular, these operators can be used in order to study properties of the extended stochastic integral and of solutions of stochastic equations with Wick-type nonlinearities. In this paper we introduce and study bounded and unbounded operators of stochastic differentiation in the Levy white noise analysis. More exactly, we consider these operators on spaces from parametrized regular rigging of the space of square integrable with respect to the measure of a Levy white noise functions, using the Lytvynov's generalization of the chaotic representation property. This gives a possibility to extend to the Levy white noise analysis and to deepen the corresponding results of the classical white noise analysis.

Prediction-Correction Scheme for Decoupled Forward Backward Stochastic Differential Equations with Jumps

2016 ◽  
Vol 6 (3) ◽  
pp. 253-277 ◽  
Author(s):  
Yu Fu ◽  
Jie Yang ◽  
Weidong Zhao
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Random Measure ◽  
Backward Stochastic Differential Equations ◽  
Explicit Scheme ◽  
Poisson Random Measure ◽  
Correction Scheme ◽  
Compensated Poisson Random Measure ◽  
Theoretical Results ◽  
General Error

AbstractBy introducing a new Gaussian process and a new compensated Poisson random measure, we propose an explicit prediction-correction scheme for solving decoupled forward backward stochastic differential equations with jumps (FBSDEJs). For this scheme, we first theoretically obtain a general error estimate result, which implies that the scheme is stable. Then using this result, we rigorously prove that the accuracy of the explicit scheme can be of second order. Finally, we carry out some numerical experiments to verify our theoretical results.

scholarly journals BSDEs with Logarithmic Growth Driven by Brownian Motion and Poisson Random Measure and Connection to Stochastic Control Problem

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Khalid Oufdil
Keyword(s):  
Optimal Control ◽  
Brownian Motion ◽  
Control Problem ◽  
Stochastic Optimal Control ◽  
Random Measure ◽  
Poisson Random Measure ◽  
One Dimensional ◽  
Stochastic Optimal Control Problem ◽  
Uniqueness Of The Solution ◽  
Logarithmic Growth

Abstract In this paper, we study one-dimensional backward stochastic differential equations under logarithmic growth in the 𝑧-variable ( | z | ⁢ | ln ⁡ | z | | ) (\lvert z\rvert\sqrt{\lvert\ln\lvert z\rvert\rvert}) . We show the existence and the uniqueness of the solution when the noise is driven by a Brownian motion and an independent Poisson random measure. In addition, we highlight the connection of such BSDEs with stochastic optimal control problem, where we show the existence of an optimal strategy for the control problem.

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