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Support of a Marcus equation in Dimension 1
2000
Electronic Communications in Probability
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Compared to the immense literature on diffusion processes, the mathematical corpus dealing with stochastic differential equations of jump-type remains somewhat poor, and it is of course very tempting to ...
the Marcus equation as a certain locally Lipschitz functional of the driving Lévy process, whose support in the Skorohod space was already described in [13] . ...
Let a : R → R be a locally Lipschitz function with linear growth and b : R → R a C 1 function with bounded and locally Lipschitz derivative. ...
doi:10.1214/ecp.v5-1028
fatcat:orr7eop6qjgx7lqkxluil4745q
Page 4028 of Mathematical Reviews Vol. , Issue 2001F
[page]
2001
Mathematical Reviews
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By means of the comparison theorem for solutions of the BSDE with jumps, a viscosity solution of a generalized Hamilton-Jacobi-Bellman equation is obtained. ...
We prove that under smooth- ness, independence, controllability and coercivity conditions at a reference solution of the continuous problem, there exists a locally unique solution to the Euler approximation ...
Page 7802 of Mathematical Reviews Vol. , Issue 2003j
[page]
2003
Mathematical Reviews
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The authors give sufficient conditions for strong mean-square convergence of certain numerical solutions of stochastic differ- ential equations to their exact solutions. ...
“Successively we propose a local method consisting of several local least squares approximations in which the forms of the local fitting surfaces are Gaussian functions. ...
An extension of the Yamada-Watanabe condition for pathwise uniqueness to stochastic differential equations with jumps
[article]
2009
arXiv
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We extend the Yamada-Watanabe condition for pathwise uniqueness to stochastic differential equations with jumps, in the special case where small jumps are summable. ...
For s ≥ 0 and in restriction to the event {T 1 + s < T 2 }, representation v) of a solution X ′ to equation
2 ( 2 on the stochastic interval [[0, T 2 ]].iii) The same argument as in ii) works successively ...
These are general conditions needed to deal with solutions of SDE with jumps. In the restricted setting (9) where small jumps are summable, we have the following result. ...
arXiv:0906.1699v3
fatcat:pcbzpbgeeban5l4yvmj73tcoee
Convergence Rate of Numerical Solutions for Nonlinear Stochastic Pantograph Equations with Markovian Switching and Jumps
2013
Abstract and Applied Analysis
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The sufficient conditions of existence and uniqueness of the solutions for nonlinear stochastic pantograph equations with Markovian switching and jumps are given. ...
For nonlinear stochastic pantograph equations with Markovian switching and pure jumps, it is best to use the mean-square convergence, and the order of mean-square convergence is close to 1/2. ...
Acknowledgments This work was supported by the National Natural Science Foundation of China (nos. 61104062 and 61174077), Jiangsu Qing Lan Project, and PAPD. ...
doi:10.1155/2013/420648
fatcat:vk5tzoi2ufcw3d4cqrvnfwzedy
Existence and stability of mild solutions to parabolic stochastic partial differential equations driven by Lévy space-time noise
2016
Electronic Journal of Qualitative Theory of Differential Equations
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equations driven by Lévy space-time noise under the local/non-Lipschitz condition. ...
by Lévy space-time noise under the local/non-Lipschitz condition. ...
This research has been supported in part by the NSFC (Grant No. 11271115), by the Doctoral Fund of Ministry of Education of China, and by the Hunan Provincial Natural Science Foundation (Grant No. 14JJ1025 ...
doi:10.14232/ejqtde.2016.1.53
fatcat:d265m3r3vjga7kxb5ihojwenba
On the Solution of Locally Lipschitz BSDE Associated to Jump Markov Process
[article]
2018
arXiv
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In this study, we consider a class of backward SDE driven by jump Markov process. An existence and uniqueness result to this kind of equations is obtained in a locally Lipschitz case. ...
Then, we show, by passing to the limits, the existence, and uniqueness of a solution to the initial problem. After that, a stability theorem is also proved in the local Lipschitz setting. ...
Introduction The history of backward stochastic differential equations driven by continuous Brownian motion goes back to the work of J.M. Bismut [8] , in 1973. ...
arXiv:1812.09723v1
fatcat:hxwoqygejzdehoemdgt3rbvq7m
On the asymptotic stability and numerical analysis of solutions to nonlinear stochastic differential equations with jumps
2016
Journal of Computational and Applied Mathematics
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The authors would also like to thank the Royal Society of Edinburgh, the National Natural Science ...
Acknowledgements The authors would like to thank the referees for their valuable comments and suggestions. ...
convergence of the approximate solution to the true solution of equation (2.1) under the local Lipschitz and nonlinear growth condition, so we generalize and improve the corresponding results of [6] ...
doi:10.1016/j.cam.2016.01.020
fatcat:mafwk3whpfayvlqzy6nvhdkwne
Almost sure convergence of numerical approximations for Piecewise Deterministic Markov Processes
[article]
2011
arXiv
pre-print
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The stochastic problem of simulating the random, path-dependent jump times of such processes is reformulated as a hitting time problem for a system of ordinary differential equations with random threshold ...
We show that the almost sure asymptotic convergence rate of the stochastic algorithm is identical to the order of the embedded deterministic method. ...
of networks with random switching" (1349/50021880). ...
arXiv:1112.1190v1
fatcat:qrsmy3zp2feylclrxbzmuhk32e
Neutral stochastic functional differential equations with Lévy jumps under the local Lipschitz condition
2017
Advances in Difference Equations
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In this paper, a general neutral stochastic functional differential equations with infinite delay and Lévy jumps (NSFDEwLJs) is studied. ...
We investigate the existence and uniqueness of solutions to NSFDEwLJs at the phase space C g under the local Carathéodory type conditions. ...
Acknowledgements The authors would like to thank the Edinburgh Mathematical Society (RKES130172) and the National Natural Science Foundation of China under NSFC grant (11401261, 11471071) for their financial ...
doi:10.1186/s13662-017-1102-9
fatcat:zs2c4w57x5aajguym7lkuutiiu
Mixed Neutral Caputo Fractional Stochastic Evolution Equations with Infinite Delay: Existence, Uniqueness and Averaging Principle
2022
Fractal and Fractional
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The aim of this article is to consider a class of neutral Caputo fractional stochastic evolution equations with infinite delay (INFSEEs) driven by fractional Brownian motion (fBm) and Poisson jumps in ...
First, we establish the local and global existence and uniqueness theorems of mild solutions for the aforementioned neutral fractional stochastic system under local and global Carathéodory conditions by ...
[39] derived the existence and optimal control for delay neutral fractional stochastic differential equations (NFSDEs) driven with Poisson jumps by using successive approximations under non-Lipschitz ...
doi:10.3390/fractalfract6020105
fatcat:one2cpq2xrgvvn5w3ff3liaa6y
The existence and uniqueness of mild solutions to stochastic differential equations with Lévy noise
2017
Advances in Difference Equations
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In this paper, we study a class of neutral stochastic differential equations (NSDEs) with the cylindrical Brownian motion and Lévy noises in an infinite-dimensional Hilbert space. ...
The existence and uniqueness of the mild solutions to these stochastic differential equations are discussed under assumptions of linear growth on the coefficients. The results of Taniguchi (J. Math. ...
of the paper. ...
doi:10.1186/s13662-017-1224-0
fatcat:yyt6zqzmuzbvdfzssv4umsoxlm
Mixed Caputo Fractional Neutral Stochastic Differential Equations with Impulses and Variable Delay
2021
Fractal and Fractional
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We utilized the Carathéodory approximation approach and stochastic calculus to present the existence and uniqueness theorem of the stochastic system under Carathéodory-type conditions with Lipschitz and ...
In this manuscript, a new class of impulsive fractional Caputo neutral stochastic differential equations with variable delay (IFNSDEs, in short) perturbed by fractional Brownain motion (fBm) and Poisson ...
Acknowledgments: We are very grateful to the anonymous referees for their suggestions and valuable advice. The Deanship of Scientific Research of King Abdulaziz University (No. ...
doi:10.3390/fractalfract5040239
fatcat:f37o5yikhrcxhmzvmotuvoqrw4
Direct simulation of the infinitesimal dynamics of semi-discrete approximations for convection–diffusion–reaction problems
2010
Mathematics and Computers in Simulation
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In this paper a scheme for approximating solutions of convection-diffusionreaction equations by Markov jump processes is studied. ...
The general principle of the method of lines reduces evolution partial differential equations to semi-discrete approximations consisting of systems of ordinary differential equations. ...
Acknowledgement The author thanks an anonymous referee for useful comments and observations which helped improving the clarity of the exposition. ...
doi:10.1016/j.matcom.2010.09.005
fatcat:7f3jeabzkvb4tjdwbnbnpi62zy
On the existence of weak solutions to stochastic Volterra equations
[article]
2023
arXiv
pre-print
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2023 pre-print arXiv:2207.01367v3 fatcat:jbl7zi5cxvbznd7gscxp2hen5eOther Versions
The existence of weak solutions is established for stochastic Volterra equations with time-inhomogeneous coefficients allowing for general kernels in the drift and convolutional or bounded kernels in the ...
The presented approach is based on a newly formulated local martingale problem associated to stochastic Volterra equations. ...
In contrast, [AJLP19] uses an approximation of the driving noise by pure jump processes with finite activity, which allows to treat convolutional SVEs with jumps. ...
arXiv:2207.01367v4
fatcat:hwaxw7r2crgyvlvmxnask5ruzu
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