scholarly journals Asymptotic Expansion for the Transition Densities of Stochastic Differential Equations Driven by the Gamma Processes

2020 ◽  
Author(s):  
Fan Jiang ◽  
Xin Zang ◽  
Jingping Yang
Keyword(s):  
Asymptotic Expansion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Transition Densities ◽  
Gamma Processes

scholarly journals ASYMPTOTIC EXPANSION FOR THE TRANSITION DENSITIES OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY THE GAMMA PROCESSES

2020 ◽  
pp. 1-32
Author(s):  
Fan Jiang ◽  
Xin Zang ◽  
Jingping Yang
Keyword(s):  
Asymptotic Expansion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Closed Form ◽  
Diffusion Processes ◽  
Transition Density ◽  
Maximum Relative Error ◽  
Dirac Delta ◽  
Transition Densities ◽  
Gamma Processes

In this paper, enlightened by the asymptotic expansion methodology developed by Li [(2013). Maximum-likelihood estimation for diffusion processes via closed-form density expansions. Annals of Statistics 41: 1350–1380] and Li and Chen [(2016). Estimating jump-diffusions using closed-form likelihood expansions. Journal of Econometrics 195(1): 51–70], we propose a Taylor-type approximation for the transition densities of the stochastic differential equations (SDEs) driven by the gamma processes, a special type of Lévy processes. After representing the transition density as a conditional expectation of Dirac delta function acting on the solution of the related SDE, the key technical method for calculating the expectation of multiple stochastic integrals conditional on the gamma process is presented. To numerically test the efficiency of our method, we examine the pure jump Ornstein–Uhlenbeck model and its extensions to two jump-diffusion models. For each model, the maximum relative error between our approximated transition density and the benchmark density obtained by the inverse Fourier transform of the characteristic function is sufficiently small, which shows the efficiency of our approximated method.

scholarly journals Some Applications of Lévy Processes to Stochastic Investment Models for Actuarial Use

1998 ◽  
Vol 28 (1) ◽  
pp. 77-93 ◽  
Author(s):  
Terence Chan
Keyword(s):  
Time Series ◽  
Brownian Motion ◽  
Differential Equations ◽  
White Noise ◽  
Stochastic Differential Equations ◽  
Continuous Time ◽  
Investment Model ◽  
Stable Processes ◽  
Gamma Processes ◽  
Investment Models

AbstractThis paper presents a continuous time version of a stochastic investment model originally due to Wilkie. The model is constructed via stochastic differential equations. Explicit distributions are obtained in the case where the SDEs are driven by Brownian motion, which is the continuous time analogue of the time series with white noise residuals considered by Wilkie. In addition, the cases where the driving “noise” are stable processes and Gamma processes are considered.

scholarly journals Dynamics of Markov chains and stable manifolds for random diffeomorphisms

1987 ◽  
Vol 7 (3) ◽  
pp. 351-374 ◽  
Author(s):  
M. Brin ◽  
Yu. Kifer
Keyword(s):  
Markov Chain ◽  
Differential Equations ◽  
Markov Chains ◽  
Stochastic Differential Equations ◽  
Compact Manifold ◽  
Diffusion Processes ◽  
Global Dynamics ◽  
Continuous Transition ◽  
Common Distribution ◽  
Transition Densities

AbstractWe consider the Markov chain on a compact manifold M generated by a sequence of random diffeomorphisms, i.e. a sequence of independent Diff2(M)-valued random variables with common distribution. Random diffeomorphisms appear for instance when diffusion processes are considered as solutions of stochastic differential equations. We discuss the global dynamics of Markov chains with continuous transition densities and construct non-random stable foliations for random diffeomorphisms.

Stochastic Differential Equations for Power Law Behaviors

2012 ◽  
Author(s):  
Bo Jiang ◽  
Roger Brockett ◽  
Weibo Gong ◽  
Don Towsley
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Power Law

scholarly journals Strong Solution Existence for a Class of Degenerate Stochastic Differential Equations

2020 ◽  
Vol 53 (2) ◽  
pp. 2220-2224
Author(s):  
William M. McEneaney ◽  
Hidehiro Kaise ◽  
Peter M. Dower ◽  
Ruobing Zhao
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Strong Solution ◽  
Solution Existence
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