scholarly journals Contrast estimation for noisy observations of diffusion processes via closed-form density expansions

2021 ◽  
Author(s):  
Salima El Kolei ◽  
Fabien Navarro
Keyword(s):  
Closed Form ◽  
Diffusion Processes ◽  
Contrast Estimation ◽  
Density Expansions

A note on the Volterra integral equation for the first-passage-time probability density

1995 ◽  
Vol 32 (03) ◽  
pp. 635-648 ◽  
Author(s):  
R. Gutiérrez Jáimez ◽  
P. Román Román ◽  
F. Torres Ruiz
Keyword(s):  
Integral Equation ◽  
Closed Form ◽  
Volterra Integral Equation ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
Closed Form Solutions ◽  
Actual Use ◽  
Probability Densities

In this paper we prove the validity of the Volterra integral equation for the evaluation of first-passage-time probability densities through varying boundaries, given by Buonocore et al. [1], for the case of diffusion processes not necessarily time-homogeneous. We study, specifically those processes that can be obtained from the Wiener process in the sense of [5]. A study of the kernel of the integral equation, in the same way as that by Buonocore et al. [1], is done. We obtain the boundaries for which closed-form solutions of the integral equation, without having to solve the equation, can be obtained. Finally, a few examples are given to indicate the actual use of our method.

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.

On some diffusion approximations to queueing systems

1986 ◽  
Vol 18 (4) ◽  
pp. 991-1014 ◽  
Author(s):  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Closed Form ◽  
Busy Period ◽  
Queueing System ◽  
Diffusion Processes ◽  
Queueing Systems ◽  
Diffusion Approximations ◽  
Transient Behaviour ◽  
Closed Form Solutions ◽  
Continuous Models ◽  
Reflecting Boundaries

For a class of models of adaptive queueing systems an exact diffusion approximation is derived with the aim of obtaining information on the evolution of the systems. Our approximating diffusion process includes the Wiener and the Ornstein–Uhlenbeck processes with reflecting boundaries at 0. The goodness of the approximations is thoroughly discussed and the closed-form solutions obtained for the diffusion processes are compared with those holding for the queueing system in order to investigate the conditions under which reliable information can be obtained from the approximating continuous models. For the latter the transient behaviour is quantitatively analysed and the distribution of the busy period is determined in closed form.

scholarly journals Bounded Variation Control of Itô Diffusions with Exogenously Restricted Intervention Times

2014 ◽  
Vol 46 (01) ◽  
pp. 102-120
Author(s):  
Jukka Lempa
Keyword(s):  
Poisson Process ◽  
Closed Form ◽  
Bounded Variation ◽  
Diffusion Processes ◽  
Present Value ◽  
One Dimensional ◽  
Controlled Diffusion ◽  
Dimensional Diffusion ◽  
Payoff Structure ◽  
Independent Poisson Process

In this paper, bounded variation control of one-dimensional diffusion processes is considered. We assume that the agent is allowed to control the diffusion only at the jump times of an observable, independent Poisson process. The agent's objective is to maximize the expected present value of the cumulative payoff generated by the controlled diffusion over its lifetime. We propose a relatively weak set of assumptions on the underlying diffusion and the instantaneous payoff structure, under which we solve the problem in closed form. Moreover, we illustrate the main results with an explicit example.

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