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Journal articles on the topic 'Ergodic Diffusion Processe'

Author: Grafiati

Published: 14 December 2024

Last updated: 10 July 2025

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1

Corradi, Valentina. "Comovements Between Diffusion Processes." Econometric Theory 13, no. 5 (1997): 646–66. http://dx.doi.org/10.1017/s0266466600006113.

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The aim of this paper is to characterize and analyze long-run comovements among diffusion processes. Broadly speaking, if X = (X1,,X2,;t ≥ 0) is a nonergodic diffusion in R2, but there exists a linear combination, say, γ′X, that is instead ergodic in R, then we say there exists a linear stochastic comovement between the components of X. Linear diffusions exhibiting stochastic comovements admit an error correction representation. Estimation of γ and hypothesis testing, under different sampling schemes, are considered.

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2

Kamarianakis, Yiannis. "Ergodic control of diffusion processes." Journal of Applied Statistics 40, no. 4 (2013): 921–22. http://dx.doi.org/10.1080/02664763.2012.750440.

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3

Wong, Bernard. "On Modelling Long Term Stock Returns with Ergodic Diffusion Processes: Arbitrage and Arbitrage-Free Specifications." Journal of Applied Mathematics and Stochastic Analysis 2009 (September 23, 2009): 1–16. http://dx.doi.org/10.1155/2009/215817.

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We investigate the arbitrage-free property of stock price models where the local martingale component is based on an ergodic diffusion with a specified stationary distribution. These models are particularly useful for long horizon asset-liability management as they allow the modelling of long term stock returns with heavy tail ergodic diffusions, with tractable, time homogeneous dynamics, and which moreover admit a complete financial market, leading to unique pricing and hedging strategies. Unfortunately the standard specifications of these models in literature admit arbitrage opportunities. We investigate in detail the features of the existing model specifications which create these arbitrage opportunities and consequently construct a modification that is arbitrage free.

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4

Swishchuk, Anatoliy, and M. Shafiqul Islam. "Diffusion Approximations of the Geometric Markov Renewal Processes and Option Price Formulas." International Journal of Stochastic Analysis 2010 (December 19, 2010): 1–21. http://dx.doi.org/10.1155/2010/347105.

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We consider the geometric Markov renewal processes as a model for a security market and study this processes in a diffusion approximation scheme. Weak convergence analysis and rates of convergence of ergodic geometric Markov renewal processes in diffusion scheme are presented. We present European call option pricing formulas in the case of ergodic, double-averaged, and merged diffusion geometric Markov renewal processes.

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5

Kutoyants, Yury A., and Nakahiro Yoshida. "Moment estimation for ergodic diffusion processes." Bernoulli 13, no. 4 (2007): 933–51. http://dx.doi.org/10.3150/07-bej1040.

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6

Kiessler, Peter C. "Statistical Inference for Ergodic Diffusion Processes." Journal of the American Statistical Association 101, no. 474 (2006): 846. http://dx.doi.org/10.1198/jasa.2006.s98.

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7

Chen, Mu Fa. "Ergodic theorems for reaction-diffusion processes." Journal of Statistical Physics 58, no. 5-6 (1990): 939–66. http://dx.doi.org/10.1007/bf01026558.

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8

Magdziarz, Marcin, and Aleksander Weron. "Ergodic properties of anomalous diffusion processes." Annals of Physics 326, no. 9 (2011): 2431–43. http://dx.doi.org/10.1016/j.aop.2011.04.015.

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9

Bel, Golan, and Ilya Nemenman. "Ergodic and non-ergodic anomalous diffusion in coupled stochastic processes." New Journal of Physics 11, no. 8 (2009): 083009. http://dx.doi.org/10.1088/1367-2630/11/8/083009.

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10

Di Masp, G. B., and Ł. Stettner. "Bayesian ergodic adaptive control of diffusion processes." Stochastics and Stochastic Reports 60, no. 3-4 (1997): 155–83. http://dx.doi.org/10.1080/17442509708834104.

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11

Galtchouk, L., and S. Pergamenshchikov. "Uniform concentration inequality for ergodic diffusion processes." Stochastic Processes and their Applications 117, no. 7 (2007): 830–39. http://dx.doi.org/10.1016/j.spa.2006.10.008.

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12

DEGREGORIO, A., and S. IACUS. "On Rényi information for ergodic diffusion processes." Information Sciences 179, no. 3 (2009): 279–91. http://dx.doi.org/10.1016/j.ins.2008.09.016.

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13

Menaldi, Jose-Luis, and Maurice Robin. "Ergodic switching control for diffusion-type processes." Probability, Uncertainty and Quantitative Risk 8, no. 1 (2023): 53–74. http://dx.doi.org/10.3934/puqr.2023003.

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14

Arapostathis, Ari, Guodong Pang, and Yi Zheng. "Exponential ergodicity and steady-state approximations for a class of markov processes under fast regime switching." Advances in Applied Probability 53, no. 1 (2021): 1–29. http://dx.doi.org/10.1017/apr.2020.47.

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AbstractWe study ergodic properties of a class of Markov-modulated general birth–death processes under fast regime switching. The first set of results concerns the ergodic properties of the properly scaled joint Markov process with a parameter that is taken to be large. Under very weak hypotheses, we show that if the averaged process is exponentially ergodic for large values of the parameter, then the same applies to the original joint Markov process. The second set of results concerns steady-state diffusion approximations, under the assumption that the ‘averaged’ fluid limit exists. Here, we establish convergence rates for the moments of the approximating diffusion process to those of the Markov-modulated birth–death process. This is accomplished by comparing the generator of the approximating diffusion and that of the joint Markov process. We also provide several examples which demonstrate how the theory can be applied.

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15

Kutoyants, Yury A. "On parameter estimation for switching ergodic diffusion processes." Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 330, no. 10 (2000): 925–30. http://dx.doi.org/10.1016/s0764-4442(00)00286-x.

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16

Negri, Ilia, and Yoichi Nishiyama. "Goodness of fit test for ergodic diffusion processes." Annals of the Institute of Statistical Mathematics 61, no. 4 (2008): 919–28. http://dx.doi.org/10.1007/s10463-007-0162-0.

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17

Uchida, Masayuki, and Nakahiro Yoshida. "Estimation for misspecified ergodic diffusion processes from discrete observations." ESAIM: Probability and Statistics 15 (2011): 270–90. http://dx.doi.org/10.1051/ps/2010001.

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18

Cherstvy, Andrey G., Aleksei V. Chechkin, and Ralf Metzler. "Ageing and confinement in non-ergodic heterogeneous diffusion processes." Journal of Physics A: Mathematical and Theoretical 47, no. 48 (2014): 485002. http://dx.doi.org/10.1088/1751-8113/47/48/485002.

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19

Fujii, Takayuki, and Masayuki Uchida. "AIC type statistics for discretely observed ergodic diffusion processes." Statistical Inference for Stochastic Processes 17, no. 3 (2014): 267–82. http://dx.doi.org/10.1007/s11203-014-9101-x.

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20

Geng, Ruoming. "Ergodic Foundations of Langevin-Based MCMC." International Journal of Applied Science 7, no. 2 (2024): p8. http://dx.doi.org/10.30560/ijas.v7n2p8.

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In this work, we provide a comprehensive theoretical analysis of Langevin diffusion and its applications to Markov Chain Monte Carlo (MCMC) methods. We establish the ergodicity of continuous-time Langevin diffusion processes, proving their convergence to target distributions under suitable regularity conditions. The analysis is then extended to discrete-time settings, examining the properties of the Unadjusted Langevin Algorithm (ULA) and the Metropolis-Adjusted Langevin Algorithm (MALA). Employing tools from stochastic processes, ergodic theory, and Markov chain theory, we establish strong convergence results using Foster-Lyapunov drift conditions, coupling arguments, and geometric ergodicity. The paper explores connections between Langevin diffusion and optimal transport theory, highlighting recent developments in adaptive methods, transport map accelerated MCMC, and applications to high-dimensional Bayesian inference. Our theoretical results provide insights into algorithm design, parameter tuning, and convergence diagnostics for Langevin-based MCMC methods, bridging the gap between theory and practice in the development of efficient sampling algorithms for complex probability distributions.

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21

Zhang, Wei. "Ergodic SDEs on submanifolds and related numerical sampling schemes." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 2 (2020): 391–430. http://dx.doi.org/10.1051/m2an/2019071.

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In many applications, it is often necessary to sample the mean value of certain quantity with respect to a probability measure μ on the level set of a smooth function ξ : ℝd → ℝk, 1 ≤ k < d. A specially interesting case is the so-called conditional probability measure, which is useful in the study of free energy calculation and model reduction of diffusion processes. By Birkhoff’s ergodic theorem, one approach to estimate the mean value is to compute the time average along an infinitely long trajectory of an ergodic diffusion process on the level set whose invariant measure is μ. Motivated by the previous work of Ciccotti et al. (Commun. Pur. Appl. Math. 61 (2008) 371–408), as well as the work of Leliévre et al. (Math. Comput. 81 (2012) 2071–2125), in this paper we construct a family of ergodic diffusion processes on the level set of ξ whose invariant measures coincide with the given one. For the conditional measure, we propose a consistent numerical scheme which samples the conditional measure asymptotically. The numerical scheme doesn’t require computing the second derivatives of ξ and the error estimates of its long time sampling efficiency are obtained.

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22

Kutoyants, Yury A. "On the goodness-of-fit testing for ergodic diffusion processes." Journal of Nonparametric Statistics 22, no. 4 (2010): 529–43. http://dx.doi.org/10.1080/10485250903359564.

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23

Galtchouk, L. I., and S. M. Pergamenshchikov. "Adaptive sequential estimation for ergodic diffusion processes in quadratic metric." Journal of Nonparametric Statistics 23, no. 2 (2011): 255–85. http://dx.doi.org/10.1080/10485252.2010.544307.

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24

Jasso-Fuentes, Héctor, and Onésimo Hernández-Lerma. "Optimal ergodic control of Markov diffusion processes with minimum variance." Stochastics 85, no. 6 (2012): 929–45. http://dx.doi.org/10.1080/17442508.2012.688973.

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25

Wang, Xudong, Weihua Deng, and Yao Chen. "Ergodic properties of heterogeneous diffusion processes in a potential well." Journal of Chemical Physics 150, no. 16 (2019): 164121. http://dx.doi.org/10.1063/1.5090594.

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26

Fujii, Takayuki. "An extension of cusp estimation problem in ergodic diffusion processes." Statistics & Probability Letters 80, no. 9-10 (2010): 779–83. http://dx.doi.org/10.1016/j.spl.2010.01.010.

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27

Negri, Ilia. "On efficient estimation of invariant density for ergodic diffusion processes." Statistics & Probability Letters 51, no. 1 (2001): 79–85. http://dx.doi.org/10.1016/s0167-7152(00)00147-4.

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28

Stojkoski, Viktor, Trifce Sandev, Ljupco Kocarev, and Arnab Pal. "Autocorrelation functions and ergodicity in diffusion with stochastic resetting." Journal of Physics A: Mathematical and Theoretical 55, no. 10 (2022): 104003. http://dx.doi.org/10.1088/1751-8121/ac4ce9.

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Abstract Diffusion with stochastic resetting is a paradigm of resetting processes. Standard renewal or master equation approach are typically used to study steady state and other transport properties such as average, mean squared displacement etc. What remains less explored is the two time point correlation functions whose evaluation is often daunting since it requires the implementation of the exact time dependent probability density functions of the resetting processes which are unknown for most of the problems. We adopt a different approach that allows us to write a stochastic solution for a single trajectory undergoing resetting. Moments and the autocorrelation functions between any two times along the trajectory can then be computed directly using the laws of total expectation. Estimation of autocorrelation functions turns out to be pivotal for investigating the ergodic properties of various observables for this canonical model. In particular, we investigate two observables (i) sample mean which is widely used in economics and (ii) time-averaged-mean-squared-displacement (TAMSD) which is of acute interest in physics. We find that both diffusion and drift–diffusion processes with resetting are ergodic at the mean level unlike their reset-free counterparts. In contrast, resetting renders ergodicity breaking in the TAMSD while both the stochastic processes are ergodic when resetting is absent. We quantify these behaviors with detailed analytical study and corroborate with extensive numerical simulations. Our results can be verified in experimental set-ups that can track single particle trajectories and thus have strong implications in understanding the physics of resetting.

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29

Suciu, N., C. Vamoş, H. Vereecken, K. Sabelfeld, and P. Knabner. "Itô equation model for dispersion of solutes in heterogeneous media." Journal of Numerical Analysis and Approximation Theory 37, no. 2 (2008): 221–38. http://dx.doi.org/10.33993/jnaat372-895.

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Transport processes in heterogeneous media such as ionized plasmas, natural porous media, and turbulent atmosphere are often modeled as diffusion processes in random velocity fields. Using the Itô formalism, we decompose the second spatial moments of the concentration and the equivalent effective dispersion coefficients in terms corresponding to various physical factors which influence the transport. We explicitly define "ergodic'' dispersion coefficients, independent of the initial conditions and completely determined by local dispersion coefficients and velocity correlations. Ergodic coefficients govern an upscaled process which describes the transport at large tine-space scales. The non-ergodic behavior at finite times shown by numerical experiments for large initial plumes is explained by "memory terms'' accounting for correlations between initial positions and velocity fluctuations on the trajectories of the solute molecules.

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30

Mao, Yong-Hua, and Tao Wang. "Lyapunov-type conditions for non-strong ergodicity of Markov processes." Journal of Applied Probability 58, no. 1 (2021): 238–53. http://dx.doi.org/10.1017/jpr.2020.84.

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AbstractWe present Lyapunov-type conditions for non-strong ergodicity of Markov processes. Some concrete models are discussed, including diffusion processes on Riemannian manifolds and Ornstein–Uhlenbeck processes driven by symmetric $\alpha$-stable processes. In particular, we show that any process of d-dimensional Ornstein–Uhlenbeck type driven by $\alpha$-stable noise is not strongly ergodic for every $\alpha\in (0,2]$.

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31

Ivanov, Leonid M., Collins A. Collins, and Tetyana Margolina. "Reconstruction of Diffusion Coefficients and Power Exponents from Single Lagrangian Trajectories." Fluids 6, no. 3 (2021): 111. http://dx.doi.org/10.3390/fluids6030111.

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Using discrete wavelets, a novel technique is developed to estimate turbulent diffusion coefficients and power exponents from single Lagrangian particle trajectories. The technique differs from the classical approach (Davis (1991)’s technique) because averaging over a statistical ensemble of the mean square displacement (<X2>) is replaced by averaging along a single Lagrangian trajectory X(t) = {X(t), Y(t)}. Metzler et al. (2014) have demonstrated that for an ergodic (for example, normal diffusion) flow, the mean square displacement is <X2> = limT→∞τX2(T,s), where τX2 (T, s) = 1/(T − s) ∫0T−s(X(t+Δt) − X(t))2 dt, T and s are observational and lag times but for weak non-ergodic (such as super-diffusion and sub-diffusion) flows <X2> = limT→∞≪τX2(T,s)≫, where ≪…≫ is some additional averaging. Numerical calculations for surface drifters in the Black Sea and isobaric RAFOS floats deployed at mid depths in the California Current system demonstrated that the reconstructed diffusion coefficients were smaller than those calculated by Davis (1991)’s technique. This difference is caused by the choice of the Lagrangian mean. The technique proposed here is applied to the analysis of Lagrangian motions in the Black Sea (horizontal diffusion coefficients varied from 105 to 106 cm2/s) and for the sub-diffusion of two RAFOS floats in the California Current system where power exponents varied from 0.65 to 0.72. RAFOS float motions were found to be strongly non-ergodic and non-Gaussian.

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32

Wen, Bao, Ming-Gen Li, Jian Liu, and Jing-Dong Bao. "Ergodic Measure and Potential Control of Anomalous Diffusion." Entropy 25, no. 7 (2023): 1012. http://dx.doi.org/10.3390/e25071012.

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In statistical mechanics, the ergodic hypothesis (i.e., the long-time average is the same as the ensemble average) accompanying anomalous diffusion has become a continuous topic of research, being closely related to irreversibility and increasing entropy. While measurement time is finite for a given process, the time average of an observable quantity might be a random variable, whose distribution width narrows with time, and one wonders how long it takes for the convergence rate to become a constant. This is also the premise of ergodic establishment, because the ensemble average is always equal to the constant. We focus on the time-dependent fluctuation width for the time average of both the velocity and kinetic energy of a force-free particle described by the generalized Langevin equation, where the stationary velocity autocorrelation function is considered. Subsequently, the shortest time scale can be estimated for a system transferring from a stationary state to an effective ergodic state. Moreover, a logarithmic spatial potential is used to modulate the processes associated with free ballistic diffusion and the control of diffusion, as well as the minimal realization of the whole power-law regime. The results presented suggest that non-ergodicity mimics the sparseness of the medium and reveals the unique role of logarithmic potential in modulating diffusion behavior.

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33

Mao, Yong-Hua. "Strong ergodicity for Markov processes by coupling methods." Journal of Applied Probability 39, no. 4 (2002): 839–52. http://dx.doi.org/10.1239/jap/1037816023.

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In this paper, we apply coupling methods to study strong ergodicity for Markov processes, and sufficient conditions are presented in terms of the expectations of coupling times. In particular, explicit criteria are obtained for one-dimensional diffusions and birth-death processes to be strongly ergodic. As a by-product, strong ergodicity implies that the essential spectra of the generators for these processes are empty.

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34

Mao, Yong-Hua. "Strong ergodicity for Markov processes by coupling methods." Journal of Applied Probability 39, no. 04 (2002): 839–52. http://dx.doi.org/10.1017/s0021900200022087.

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In this paper, we apply coupling methods to study strong ergodicity for Markov processes, and sufficient conditions are presented in terms of the expectations of coupling times. In particular, explicit criteria are obtained for one-dimensional diffusions and birth-death processes to be strongly ergodic. As a by-product, strong ergodicity implies that the essential spectra of the generators for these processes are empty.

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35

Metzler, Ralf. "Weak ergodicity breaking and ageing in anomalous diffusion." International Journal of Modern Physics: Conference Series 36 (January 2015): 1560007. http://dx.doi.org/10.1142/s2010194515600071.

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Modern single particle tracking techniques and many large scale simulations produce time series r(t) of the position of a tracer particle. Standardly these are evaluated in terms of the time averaged mean squared displacement. For ergodic processes such as Brownian motion, one can interpret the results of such an analysis in terms of the known theories for the corresponding ensemble averaged mean squared displacement, if only the measurement time is sufficiently long. In anomalous diffusion processes, that are widely observed over many orders of magnitude, the equivalence between (long) time and ensemble averages may be broken (weak ergodicity breaking). In such cases the time averages may no longer be interpreted in terms of ensemble theories. Here we collect some recent results on weakly non-ergodic systems with respect to the time averaged mean squared displacement and the inherent irreproducibility of individual measurements. We also address the phenomenon of ageing, the dependence of physical observables on the time span between initial preparation of the system and the start of the measurement.

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36

Кутоянц, Юрий Артемович, Yurii Artemovich Kutoyants, Ilia Negri, and Ilia Negri. "On $L_2$ Efficiency of an Empiric Distribution for Ergodic Diffusion Processes." Teoriya Veroyatnostei i ee Primeneniya 46, no. 1 (2001): 164–69. http://dx.doi.org/10.4213/tvp4034.

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37

Kutoyants, Yu A., and I. Negri. "On L2 Efficiency of an Empiric Distribution for Ergodic Diffusion Processes." Theory of Probability & Its Applications 46, no. 1 (2002): 140–46. http://dx.doi.org/10.1137/s0040585x97978816.

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38

Jasso-Fuentes, Héctor, and Onésimo Hernández-Lerma. "Ergodic Control, Bias, and Sensitive Discount Optimality for Markov Diffusion Processes." Stochastic Analysis and Applications 27, no. 2 (2009): 363–85. http://dx.doi.org/10.1080/07362990802679034.

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39

Galtchouk, L., and S. Pergamenshchikov. "Uniform concentration inequality for ergodic diffusion processes observed at discrete times." Stochastic Processes and their Applications 123, no. 1 (2013): 91–109. http://dx.doi.org/10.1016/j.spa.2012.09.004.

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40

Kitagawa, Hayato, and Masayuki Uchida. "Adaptive test statistics for ergodic diffusion processes sampled at discrete times." Journal of Statistical Planning and Inference 150 (July 2014): 84–110. http://dx.doi.org/10.1016/j.jspi.2014.03.003.

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41

Uchida, Masayuki. "Contrast-based information criterion for ergodic diffusion processes from discrete observations." Annals of the Institute of Statistical Mathematics 62, no. 1 (2009): 161–87. http://dx.doi.org/10.1007/s10463-009-0245-1.

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42

Uchida, Masayuki, and Nakahiro Yoshida. "Adaptive Bayes type estimators of ergodic diffusion processes from discrete observations." Statistical Inference for Stochastic Processes 17, no. 2 (2014): 181–219. http://dx.doi.org/10.1007/s11203-014-9095-4.

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43

Kutoyants, Yury A. "On asymptotic distribution of parameter free tests for ergodic diffusion processes." Statistical Inference for Stochastic Processes 17, no. 2 (2014): 139–61. http://dx.doi.org/10.1007/s11203-014-9097-2.

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44

Kaino, Yusuke, Shogo H. Nakakita, and Masayuki Uchida. "Hybrid estimation for ergodic diffusion processes based on noisy discrete observations." Statistical Inference for Stochastic Processes 23, no. 1 (2019): 171–98. http://dx.doi.org/10.1007/s11203-019-09203-2.

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45

BHATTACHARYA, RABI, and ARAMIAN WASIELAK. "ON THE SPEED OF CONVERGENCE OF MULTIDIMENSIONAL DIFFUSIONS TO EQUILIBRIUM." Stochastics and Dynamics 12, no. 01 (2012): 1150003. http://dx.doi.org/10.1142/s0219493712003638.

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We obtain new criteria for polynomial rates of convergence of ergodic multidimensional diffusions to equilibrium. For this, (1) a method is provided for adapting to continuous-time Markov processes coupling techniques for discrete parameter Harris processes, and (2) estimates of moments of return times to a ball are derived.

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46

Monthus, Cécile. "Large deviations for the Pearson family of ergodic diffusion processes involving a quadratic diffusion coefficient and a linear force." Journal of Statistical Mechanics: Theory and Experiment 2023, no. 8 (2023): 083204. http://dx.doi.org/10.1088/1742-5468/ace431.

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Abstract The Pearson family of ergodic diffusions with a quadratic diffusion coefficient and a linear force is characterized by explicit dynamics of their integer moments and by explicit relaxation of spectral properties towards their steady state. Besides the Ornstein–Uhlenbeck process with a Gaussian steady state, other representative examples of the Pearson family are the square root or the Cox–Ingersoll–Ross process converging towards the gamma distribution, the Jacobi process converging towards the beta distribution, the reciprocal gamma process (corresponding to an exponential functional of the Brownian motion) that converges towards the inverse-gamma distribution, the Fisher–Snedecor process and the Student process. The last three steady states display heavy tails. The goal of the present paper is to analyze the large deviation properties of these various diffusion processes in a unified framework. We first consider level 1 concerning time-averaged observables over a large time window T. We write the first rescaled cumulants for generic observables and identify specific observables whose large deviations can be explicitly computed from the dominant eigenvalue of the appropriate deformed generator. The explicit large deviations at level 2 concerning the time-averaged density are then used to analyze the statistical inference of model parameters from data on a very long stochastic trajectory in order to obtain the explicit rate function for the two inferred parameters of the Pearson linear force.

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47

Benfatto, Maurizio, Elisabetta Pace, Catalina Curceanu, et al. "Biophotons and Emergence of Quantum Coherence—A Diffusion Entropy Analysis." Entropy 23, no. 5 (2021): 554. http://dx.doi.org/10.3390/e23050554.

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We study the emission of photons from germinating seeds using an experimental technique designed to detect light of extremely small intensity. We analyze the dark count signal without germinating seeds as well as the photon emission during the germination process. The technique of analysis adopted here, called diffusion entropy analysis (DEA) and originally designed to measure the temporal complexity of astrophysical, sociological and physiological processes, rests on Kolmogorov complexity. The updated version of DEA used in this paper is designed to determine if the signal complexity is generated either by non-ergodic crucial events with a non-stationary correlation function or by the infinite memory of a stationary but non-integrable correlation function or by a mixture of both processes. We find that dark count yields the ordinary scaling, thereby showing that no complexity of either kinds may occur without any seeds in the chamber. In the presence of seeds in the chamber anomalous scaling emerges, reminiscent of that found in neuro-physiological processes. However, this is a mixture of both processes and with the progress of germination the non-ergodic component tends to vanish and complexity becomes dominated by the stationary infinite memory. We illustrate some conjectures ranging from stress induced annihilation of crucial events to the emergence of quantum coherence.

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48

Ramos-Ábalos, Eva María, Ramón Gutiérrez-Sánchez, and Ahmed Nafidi. "Powers of the Stochastic Gompertz and Lognormal Diffusion Processes, Statistical Inference and Simulation." Mathematics 8, no. 4 (2020): 588. http://dx.doi.org/10.3390/math8040588.

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In this paper, we study a new family of Gompertz processes, defined by the power of the homogeneous Gompertz diffusion process, which we term the powers of the stochastic Gompertz diffusion process. First, we show that this homogenous Gompertz diffusion process is stable, by power transformation, and determine the probabilistic characteristics of the process, i.e., its analytic expression, the transition probability density function and the trend functions. We then study the statistical inference in this process. The parameters present in the model are studied by using the maximum likelihood estimation method, based on discrete sampling, thus obtaining the expression of the likelihood estimators and their ergodic properties. We then obtain the power process of the stochastic lognormal diffusion as the limit of the Gompertz process being studied and go on to obtain all the probabilistic characteristics and the statistical inference. Finally, the proposed model is applied to simulated data.

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49

Borkar, V. S. "The value function in ergodic control of diffusion processes with partial observations." Stochastics and Stochastic Reports 67, no. 3-4 (1999): 255–66. http://dx.doi.org/10.1080/17442509908834213.

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Amorino, Chiara, and Arnaud Gloter. "Invariant density adaptive estimation for ergodic jump–diffusion processes over anisotropic classes." Journal of Statistical Planning and Inference 213 (July 2021): 106–29. http://dx.doi.org/10.1016/j.jspi.2020.11.006.

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