On the machine swing dynamics: a perspective

Nonlinear Filtering ◽

Applied Mathematics ◽

Formal Approach ◽

Augmented State ◽

Swing Equation

Abstract A formal approach to rephrase nonlinear filtering of stochastic differential equations is the Kushner setting in applied mathematics and dynamical systems. Thanks to the ability of the Carleman linearization, the ‘nonlinear’ stochastic differential equation can be equivalently expressed as a finite system of ‘bilinear’ stochastic differential equations with the augmented state under the finite closure. Interestingly, the novelty of this paper is to embed the Carleman linearization into a stochastic evolution of the Markov process. To illustrate the Carleman linearization of the Markov process, this paper embeds the Carleman linearization into a nonlinear swing stochastic differential equation. Furthermore, we achieve the nonlinear swing equation filtering in the Carleman setting. Filtering in the Carleman setting has simplified algorithmic procedure. The concerning augmented state accounts for the nonlinearity as well as stochasticity. We show that filtering of the nonlinear stochastic swing equation in the Carleman framework is more refined as well as sharper in contrast to benchmark nonlinear EKF. This paper suggests the usefulness of the Carleman embedding into the stochastic differential equation to filter the concerning nonlinear stochastic differential system. This paper will be of interest to nonlinear stochastic dynamists exploring and unfolding linearization embedding techniques to their research.

TOPOLOGICAL EQUIVALENCE FOR DISCONTINUOUS RANDOM DYNAMICAL SYSTEMS AND APPLICATIONS

Stochastics and Dynamics ◽

10.1142/s021949371350007x ◽

2013 ◽

Vol 14 (01) ◽

pp. 1350007 ◽

Cited By ~ 5

Author(s):

HUIJIE QIAO ◽

JINQIAO DUAN

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Lévy Processes ◽

Levy Processes ◽

Topological Equivalence ◽

Random Differential Equation ◽

Jump Discontinuities ◽

Non Gaussian

After defining non-Gaussian Lévy processes for two-sided time, stochastic differential equations with such Lévy processes are considered. Solution paths for these stochastic differential equations have countable jump discontinuities in time. Topological equivalence (or conjugacy) for such an Itô stochastic differential equation and its transformed random differential equation is established. Consequently, a stochastic Hartman–Grobman theorem is proved for the linearization of the Itô stochastic differential equation. Furthermore, for Marcus stochastic differential equations, this topological equivalence is used to prove the existence of global random attractors.

Mean reflected stochastic differential equations with jumps

Advances in Applied Probability ◽

10.1017/apr.2020.11 ◽

2020 ◽

Vol 52 (2) ◽

pp. 523-562

Author(s):

Phillippe Briand ◽

Abir Ghannoum ◽

Céline Labart

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Numerical Scheme ◽

The Law

AbstractIn this paper, a reflected stochastic differential equation (SDE) with jumps is studied for the case where the constraint acts on the law of the solution rather than on its paths. These reflected SDEs have been approximated by Briand et al. (2016) using a numerical scheme based on particles systems, when no jumps occur. The main contribution of this paper is to prove the existence and the uniqueness of the solutions to this kind of reflected SDE with jumps and to generalize the results obtained by Briand et al. (2016) to this context.

Lie-Point Symmetries and Backward Stochastic Differential Equations

Symmetry ◽

10.3390/sym11091153 ◽

2019 ◽

Vol 11 (9) ◽

pp. 1153

Author(s):

Na Zhang ◽

Guangyan Jia

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Backward Stochastic Differential Equation ◽

Backward Stochastic Differential Equations ◽

Point Symmetry ◽

Lie Point Symmetries ◽

Lie Point Symmetry ◽

Symmetry Method

In this paper, we introduce the Lie-point symmetry method into backward stochastic differential equation and forward–backward stochastic differential equations, and get the corresponding deterministic equations.

An optimal control of a risk-sensitive problem for backward doubly stochastic differential equations with applications

Random Operators and Stochastic Equations ◽

10.1515/rose-2020-2024 ◽

2020 ◽

Vol 28 (1) ◽

pp. 1-18

Author(s):

Dahbia Hafayed ◽

Adel Chala

Keyword(s):

Differential Equation ◽

Optimal Control ◽

Maximum Principle ◽

Differential Equations ◽

Control Problem ◽

Doubly Stochastic ◽

Risk Sensitive ◽

Performance Functional

AbstractIn this paper, we are concerned with an optimal control problem where the system is driven by a backward doubly stochastic differential equation with risk-sensitive performance functional. We generalized the result of Chala [A. Chala, Pontryagin’s risk-sensitive stochastic maximum principle for backward stochastic differential equations with application, Bull. Braz. Math. Soc. (N. S.) 48 2017, 3, 399–411] to a backward doubly stochastic differential equation by using the same contribution of Djehiche, Tembine and Tempone in [B. Djehiche, H. Tembine and R. Tempone, A stochastic maximum principle for risk-sensitive mean-field type control, IEEE Trans. Automat. Control 60 2015, 10, 2640–2649]. We use the risk-neutral model for which an optimal solution exists as a preliminary step. This is an extension of an initial control system in this type of problem, where an admissible controls set is convex. We establish necessary as well as sufficient optimality conditions for the risk-sensitive performance functional control problem. We illustrate the paper by giving two different examples for a linear quadratic system, and a numerical application as second example.

Convergence of Recent Multistep Schemes for a Forward-Backward Stochastic Differential Equation

East Asian Journal on Applied Mathematics ◽

10.4208/eajam.280515.211015a ◽

2015 ◽

Vol 5 (4) ◽

pp. 387-404 ◽

Cited By ~ 4

Author(s):

Jie Yang ◽

Weidong Zhao

Keyword(s):

Differential Equation ◽

Exact Solution ◽

Differential Equations ◽

Convergence Rate ◽

Convergence Analysis ◽

Stochastic Control ◽

Backward Stochastic Differential Equation ◽

Order Convergence

AbstractConvergence analysis is presented for recently proposed multistep schemes, when applied to a special type of forward-backward stochastic differential equations (FB-SDEs) that arises in finance and stochastic control. The corresponding k-step scheme admits a k-order convergence rate in time, when the exact solution of the forward stochastic differential equation (SDE) is given. Our analysis assumes that the terminal conditions and the FBSDE coefficients are sufficiently regular.

On a non-standard two-species stochastic competing system and a related degenerate parabolic equation

ANZIAM Journal ◽

10.21914/anziamj.v61i0.15040 ◽

2020 ◽

Vol 61 ◽

pp. C1-C14

Author(s):

Hidekazu Yoshioka ◽

Yumi Yoshioka

Keyword(s):

Differential Equation ◽

Population Dynamics ◽

Differential Equations ◽

Growth Rates ◽

Epidemic Model ◽

Volterra Model ◽

Sis Epidemic Model ◽

Sis Epidemic

We propose and analyse a new stochastic competing two-species population dynamics model. Competing algae population dynamics in river environments, an important engineering problem, motivates this model. The algae dynamics are described by a system of stochastic differential equations with the characteristic that the two populations are competing with each other through the environmental capacities. Unique existence of the uniformly bounded strong solution is proven and an attractor is identified. The Kolmogorov backward equation associated with the population dynamics is formulated and its unique solvability in a Banach space with a weighted norm is discussed. Our mathematical analysis results can be effectively utilized for a foundation of modelling, analysis, and control of the competing algae population dynamics. References S. Cai, Y. Cai, and X. Mao. A stochastic differential equation SIS epidemic model with two correlated brownian motions. Nonlin. Dyn., 97(4):2175–2187, 2019. doi:10.1007/s11071-019-05114-2. S. Cai, Y. Cai, and X. Mao. A stochastic differential equation SIS epidemic model with two independent brownian motions. J. Math. Anal. App., 474(2):1536–1550, 2019. doi:10.1016/j.jmaa.2019.02.039. U. Callies, M. Scharfe, and M. Ratto. Calibration and uncertainty analysis of a simple model of silica-limited diatom growth in the Elbe river. Ecol. Mod., 213(2):229–244, 2008. doi:10.1016/j.ecolmodel.2007.12.015. M. G. Crandall, H. Ishii, and P. L. Lions. User's guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc., 27(1):229–244, 1992. doi:10.1090/S0273-0979-1992-00266-5. N. H. Du and V. H. Sam. Dynamics of a stochastic Lotka–Volterra model perturbed by white noise. J. Math. Anal. App., 324(1):82–97, 2006. doi:10.1016/j.jmaa.2005.11.064. P. Grandits, R. M. Kovacevic, and V. M. Veliov. Optimal control and the value of information for a stochastic epidemiological SIS model. J. Math. Anal. App., 476(2):665–695, 2019. doi:10.1016/j.jmaa.2019.04.005. B. Horvath and O. Reichmann. Dirichlet forms and finite element methods for the SABR model. SIAM J. Fin. Math., 9(2):716–754, 2018. doi:10.1137/16M1066117. J. Hozman and T. Tichy. DG framework for pricing european options under one-factor stochastic volatility models. J. Comput. Appl. Math., 344:585–600, 2018. doi:10.1016/j.cam.2018.05.064. G. Lan, Y. Huang, C. Wei, and S. Zhang. A stochastic SIS epidemic model with saturating contact rate. Physica A, 529(121504):1–14, 2019. doi:10.1016/j.physa.2019.121504. J. L. Lions and E. Magenes. Non-homogeneous Boundary Value Problems and Applications (Vol. 1). Springer Berlin Heidelberg, 1972. doi:10.1007/978-3-642-65161-8. J. Lv, X. Zou, and L. Tian. A geometric method for asymptotic properties of the stochastic Lotka–Volterra model. Commun. Nonlin. Sci. Numer. Sim., 67:449–459, 2019. doi:10.1016/j.cnsns.2018.06.031. S. Morin, M. Coste, and F. Delmas. A comparison of specific growth rates of periphytic diatoms of varying cell size under laboratory and field conditions. Hydrobiologia, 614(1):285–297, 2008. doi:10.1007/s10750-008-9513-y. B. \T1\O ksendal. Stochastic Differential Equations. Springer Berlin Heidelberg, 2003. doi:10.1007/978-3-642-14394-6. O. Oleinik and E. V. Radkevic. Second-order Equations with Nonnegative Characteristic Form. Springer Boston, 1973. doi:10.1007/978-1-4684-8965-1. S. Peng. Nonlinear Expectations and Stochastic Calculus under Uncertainty: with Robust CLT and G-Brownian Motion. Springer-Verlag Berlin Heidelberg, 2019. doi:10.1007/978-3-662-59903-7. T. S. Schmidt, C. P. Konrad, J. L. Miller, S. D. Whitlock, and C. A. Stricker. Benthic algal (periphyton) growth rates in response to nitrogen and phosphorus: parameter estimation for water quality models. J. Am. Water Res. Ass., 2019. doi:10.1111/1752-1688.12797. Y. Toda and T. Tsujimoto. Numerical modeling of interspecific competition between filamentous and nonfilamentous periphyton on a flat channel bed. Landscape Ecol. Eng., 6(1):81–88, 2010. doi:10.1007/s11355-009-0093-4. H. Yoshioka, Y. Yaegashi, Y. Yoshioka, and K. Tsugihashi. Optimal harvesting policy of an inland fishery resource under incomplete information. Appl. Stoch. Models Bus. Ind., 35(4):939–962, 2019. doi:10.1002/asmb.2428.

Stochastic Measure Diffusion Processes

Canadian Mathematical Bulletin ◽

10.4153/cmb-1979-020-3 ◽

1979 ◽

Vol 22 (2) ◽

pp. 129-138 ◽

Cited By ~ 3

Author(s):

Donald A. Dawson

Keyword(s):

Differential Equation ◽

Partial Differential Equations ◽

Differential Equations ◽

Stochastic Partial Differential Equations ◽

Diffusion Processes ◽

Partial Differential ◽

Stochastic Measure ◽

Recent Developments

The purpose of this article is to give an introduction to the study of a class of stochastic partial differential equations and to give a brief review of some of the recent developments in this field. This study has evolved naturally out of the theory of stochastic differential equations initiated in a pioneering paper of K. Itô [13]. In order to set this review in its appropriate setting we begin by considering a simple scalar stochastic differential equation.

An Averaging Principle for Mckean–Vlasov-Type Caputo Fractional Stochastic Differential Equations

Journal of Mathematics ◽

10.1155/2021/8742330 ◽

2021 ◽

Vol 2021 ◽

pp. 1-11

Author(s):

Weifeng Wang ◽

Lei Yan ◽

Junhao Hu ◽

Zhongkai Guo

Keyword(s):

Differential Equation ◽

Brownian Motion ◽

Differential Equations ◽

Fractional Stochastic Differential Equations

Averaging Principle ◽

Convergence Results ◽

In this paper, we want to establish an averaging principle for Mckean–Vlasov-type Caputo fractional stochastic differential equations with Brownian motion. Compared with the classic averaging condition for stochastic differential equation, we propose a new averaging condition and obtain the averaging convergence results for Mckean–Vlasov-type Caputo fractional stochastic differential equations.

Nonexplosion and Pathwise Uniqueness of Stochastic Differential Equation Driven by Continuous Semimartingale with Non-Lipschitz Coefficients

Journal of Mathematics ◽

10.1155/2015/105784 ◽

2015 ◽

Vol 2015 ◽

pp. 1-5 ◽

Cited By ~ 1

Author(s):

Jinxia Wang

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Sufficient Conditions ◽

Pathwise Uniqueness ◽

Strong Uniqueness ◽

Continuous Semimartingale

We study a class of stochastic differential equations driven by semimartingale with non-Lipschitz coefficients. New sufficient conditions on the strong uniqueness and the nonexplosion are derived ford-dimensional stochastic differential equations onRd(d>2)with non-Lipschitz coefficients, which extend and improve Fei’s results.

The Euler Scheme for a Stochastic Differential Equation Driven by Pure Jump Semimartingales

Journal of Applied Probability ◽

10.1017/s0021900200012250 ◽

2015 ◽

Vol 52 (01) ◽

pp. 149-166 ◽

Cited By ~ 2

Author(s):

Hanchao Wang

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Lévy Processes ◽

Levy Processes ◽

Euler Scheme ◽

Asymptotic Error ◽

Error Distributions ◽

Itô Semimartingale

In this paper we propose the asymptotic error distributions of the Euler scheme for a stochastic differential equation driven by Itô semimartingales. Jacod (2004) studied this problem for stochastic differential equations driven by pure jump Lévy processes and obtained quite sharp results. We extend his results to a more general pure jump Itô semimartingale.