The Semimartingale Approach to Almost Sure Stability Analysis of a Two-Stage Numerical Method for Stochastic Delay Differential Equation by Qian Guo, Xueyin Tao

“…Almost sure exponential stability of the split-step backward Euler (SSBE) method applied to an Itô-type stochastic differential equation with time-varying delay is discussed by the techniques based on Doob-Mayer decomposition and semimartingale convergence theorem. …”
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Almost Sure and Lp Convergence of Split-Step Backward Euler Method for Stochastic Delay Differential Equation by Qian Guo, Xueyin Tao

“…The convergence of the split-step backward Euler (SSBE) method applied to stochastic differential equation with variable delay is proven in Lp-sense. …”
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The Pricing of Vulnerable Options in a Fractional Brownian Motion Environment by Chao Wang, Shengwu Zhou, Jingyuan Yang

“…Under the assumption of the stock price, interest rate, and default intensity obeying the stochastic differential equation driven by fractional Brownian motion, the jump-diffusion model is established for the financial market in fractional Brownian motion setting. …”
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BSDE with Jumps When Mean Reflection Is Nonlinear by Winfrida Felix Mwigilwa, Farai Julius Mhlanga

“…In this paper, we investigate a reflected backward stochastic differential equation (RBSDE) with jumps, focusing on cases where the mean reflection is nonlinear. …”
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Nonlinear Decomposition of Doob-Meyer's Type for Continuous g-Supermartingale with Uniformly Continuous Coefficient by Xuejun Shi, Long Jiang, Ronglin Ji

“…We prove that a continuous g-supermartingale with uniformly continuous coeffcient g on finite or infinite horizon, is a g-supersolution of the corresponding backward stochastic differential equation. It is a new nonlinear Doob-Meyer decomposition theorem for the g-supermartingale with continuous trajectory.…”
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Stability of a Three-Species Cooperative System with Time Delays and Stochastic Perturbations by Jinxing Zhao, Yuanfu Shao

“…We establish the sufficient criteria of the asymptotical stability and stability in probability by constructing a neutral stochastic differential equation and some suitable functionals. …”
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The Implementation of Milstein Scheme in Two-Dimensional SDEs Using the Fourier Method by Yousef Alnafisah

“…In this paper we describe how the Fourier series expansion of Wiener process can be used to simulate a two-dimensional stochastic differential equation (SDE) using Matlab program. Our numerical experiments use Matlab to show how our truncation of Itô’-Taylor expansion at an appropriate point produces Milstein method for the SDE.…”
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Stochastic Periodic Solution and Persistence of a Nonautonomous Impulsive System with Nonlinear Self-Interaction by Weili Kong, Yuanfu Shao

“…Secondly, by comparison theory of the stochastic differential equation, we study the extinction and permanence in the mean of all species. …”
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Stability of a positive equilibrium state for a stochastically perturbed mathematical model ofglassy-winged sharpshooter population by Leonid Shaikhet

“…A necessary and sufficient condition for asymptotic mean square stability of the equilibrium point of the linear part of the considered stochastic differential equation is obtained. This condition is at the same time a sufficient one for stability in probability of the equilibrium point of the initial nonlinear equation. …”
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Identification of a Stochastic Dynamic Model for Aircraft Flight Attitude Based on Measured Data by Haiquan Li, Xiaoqian Chen, Jiatu Zhang, Bochen Wang, Jiahui Peng, Liang Wang

“…Therefore, a stochastic differential equation for aircraft flight attitude is modeled based on the traditional one in this paper. …”
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Asymptotic and numerical solutions for diffusion models for compounded risk reserves with dividend payments by S. Shao, C. L. Chang

“…After defining the process of conditional probability in finite time, martingale theory turns the nonlinear stochastic differential equation to a special class of boundary value problems defined by a parabolic equation with a nonsmooth coefficient of the convection term. …”
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Three-Party Stochastic Evolutionary Game Analysis of Reward and Punishment Mechanism for Green Credit by Qingfeng Zhu, Kaimin Zheng, Yilin Wei

“…We gave sufficient conditions for the stability of strategy based on the stability discriminant theorem of Ito^'s stochastic differential equation (SDE). Then, we discussed the impacts of incentive and penalty parameters on green credit. …”
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Growth-rate distributions of gut microbiota time series by E. Brigatti, S. Azaele

“…Here a neutral model based on a stochastic differential equation with demographic noise, which presents a closed form for these distributions, is used to describe the population dynamics of microbiota. …”
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A stochastic model for water-vegetation systems and the effect of decreasing precipitation on semi-arid environments by Shannon Dixon, Nancy Huntly, Priscilla E. Greenwood, Luis F. Gordillo

“…The model, a stochastic differential equation approximation derived from a Markov jump process, is used to generate extensive simulations that suggest a relationship between precipitation reduction and the desertification process, which might take several years in some instances.…”
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Deterministic and Stochastic Study for an Infected Computer Network Model Powered by a System of Antivirus Programs by Youness El Ansari, Ali El Myr, Lahcen Omari

“…The system is analyzed by using the stability theory of stochastic differential equations and the computer simulations. …”
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Trivariate Stochastic Weather Model for Predicting Maize Yield by Patrick Chidzalo, Phillip O. Ngare, Joseph K. Mung’atu

“…Stochastic analyses are applied on the pdf and process to account for nonlinearity and nonstationarity, and also establish a corresponding stochastic differential equation (SDE) for maize yield. The trivariate stochastic process predicts maize yield with R2=0.8389 and MAPE=4.31% under a deep learning framework. …”
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Impact of Cell Size Effect on Nutrient-Phytoplankton Dynamics by Tiancai Liao, Hengguo Yu, Chuanjun Dai, Min Zhao

“…In this paper, a nutrient-phytoplankton model, which is described by a system of ordinary differential equations incorporating the effect of cell size, and its corresponding stochastic differential equation version are studied analytically and numerically. …”
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Integrated movement models for individual tracking and species distribution data by Frances E. Buderman, Ephraim M. Hanks, Viviana Ruiz‐Gutierrez, Michael Shull, Robert K. Murphy, David A. W. Miller

“…We outline a general IMM framework and develop and apply a specific stochastic differential equation model to a case study of telemetry and species distribution data for golden eagles in western North American during spring migration. …”
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SPDIEs and BSDEs Driven by Lévy Processes and Countable Brownian Motions by Pengju Duan

“…The paper is devoted to solving a new class of backward stochastic differential equations driven by Lévy process and countable Brownian motions. …”
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Exact Finite-Difference Schemes for d-Dimensional Linear Stochastic Systems with Constant Coefficients by Peng Jiang, Xiaofeng Ju, Dan Liu, Shaoqun Fan

“…The authors attempt to construct the exact finite-difference schemes for linear stochastic differential equations with constant coefficients. The explicit solutions to Itô and Stratonovich linear stochastic differential equations with constant coefficients are adopted with the view of providing exact finite-difference schemes to solve them. …”
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