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Journal articles on the topic 'Stochastic Difference Equation'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

S., Elizabeth, and Nirmal Veena S. "STABILIZATION OF DISCRETE STOCHASTIC DYNAMIC SYSTEM WITH DELAY." International Journal of Current Research and Modern Education, Special Issue (August 13, 2017): 53–56. https://doi.org/10.5281/zenodo.842234.

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In this paper, we discuss the stability of stochastic type differential equations through obtaining the stability condition for the respective stochastic difference equation. The system formulation is done by considering the stochastic differential equation that describes the dynamics of single isolated neuron involving delay. Here the discretization of the stochastic differential equation is done through the Euler- Maruyama Method. And the desired stability is obtained by applying suitable assumptions and through the help of theorems. The obtained theoretical results are represented through n

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2

Ding, Xiaohua. "Exponential stability of a kind of stochastic delay difference equations." Discrete Dynamics in Nature and Society 2006 (2006): 1–9. http://dx.doi.org/10.1155/ddns/2006/94656.

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We present a Razumilchin-type theorem for stochastic delay difference equation, and use it to investigate the mean square exponential stability of a kind of nonautonomous stochastic difference equation which may also be viewed as an approximation of a nonautonomous stochastic delay integrodifferential equations (SDIDEs), and of a difference equation arises from some of the earliest mathematical models of the macroeconomic trade cycle with the environmental noise.

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3

Galal, O. H. "A Proposed Stochastic Finite Difference Approach Based on Homogenous Chaos Expansion." Journal of Applied Mathematics 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/950469.

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This paper proposes a stochastic finite difference approach, based on homogenous chaos expansion (SFDHC). The said approach can handle time dependent nonlinear as well as linear systems with deterministic or stochastic initial and boundary conditions. In this approach, included stochastic parameters are modeled as second-order stochastic processes and are expanded using Karhunen-Loève expansion, while the response function is approximated using homogenous chaos expansion. Galerkin projection is used in converting the original stochastic partial differential equation (PDE) into a set of coupled

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4

Allen, E. J. "Stochastic difference equations and a stochastic partial differential equation for neutron transport." Journal of Difference Equations and Applications 18, no. 8 (2012): 1267–85. http://dx.doi.org/10.1080/10236198.2010.488229.

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5

Abdallah, Lallouche, Aoufi Amani, and Tedjine Meriem. "Stability of equilibrium states for a stochastic difference equation of exponential form." STUDIES IN ENGINEERING AND EXACT SCIENCES 5, no. 2 (2024): e6475. http://dx.doi.org/10.54021/seesv5n2-080.

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It is well know that many process and problems in different fields of sciences and engineering can be modelled by linear and nonlinear difference equations. Particularly, in case of systems biology, ecology, biochemistry, genetics and physiology dynamics, many population models are governed by exponential difference equations, and lot of papers were published in this matter. The goal of this work is to study a nonlinear second order difference equation of exponential form: Xn+1 = bXn + cXne−σXn−1, n ≥ 0 where the parameters b, c have arbitrary values, σ > 0 and the initial values, X0, X−1 a

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6

SHAIKHET, LEONID. "GENERAL METHOD OF LYAPUNOV FUNCTIONALS CONSTRUCTION IN STABILITY INVESTIGATIONS OF NONLINEAR STOCHASTIC DIFFERENCE EQUATIONS WITH CONTINUOUS TIME." Stochastics and Dynamics 05, no. 02 (2005): 175–88. http://dx.doi.org/10.1142/s0219493705001377.

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The general method of Lyapunov functionals construction has been developed during the last decade for stability investigations of stochastic differential equations with after-effect and stochastic difference equations. After some modification of the basic Lyapunov type theorem this method was successfully used also for difference Volterra equations with continuous time. The latter often appear as useful mathematical models. Here this method is used for a stability investigation of some nonlinear stochastic difference equation with continuous time.

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7

A. Lallouche. "Stability Of Equilibrium States For A Stochastically Perturbed Pielou’s Equation [." Communications on Applied Nonlinear Analysis 31, no. 7s (2024): 631–40. http://dx.doi.org/10.52783/cana.v31.1403.

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It is widely recognized that the theory of stochastic difference equations is a mathematical area of great interest. Concrete systems and processes is one of its subareas, which is of some interest nowadays. Specifically, in fields such as systems biology, ecology, biochemistry, genetics, and physiology dynamics, numerous population models are described by studying linear and nonlinear difference equations and systems. These studies have been very productive and helpful to develop the basic theory of the qualitative behaviour of nonlinear rational and exponential difference equations and syste

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8

Ungureanu, Viorica Mariela, and Sui Sun Cheng. "Mean stability of a stochastic difference equation." Annales Polonici Mathematici 93, no. 1 (2008): 33–52. http://dx.doi.org/10.4064/ap93-1-3.

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9

KUSKE, R., and J. B. KELLER. "Large deviation theory for stochastic difference equations." European Journal of Applied Mathematics 8, no. 6 (1997): 567–80. http://dx.doi.org/10.1017/s095679259700332x.

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The probability density for the solution yn of a stochastic difference equation is considered. Following Knessl et al. [1], it is shown to satisfy a master equation, which is solved asymptotically for large values of the index n. The method is illustrated by deriving the large deviation results for a sum of independent identically distributed random variables and for the joint density of two dependent sums. Then it is applied to a difference approximation to the Helmholtz equation in a random medium. A large deviation result is obtained for the probability density of the decay rate of a soluti

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10

Wu, Kaining, Xiaohua Ding, and Liming Wang. "Stability and Stabilization of Impulsive Stochastic Delay Difference Equations." Discrete Dynamics in Nature and Society 2010 (2010): 1–15. http://dx.doi.org/10.1155/2010/592036.

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When an impulsive control is adopted for a stochastic delay difference system (SDDS), there are at least two situations that should be contemplated. If the SDDS is stable, then what kind of impulse can the original system tolerate to keep stable? If the SDDS is unstable, then what kind of impulsive strategy should be taken to make the system stable? Using the Lyapunov-Razumikhin technique, we establish criteria for the stability of impulsive stochastic delay difference equations and these criteria answer those questions. As for applications, we consider a kind of impulsive stochastic delay dif

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11

Iizuka, Masaru. "Weak convergence of a sequence of stochastic difference equations to a stochastic ordinary differential equation." Journal of Mathematical Biology 25, no. 6 (1987): 643–52. http://dx.doi.org/10.1007/bf00275500.

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12

Shaikhet, Leonid. "About Some Unsolved Problems in the Stability Theory of Stochastic Differential and Difference Equations." Axioms 14, no. 6 (2025): 452. https://doi.org/10.3390/axioms14060452.

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This paper continues a series of papers by the author devoted to unsolved problems in the theory of stability and optimal control for stochastic systems. A delay differential equation with stochastic perturbations of the white noise and Poisson’s jump types is considered. In contrast with the known stability condition, in which it is assumed that stochastic perturbations fade on the infinity quickly enough, a new situation is studied, in which stochastic perturbations can either fade on the infinity slowly or not fade at all. Some unsolved problem in this connection is brought to readers’ atte

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13

Li, Ning, Bo Meng, Xinlong Feng, and Dongwei Gui. "A Numerical Comparison of Finite Difference and Finite Element Methods for a Stochastic Differential Equation with Polynomial Chaos." East Asian Journal on Applied Mathematics 5, no. 2 (2015): 192–208. http://dx.doi.org/10.4208/eajam.250714.020515a.

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AbstractA numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

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14

Tanaka, Hideyuki. "On the Riccati difference equation for stochastic realization." Proceedings of the ISCIE International Symposium on Stochastic Systems Theory and its Applications 2009 (May 5, 2009): 30–35. http://dx.doi.org/10.5687/sss.2009.30.

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15

Shaikhet, Leonid. "Some Unsolved Problems in Stability and Optimal Control Theory of Stochastic Systems." Mathematics 10, no. 3 (2022): 474. http://dx.doi.org/10.3390/math10030474.

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In spite of the fact that the theory of stability and optimal control for different types of stochastic systems is well developed and very popular in research, there are some simply and clearly formulated problems, solutions of which have not been found so far. To the readers’ attention six open stability problems for stochastic differential equations with delay, for stochastic difference equations with discrete and continuous time and one open optimal control problem for stochastic hyperbolic equation with two-parameter white noise are offered.

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16

Chandiraraj, Robinson Edward Raja. "A Stochastic Difference Equation with Stationary Noise on Groups." Canadian Journal of Mathematics 64, no. 5 (2012): 1075–89. http://dx.doi.org/10.4153/cjm-2011-094-6.

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AbstractWe consider the stochastic difference equation on a locally compact group G, where is an automorphism of G, ξk are given G-valued random variables and ηk are unknown G-valued random variables. This equation was considered by Tsirelson and Yor on a one-dimensional torus. We consider the case when ξk have a common law μ and prove that if G is a distal group and is a distal automorphism of G and if the equation has a solution, then extremal solutions of the equation are in one-to-one correspondence with points on the coset space K\G for some compact subgroup K of G such that μ is supporte

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17

Goelz, J. C. G., and T. E. Burk. "Development of a well-behaved site index equation: jack pine in north central Ontario." Canadian Journal of Forest Research 22, no. 6 (1992): 776–84. http://dx.doi.org/10.1139/x92-106.

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A base-age invariant site index equation for jack pine based on the Chapman–Richards function was produced that satisfied nine criteria of preferred behavior for site index equations. A difference form of the Chapman–Richards equation produced the best behavior; height equalled site index at base age, and the shape of the curves reflected the data. The data structure used to fit the difference equation was all possible differences rather than the conventional nonoverlapping sequential intervals because this improved the behavior of the model. Height-prediction equations typically use height at

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18

Zhang, Yinghan, Xiaoyuan Yang, and Ruisheng Qi. "Difference Approximation of Stochastic Elastic Equation Driven by Infinite Dimensional Noise." Numerical Mathematics: Theory, Methods and Applications 9, no. 1 (2016): 123–46. http://dx.doi.org/10.4208/nmtma.2015.y14002.

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AbstractAn explicit difference scheme is described, analyzed and tested for numerically approximating stochastic elastic equation driven by infinite dimensional noise. The noise processes are approximated by piecewise constant random processes and the integral formula of the stochastic elastic equation is approximated by a truncated series. Error analysis of the numerical method yields estimate of convergence rate. The rate of convergence is demonstrated with numerical experiments.

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19

Karlsen, Hans Arnfinn. "Existence of moments in a stationary stochastic difference equation." Advances in Applied Probability 22, no. 1 (1990): 129–46. http://dx.doi.org/10.2307/1427601.

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The stationary stochastic difference equation Xt = YtXt–1 + Wt is analyzed with emphasis on conditions ensuring that ||Xt||p <∞. Some general results are obtained and then applied to different classes of input processes {(Yt, Wt)}. Especially both necessary and sufficient conditions are given in the Gaussian case. We also obtain results concerning moments of products of dependent variables.

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20

Pereira, Pedro L. Valls. "Missing observations in stochastic difference equation with arma errors." Brazilian Review of Econometrics 7, no. 1 (1987): 5. http://dx.doi.org/10.12660/bre.v7n11987.3102.

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21

Derman, Emanuel, Kun Soo Park, and Ward Whitt. "A stochastic-difference-equation model for hedge-fund returns." Quantitative Finance 10, no. 7 (2010): 701–33. http://dx.doi.org/10.1080/14697680903200739.

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22

Karlsen, Hans Arnfinn. "Existence of moments in a stationary stochastic difference equation." Advances in Applied Probability 22, no. 01 (1990): 129–46. http://dx.doi.org/10.1017/s0001867800019376.

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The stationary stochastic difference equation Xt = YtXt –1 + Wt is analyzed with emphasis on conditions ensuring that ||Xt || p &lt;∞. Some general results are obtained and then applied to different classes of input processes {(Yt, Wt )}. Especially both necessary and sufficient conditions are given in the Gaussian case. We also obtain results concerning moments of products of dependent variables.

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23

Ferrante, Marco, and David Nualart. "On the Markov property of a stochastic difference equation." Stochastic Processes and their Applications 52, no. 2 (1994): 239–50. http://dx.doi.org/10.1016/0304-4149(94)90027-2.

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24

Zhu, Yunmin. "Asymptotic Normality for a Vector Stochastic Difference Equation with Applications in Stochastic Approximation." Journal of Multivariate Analysis 57, no. 1 (1996): 101–18. http://dx.doi.org/10.1006/jmva.1996.0024.

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25

Zeng, Jun Shan, Ya Bo Wang, and Su Fang Han. "Oscillation of Solutions to Second Order Stochastic Difference Equations." Applied Mechanics and Materials 666 (October 2014): 282–87. http://dx.doi.org/10.4028/www.scientific.net/amm.666.282.

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26

Mario, Di Paola, and Nguyen Dong Anh. "On an extension of the stochastic linearization." Vietnam Journal of Mechanics 15, no. 4 (1993): 1–6. http://dx.doi.org/10.15625/0866-7136/10213.

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Stochastic linearization method is one of the most useful tools for analysis of nonlinear systems under random excitation. The fundamental idea of the classical stochastic linearization consists in replacing the original nonlinear equation by a linear one in such a way that the difference between two equations is minimized in the mean square value. In this paper a new version of the stochastic linearization is proposed. It is shown that for two nonlinear systems considered the new version gives good results for both the weak and strong nonlinearities.

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27

Ashyralyev, A. "On Modified Crank–Nicholson Difference Schemes for Stochastic Parabolic Equation." Numerical Functional Analysis and Optimization 29, no. 3-4 (2008): 268–82. http://dx.doi.org/10.1080/01630560801998138.

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28

de Haan, L., and R. L. Karandikar. "Embedding a stochastic difference equation into a continuous-time process." Stochastic Processes and their Applications 32, no. 2 (1989): 225–35. http://dx.doi.org/10.1016/0304-4149(89)90077-x.

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29

Geiss, Christel, Céline Labart, and Antti Luoto. "Mean square rate of convergence for random walk approximation of forward-backward SDEs." Advances in Applied Probability 52, no. 3 (2020): 735–71. http://dx.doi.org/10.1017/apr.2020.17.

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AbstractLet (Y, Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk $B^n$ from the underlying Brownian motion B by Skorokhod embedding, one can show $L_2$-convergence of the corresponding solutions $(Y^n,Z^n)$ to $(Y, Z).$ We estimate the rate of convergence based on smoothness properties, especially for a terminal condition function in $C^{2,\alpha}$. The proof relies on an approximative representation of $Z^n$ and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the part

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30

Wu, Yongjun, Changshui Feng, and Ronghua Huan. "Optimal Bounded Control for Stationary Response of Strongly Nonlinear Oscillators under Combined Harmonic and Wide-Band Noise Excitations." Mathematical Problems in Engineering 2011 (2011): 1–21. http://dx.doi.org/10.1155/2011/635823.

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We study the stochastic optimal bounded control for minimizing the stationary response of strongly nonlinear oscillators under combined harmonic and wide-band noise excitations. The stochastic averaging method and the dynamical programming principle are combined to obtain the fully averaged Itô stochastic differential equations which describe the original controlled strongly nonlinear system approximately. The stationary joint probability density of the amplitude and phase difference of the optimally controlled systems is obtained from solving the corresponding reduced Fokker-Planck-Kolmogorov

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31

El-Nadi, Khairia El-Said. "On some stochastic parabolic differential equations in a Hilbert space." Journal of Applied Mathematics and Stochastic Analysis 2005, no. 2 (2005): 167–73. http://dx.doi.org/10.1155/jamsa.2005.167.

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We consider some stochastic difference partial differential equations of the form du(x,t,c)=L(x,t,D)u(x,t,c)dt+M(x,t,D)u(x,t−a,c)dw(t), where L(x,t,D) is a linear uniformly elliptic partial differential operator of the second order, M(x,t,D) is a linear partial differential operator of the first order, and w(t) is a Weiner process. The existence and uniqueness of the solution of suitable mixed problems are studied for the considered equation. Some properties are also studied. A more general stochastic problem is considered in a Hilbert space and the results concerning stochastic partial differ

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32

Song, Wanqing, Dongdong Chen, and Enrico Zio. "Heavy Tail and Long-Range Dependence for Skewed Time Series Prediction Based on a Fractional Weibull Process." Fractal and Fractional 8, no. 1 (2023): 7. http://dx.doi.org/10.3390/fractalfract8010007.

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In this paper, a fractional Weibull process is utilized in a predictive stochastic differential equation model to allow for skewness and heavy-tailed characteristics. To this aim, a fractional Weibull process with non-Gaussian characteristics and a long memory effect is proposed to drive the predictive stochastic differential equation. The difference iterative forecasting model is proposed as its stochastic difference scheme. The consistency, stability, and convergence of the model are analyzed. In the proposed model, variational mode decomposition is utilized as the data preprocessing approac

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33

Liu, Xikui, Guiling Li, and Yan Li. "Stochastic Linear Quadratic Optimal Control with Indefinite Control Weights and Constraint for Discrete-Time Systems." Mathematical Problems in Engineering 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/476545.

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The Karush-Kuhn-Tucker (KKT) theorem is used to study stochastic linear quadratic optimal control with terminal constraint for discrete-time systems, allowing the control weighting matrices in the cost to be indefinite. A generalized difference Riccati equation is derived, which is different from those without constraint case. It is proved that the well-posedness and the attainability of stochastic linear quadratic optimal control problem are equivalent. Moreover, an optimal control can be denoted by the solution of the generalized difference Riccati equation.

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34

Oduselu–Hassan, Emmanuel Oladayo, Ignatius N. Njoseh, and Jonathan Tsetimi. "A Numerical Approximation of the Stochastic Ito-Volterra Integral Equation." Asian Research Journal of Mathematics 19, no. 11 (2023): 61–68. http://dx.doi.org/10.9734/arjom/2023/v19i11753.

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A stochastic differential equation (SDE) is a differential equation in which one or more of the terms and the solution are stochastic processes. Numerous studies have employed orthogonal polynomials, however most of them focus on deterministic rather than stochastic systems. This is the reason why in this study, we looked into a numerical solution for the stochastic Ito-Volterra integral equation using the explicit finite difference scheme and Bernstein polynomials as trial functions. The equidistant collocation procedure was used to calculate the unknown constant parameters in between and rea

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35

Lin, Jiamian, Xi Li, SingRu (Celine) Hoe, and Zhongfeng Yan. "A Generalized Finite Difference Method for Solving Hamilton–Jacobi–Bellman Equations in Optimal Investment." Mathematics 11, no. 10 (2023): 2346. http://dx.doi.org/10.3390/math11102346.

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This paper studies the numerical algorithm of stochastic control problems in investment optimization. Investors choose the optimal investment to maximize the expected return under uncertainty. The optimality condition, the Hamilton–Jacobi–Bellman (HJB) equation, satisfied by the value function and obtained by the dynamic programming method, is a partial differential equation coupled with optimization. One of the major computational difficulties is the irregular boundary conditions presented in the HJB equation. In this paper, two mesh-free algorithms are proposed to solve two different cases o

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36

Bonnans, J. Frédéric, and Housnaa Zidani. "Consistency of Generalized Finite Difference Schemes for the Stochastic HJB Equation." SIAM Journal on Numerical Analysis 41, no. 3 (2003): 1008–21. http://dx.doi.org/10.1137/s0036142901387336.

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37

Berkolaiko, Gregory, and Alexandra Rodkina. "Almost sure convergence of solutions to non-homogeneous stochastic difference equation." Journal of Difference Equations and Applications 12, no. 6 (2006): 535–53. http://dx.doi.org/10.1080/10236190600574093.

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38

陈, 会利. "Application of Stochastic Difference Equation in a Class of Economic Model." Advances in Applied Mathematics 07, no. 09 (2018): 1153–58. http://dx.doi.org/10.12677/aam.2018.79134.

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39

Arora, Karunesh, and Tamar Schlick. "Deoxyadenosine sugar puckering pathway simulated by the stochastic difference equation algorithm." Chemical Physics Letters 378, no. 1-2 (2003): 1–8. http://dx.doi.org/10.1016/s0009-2614(03)01195-3.

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40

Cárdenas, Alfredo E., and Ron Elber. "Atomically Detailed Simulations of Helix Formation with the Stochastic Difference Equation." Biophysical Journal 85, no. 5 (2003): 2919–39. http://dx.doi.org/10.1016/s0006-3495(03)74713-4.

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41

BOSWIJK, H. P., H. NEUDECKER, and L. SHUANGZHE. "A note on the asymptotics of a stochastic vector difference equation." Biometrika 81, no. 1 (1994): 216–18. http://dx.doi.org/10.1093/biomet/81.1.216.

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42

Bogoi, Alina, Sergiu Strătilă, Grigore Cican, Daniel-Eugeniu Crunțeanu, and Constatin Levențiu. "Impact of Stochastic Atmospheric Density on Satellite Orbit Stability." Symmetry 17, no. 3 (2025): 402. https://doi.org/10.3390/sym17030402.

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The orbit stability of a satellite is a crucial aspect in its design and maintenance. Without an analysis of orbital trajectories, satellites, much like any small celestial objects, are prone to orbital decay, collision with other orbiting objects, or even variations in trajectory, leading to the impossibility of performing their tasks. Starting from an equation of angular momentum variation applied to a satellite in a circular orbit around Earth, the system of second-order ordinary differential equations of motion for the satellite can be determined. By introducing this term into the satellit

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43

Geiser, Jürgen, and Amirbahador Nasari. "Comparison of Splitting Methods for Deterministic/Stochastic Gross–Pitaevskii Equation." Mathematical and Computational Applications 24, no. 3 (2019): 76. http://dx.doi.org/10.3390/mca24030076.

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In this paper, we discuss the different splitting approaches to numerically solvethe Gross–Pitaevskii equation (GPE). The models are motivated from spinor Bose–Einsteincondensate (BEC). This system is formed of coupled mean-field equations, which are based oncoupled Gross–Pitaevskii equations. We consider conservative finite-difference schemes and spectralmethods for the spatial discretisation. Furthermore, we apply implicit or explicit time-integrators andcombine these schemes with different splitting approaches. The numerical solutions are comparedbased on the conservation of the L2-norm wit

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44

Sepahvandzadeh, Amaneh, Bahman Ghazanfari, and Nader Asadian. "Numerical Solution of Stochastic Generalized Fractional Diffusion Equation by Finite Difference Method." Mathematical and Computational Applications 23, no. 4 (2018): 53. http://dx.doi.org/10.3390/mca23040053.

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The present study aimed at solving the stochastic generalized fractional diffusion equation (SGFDE) by means of the random finite difference method (FDM). Moreover, the conditions of mean square convergence of the numerical solution are studied and numerical examples are presented to demonstrate the validity and accuracy of the method.

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45

Mancini, C. "Uniqueness of the Solution to a Difference-Partial Differential Equation for Finance." Mathematical Models and Methods in Applied Sciences 13, no. 07 (2003): 919–43. http://dx.doi.org/10.1142/s0218202503002775.

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In this paper we study a difference partial differential equation, arising from a financial model, whose solution represents the price of a security linked to a dividend-paying stock. The market model consists of a jump-diffusion process modelling the stock evolution and an independent diffusion modelling the stochastic spot interest rate. We establish the desirable property of the uniqueness of solution to the equation and, since the specialized model is complete, we can consistently price any contingent claim.

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46

Huang, Dongmei, Shengxi Zhou, and Zhichun Yang. "Resonance Mechanism of Nonlinear Vibrational Multistable Energy Harvesters under Narrow-Band Stochastic Parametric Excitations." Complexity 2019 (December 14, 2019): 1–20. http://dx.doi.org/10.1155/2019/1050143.

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To improve energy harvesting performance, this paper investigates the resonance mechanism of nonlinear vibrational multistable energy harvesters under narrow-band stochastic parametric excitations. Based on the method of multiple scales, the largest Lyapunov exponent which determines the stability of the trivial steady-state solutions is derived. The first kind modified Bessel function is utilized to derive the solutions of the responses of multistable energy harvesters. Then, the first-order and second-order nontrivial steady-state moments of multistable energy harvesters are considered. To e

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47

Ruan, Xinfeng, Wenli Zhu, Shuang Li, and Jiexiang Huang. "Option Pricing under Risk-Minimization Criterion in an Incomplete Market with the Finite Difference Method." Mathematical Problems in Engineering 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/165727.

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We study option pricing with risk-minimization criterion in an incomplete market where the dynamics of the risky underlying asset is governed by a jump diffusion equation with stochastic volatility. We obtain the Radon-Nikodym derivative for the minimal martingale measure and a partial integro-differential equation (PIDE) of European option. The finite difference method is employed to compute the European option valuation of PIDE.

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48

Bezandry, Paul H. "Almost periodic stochastic Beverton-Holt difference equation with higher delays and with competition between overlapping generations." Nonautonomous Dynamical Systems 7, no. 1 (2020): 118–25. http://dx.doi.org/10.1515/msds-2020-0105.

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AbstractThe paper studies the existence of an almost periodic solution of some system of stochastic Beverton-Holt equation with higher delays and with competition between overlapping generations under some reasonable assumptions.

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49

Kaur, Manwinder, Arvind Kumar Lal, Satvinder Singh Bhatia, and Akepati Sivarami Reddy. "Numerical solution of stochastic partial differential difference equation arising in reliability engineering." Applied Mathematics and Computation 219, no. 14 (2013): 7645–52. http://dx.doi.org/10.1016/j.amc.2013.01.052.

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Rodkina, Alexandra, Nikolai Dokuchaev, and John Appleby. "On limit periodicity of discrete time stochastic processes." Stochastics and Dynamics 14, no. 04 (2014): 1450011. http://dx.doi.org/10.1142/s0219493714500117.

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Abstract:

We consider a discrete time dynamic system described by a difference equation with periodic coefficients and with additive stochastic noise. We investigate the possibility of the periodicity of the solution. In particular, we established sufficient conditions for convergence of the solution in mean square or almost surely to some stochastic periodic process.

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