Journal articles: 'Stochastic Differential Equations (SDE)'

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Eliazar, Iddo. "Selfsimilar stochastic differential equations." Europhysics Letters 136, no. 4 (November 1, 2021): 40002. http://dx.doi.org/10.1209/0295-5075/ac4dd4.

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Abstract Diffusion in a logarithmic potential (DLP) attracted significant interest in physics recently. The dynamics of DLP are governed by a Langevin stochastic differential equation (SDE) whose underpinning potential is logarithmic, and that is driven by Brownian motion. The SDE that governs DLP is a particular case of a selfsimilar SDE: one that is driven by a selfsimilar motion, and that produces a selfsimilar motion. This paper establishes the pivotal role of selfsimilar SDEs via two novel universality results. I) Selfsimilar SDEs emerge universally, on the macro level, when applying scaling limits to micro-level SDEs. II) Selfsimilar SDEs emerge universally when applying the Lamperti transformation to stationary SDEs. Using the universality results, this paper further establishes: a novel statistical-analysis approach to selfsimilar Ito diffusions; and the focal importance of DLP.

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Iddrisu, Wahab A., Inusah Iddrisu, and Abdul-Karim Iddrisu. "Modeling Cholera Epidemiology Using Stochastic Differential Equations." Journal of Applied Mathematics 2023 (May 9, 2023): 1–17. http://dx.doi.org/10.1155/2023/7232395.

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In this study, we extend Codeço’s classical SI-B epidemic and endemic model from a deterministic framework into a stochastic framework. Then, we formulated it as a stochastic differential equation for the number of infectious individuals I t under the role of the aquatic environment. We also proved that this stochastic differential equation (SDE) exists and is unique. The reproduction number, R 0 , was derived for the deterministic model, and qualitative features such as the positivity and invariant region of the solution, the two equilibrium points (disease-free and endemic equilibrium), and stabilities were studied to ensure the biological meaningfulness of the model. Numerical simulations were also carried out for the stochastic differential equation (SDE) model by utilizing the Euler-Maruyama numerical method. The method was used to simulate the sample path of the SI-B stochastic differential equation for the number of infectious individuals I t , and the findings showed that the sample path or trajectory of the stochastic differential equation for the number of infectious individuals I t is continuous but not differentiable and that the SI-B stochastic differential equation model for the number of infectious individuals I t fluctuates inside the solution of the SI-B ordinary differential equation model. Another significant feature of our proposed SDE model is its simplicity.

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IMKELLER, PETER, and CHRISTIAN LEDERER. "THE COHOMOLOGY OF STOCHASTIC AND RANDOM DIFFERENTIAL EQUATIONS, AND LOCAL LINEARIZATION OF STOCHASTIC FLOWS." Stochastics and Dynamics 02, no. 02 (June 2002): 131–59. http://dx.doi.org/10.1142/s021949370200039x.

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Random dynamical systems can be generated by stochastic differential equations (sde) on the one hand, and by random differential equations (rde), i.e. randomly parametrized ordinary differential equations on the other hand. Due to conflicting concepts in stochastic calculus and ergodic theory, asymptotic problems for systems associated with sde are harder to treat. We show that both objects are basically identical, modulo a stationary coordinate change (cohomology) on the state space. This observation opens completely new opportunities for the treatment of asymptotic problems for systems related to sde: just study them for the conjugate rde, which is often possible by simple path-by-path classical arguments. This is exemplified for the problem of local linearization of random dynamical systems, the classical analogue of which leads to the Hartman–Grobman theorem.

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Briand, Phillippe, Abir Ghannoum, and Céline Labart. "Mean reflected stochastic differential equations with jumps." Advances in Applied Probability 52, no. 2 (June 2020): 523–62. http://dx.doi.org/10.1017/apr.2020.11.

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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.

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Armstrong, J., and D. Brigo. "Intrinsic stochastic differential equations as jets." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, no. 2210 (February 2018): 20170559. http://dx.doi.org/10.1098/rspa.2017.0559.

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We explain how Itô stochastic differential equations (SDEs) on manifolds may be defined using 2-jets of smooth functions. We show how this relationship can be interpreted in terms of a convergent numerical scheme. We also show how jets can be used to derive graphical representations of Itô SDEs, and we show how jets can be used to derive the differential operators associated with SDEs in a coordinate-free manner. We relate jets to vector flows, giving a geometric interpretation of the Itô–Stratonovich transformation. We show how percentiles can be used to give an alternative coordinate-free interpretation of the coefficients of one-dimensional SDEs. We relate this to the jet approach. This allows us to interpret the coefficients of SDEs in terms of ‘fan diagrams’. In particular, the median of an SDE solution is associated with the drift of the SDE in Stratonovich form for small times.

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Bahlali, K., A. Elouaflin, and M. N'zi. "Backward stochastic differential equations with stochastic monotone coefficients." Journal of Applied Mathematics and Stochastic Analysis 2004, no. 4 (January 1, 2004): 317–35. http://dx.doi.org/10.1155/s1048953304310038.

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We prove an existence and uniqueness result for backward stochastic differential equations whose coefficients satisfy a stochastic monotonicity condition. In this setting, we deal with both constant and random terminal times. In the random case, the terminal time is allowed to take infinite values. But in a Markovian framework, that is coupled with a forward SDE, our result provides a probabilistic interpretation of solutions to nonlinear PDEs.

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Rezaeyan, Ramzan. "Application of Stochastic Differential Equation and Optimal Control for Engineering Problems." Advanced Materials Research 383-390 (November 2011): 972–75. http://dx.doi.org/10.4028/www.scientific.net/amr.383-390.972.

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Stochastic differential equations(SDEs) is fundamental for the modeling in engineering and science. The goal of this paper is study optimal control of the solution a SDE. We consider the optimal control for risky stocks stochastic model with using of the SDE.

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Fekete, Dorottya, Joaquin Fontbona, and Andreas E. Kyprianou. "Skeletal stochastic differential equations for superprocesses." Journal of Applied Probability 57, no. 4 (November 23, 2020): 1111–34. http://dx.doi.org/10.1017/jpr.2020.53.

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AbstractIt is well understood that a supercritical superprocess is equal in law to a discrete Markov branching process whose genealogy is dressed in a Poissonian way with immigration which initiates subcritical superprocesses. The Markov branching process corresponds to the genealogical description of prolific individuals, that is, individuals who produce eternal genealogical lines of descent, and is often referred to as the skeleton or backbone of the original superprocess. The Poissonian dressing along the skeleton may be considered to be the remaining non-prolific genealogical mass in the superprocess. Such skeletal decompositions are equally well understood for continuous-state branching processes (CSBP).In a previous article [16] we developed an SDE approach to study the skeletal representation of CSBPs, which provided a common framework for the skeletal decompositions of supercritical and (sub)critical CSBPs. It also helped us to understand how the skeleton thins down onto one infinite line of descent when conditioning on survival until larger and larger times, and eventually forever.Here our main motivation is to show the robustness of the SDE approach by expanding it to the spatial setting of superprocesses. The current article only considers supercritical superprocesses, leaving the subcritical case open.

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Stoyanov, Jordan, and Dobrin Botev. "Quantitative results for perturbed stochastic differential equations." Journal of Applied Mathematics and Stochastic Analysis 9, no. 3 (January 1, 1996): 255–61. http://dx.doi.org/10.1155/s104895339600024x.

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The paper is devoted to Itô type stochastic differential equations (SDE's) with small perturbations. Our goal is to present strong results showing how close are the 2m-order moments of the solutions of the perturbed SDE's and the unperturbed SDE.

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Chaharpashlou, Reza, Reza Saadati, and António M. Lopes. "Fuzzy Mittag–Leffler–Hyers–Ulam–Rassias Stability of Stochastic Differential Equations." Mathematics 11, no. 9 (May 4, 2023): 2154. http://dx.doi.org/10.3390/math11092154.

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Stability is the most relevant property of dynamical systems. The stability of stochastic differential equations is a challenging and still open problem. In this article, using a fuzzy Mittag–Leffler function, we introduce a new fuzzy controller function to stabilize the stochastic differential equation (SDE) ν′(γ,μ)=Fγ,μ,ν(γ,μ). By adopting the fixed point technique, we are able to prove the fuzzy Mittag–Leffler–Hyers–Ulam–Rassias stability of the SDE.

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SANTOS, L. F., and C. O. ESCOBAR. "STOCHASTIC DIFFERENTIAL EQUATIONS FOR THE CONTINUOUS SPONTANEOUS LOCALIZATION MODEL." Modern Physics Letters A 15, no. 30 (September 28, 2000): 1833–42. http://dx.doi.org/10.1142/s0217732300001997.

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We extend Vink's method [J. C. Vink, Phys. Rev.A48, 1808 (1993)], developed for an isolated quantum system, to an open quantum system consisting of a free particle interacting with its surrounding through a random potential, which causes the spontaneous localization of its wave function. We then obtain the stochastic differential equations (SDE) underlying its evolution. These SDE help us to observe the effects of the environment upon the movement of the particle.

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Decreusefond, L. "Time Reversal of Volterra Processes Driven Stochastic Differential Equations." International Journal of Stochastic Analysis 2013 (February 27, 2013): 1–13. http://dx.doi.org/10.1155/2013/790709.

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We consider stochastic differential equations driven by some Volterra processes. Under time reversal, these equations are transformed into past-dependent stochastic differential equations driven by a standard Brownian motion. We are then in position to derive existence and uniqueness of solutions of the Volterra driven SDE considered at the beginning.

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Sun, Xu, Xiaofan Li, and Yayun Zheng. "Governing equations for probability densities of Marcus stochastic differential equations with Lévy noise." Stochastics and Dynamics 17, no. 05 (September 23, 2016): 1750033. http://dx.doi.org/10.1142/s0219493717500332.

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Marcus stochastic differential equations (SDEs) often are appropriate models for stochastic dynamical systems driven by non-Gaussian Lévy processes and have wide applications in engineering and physical sciences. The probability density of the solution to an SDE offers complete statistical information on the underlying stochastic process. Explicit formula for the Fokker–Planck equation, the governing equation for the probability density, is well-known when the SDE is driven by a Brownian motion. In this paper, we address the open question of finding the Fokker–Planck equations for Marcus SDEs in arbitrary dimensions driven by non-Gaussian Lévy processes. The equations are given in a simple form that facilitates theoretical analysis and numerical computation. Several examples are presented to illustrate how the theoretical results can be applied to obtain Fokker–Planck equations for Marcus SDEs driven by Lévy processes.

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Michelot, Théo, Richard Glennie, Catriona Harris, and Len Thomas. "Varying-Coefficient Stochastic Differential Equations with Applications in Ecology." Journal of Agricultural, Biological and Environmental Statistics 26, no. 3 (March 26, 2021): 446–63. http://dx.doi.org/10.1007/s13253-021-00450-6.

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AbstractStochastic differential equations (SDEs) are popular tools to analyse time series data in many areas, such as mathematical finance, physics, and biology. They provide a mechanistic description of the phenomenon of interest, and their parameters often have a clear interpretation. These advantages come at the cost of requiring a relatively simple model specification. We propose a flexible model for SDEs with time-varying dynamics where the parameters of the process are nonparametric functions of covariates, similar to generalized additive models. Combining the SDE and nonparametric approaches allows for the SDE to capture more detailed, non-stationary, features of the data-generating process. We present a computationally efficient method of approximate inference, where the SDE parameters can vary according to fixed covariate effects, random effects, or basis-penalty smoothing splines. We demonstrate the versatility and utility of this approach with three applications in ecology, where there is often a modelling trade-off between interpretability and flexibility.Supplementary materials accompanying this paper appear online.

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CARABALLO, TOMÁS, PETER E. KLOEDEN, and ANDREAS NEUENKIRCH. "SYNCHRONIZATION OF SYSTEMS WITH MULTIPLICATIVE NOISE." Stochastics and Dynamics 08, no. 01 (March 2008): 139–54. http://dx.doi.org/10.1142/s0219493708002184.

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The synchronization of Stratonovich stochastic differential equations (SDE) with a one-sided dissipative Lipschitz drift and linear multiplicative noise is investigated by transforming the SDE to random ordinary differential equations (RODE) and synchronizing their dynamics. In terms of the original SDE, this gives synchronization only when the driving noises are the same. Otherwise, the synchronization is modulo exponential factors involving Ornstein–Uhlenbeck processes corresponding to the driving noises. Moreover, this occurs no matter how large the intensity coefficients of the noise.

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Jafari, Hossein, and Ghazaleh Rahimi. "Forecasting dirty tanker freight rate index by using stochastic differential equations." International Journal of Financial Engineering 05, no. 04 (December 2018): 1850034. http://dx.doi.org/10.1142/s2424786318500342.

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The accurate forecasting of freight rate index is one of the most important issues in shipping market. The continuous and jump-diffusion stochastic differential equations are used for modeling and forecasting of Baltic exchange Dirty Tanker Index (BDTI). Actual observations and simulated data are applied to estimate the best stochastic model. The comparison of forecasting between SDE methods and the ARIMA time series models show that SDE models have better accuracy than the time series techniques.

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Nabati, Parisa, Hadiseh Babazadeh, and Hamed Azadfar. "Noise analysis of band pass filters using stochastic differential equations." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 38, no. 2 (March 4, 2019): 693–702. http://dx.doi.org/10.1108/compel-06-2018-0253.

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Purpose The purpose of this paper is to analyze the effects of white noise perturbations of the input voltage on the band pass filter response, both on pass band and reject band. Design/methodology/approach By adding white noise term in the input voltage of the filter circuit, the deterministic ordinary differential equation (ODE) is replaced by a stochastic differential equation (SDE). With the application of Ito lemma, the analytical solution of SDE has been obtained. Furthermore, based on the Euler–Maruyama approximation, the numerical simulation for SDE has been done. Practical implications Numerical examples are performed using MATLAB programming to show the efficiency and accuracy of the present work. Originality/value All previous noise analyses of filter circuits were done using ODEs or component noise formulas in the electrical domain. The stochastic perspective for these circuits is adopted for the first time in this paper.

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Zhang, Wei, and Hui Min. "Weak Convergence Analysis and Improved Error Estimates for Decoupled Forward-Backward Stochastic Differential Equations." Mathematics 9, no. 8 (April 13, 2021): 848. http://dx.doi.org/10.3390/math9080848.

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In this paper, we mainly investigate the weak convergence analysis about the error terms which are determined by the discretization for solving the stochastic differential equation (SDE, for short) in forward-backward stochastic differential equations (FBSDEs, for short), which is on the basis of Itô Taylor expansion, the numerical SDE theory, and numerical FBSDEs theory. Under the weak convergence analysis of FBSDEs, we further establish better error estimates of recent numerical schemes for solving FBSDEs.

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Ekanayake, Amy J. "Stochastic SIS metapopulation models for the spread of disease among species in a fragmented landscape." International Journal of Biomathematics 09, no. 04 (April 22, 2016): 1650055. http://dx.doi.org/10.1142/s1793524516500558.

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Two stochastic models are derived for a susceptible–infectious–susceptible epidemic spreading through a metapopulation: a continuous time Markov chain (CTMC) model and an Itô stochastic differential equation (SDE) model. The stochastic models are numerically compared. Close agreement suggests that computationally intense CTMC simulations can be approximated by simpler SDE simulations. Differential equations for the moments of the SDE probability distribution are also derived, the steady states are solved numerically using a moment closure technique, and these results are compared to simulations. The moment closure technique only coarsely approximates simulation results. The effect of model parameters on stability of the disease-free equilibrium is also numerically investigated.

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Gliklikh, Yuri E., and Lora A. Morozova. "Conditions for global existence of solutions of ordinary differential, stochastic differential, and parabolic equations." International Journal of Mathematics and Mathematical Sciences 2004, no. 17 (2004): 901–12. http://dx.doi.org/10.1155/s016117120430503x.

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First, we prove a necessary and sufficient condition for global in time existence of all solutions of an ordinary differential equation (ODE). It is a condition of one-sided estimate type that is formulated in terms of so-called proper functions on extended phase space. A generalization of this idea to stochastic differential equations (SDE) and parabolic equations (PE) allows us to prove similar necessary and sufficient conditions for global in time existence of solutions of special sorts:L1-complete solutions of SDE (this means that they belong to a certain functional space ofL1type) and the so-called complete Feller evolution families giving solutions of PE. The general case of equations on noncompact smooth manifolds is under consideration.

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Kubilius, Kęstutis, and Aidas Medžiūnas. "A Class of Fractional Stochastic Differential Equations with a Soft Wall." Fractal and Fractional 7, no. 2 (January 21, 2023): 110. http://dx.doi.org/10.3390/fractalfract7020110.

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In this paper we are interested in fractional stochactic differential equations (SDEs) with a soft wall. What do we mean by such a type of equation? It has been established that SDE with reflection can be imagined as equations having a hard wall. Now, by introducing repulsion instead of reflection, one obtains an SDE with a soft wall. In contrast to the SDE with reflection, where the process cannot pass the hard wall, the soft wall is repulsive but not impenetrable. As the process crosses the soft wall boundary, it experiences the force of a chosen magnitude in the opposite direction. When the process is far from the wall, the force acts weakly. We find conditions under which SDE with a soft wall has a unique solution and construct an implicit Euler approximation with a rate of convergence for this equation. Using the example of the fractional Vasicek process with soft walls, we illustrate the dependence of the behaviour of the solution on the repulsion force.

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P, Govindaraju, and Senthil Kumar. "A study on stochastic differential equation." Journal of Computational Mathematica 5, no. 2 (December 20, 2021): 68–75. http://dx.doi.org/10.26524/cm109.

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In this paper we study solutions to stochastic differential equations (SDEs) with discontinuous drift. In this paper we discussed The Euler-Maruyama method and this shows that a candidate density function based on the Euler-Maruyama method. The point of departure for this work is a particular SDE with discontinuous drift.

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Draouil, Olfa, and Bernt Øksendal. "Optimal insider control of stochastic partial differential equations." Stochastics and Dynamics 18, no. 01 (November 6, 2017): 1850014. http://dx.doi.org/10.1142/s0219493718500144.

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We study the problem of optimal insider control of an SPDE (a stochastic evolution equation) driven by a Brownian motion and a Poisson random measure. Our optimal control problem is new in two ways: (i) The controller has access to inside information, i.e. access to information about a future state of the system. (ii) The integro-differential operator of the SPDE might depend on the control. In the first part of the paper, we formulate a sufficient and necessary maximum principle for this type of control problem, in two cases: The control is allowed to depend both on time [Formula: see text] and on the space variable [Formula: see text]. The control is not allowed to depend on [Formula: see text]. In the second part of the paper, we apply the results above to the problem of optimal control of an SDE system when the inside controller has only noisy observations of the state of the system. Using results from non-linear filtering, we transform this noisy observation SDE inside control problem into a full observation SPDE insider control problem. The results are illustrated by explicit examples.

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Moon, Jun, and Jin-Ho Chung. "Indefinite Linear-Quadratic Stochastic Control Problem for Jump-Diffusion Models with Random Coefficients: A Completion of Squares Approach." Mathematics 9, no. 22 (November 16, 2021): 2918. http://dx.doi.org/10.3390/math9222918.

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In this paper, we study the indefinite linear-quadratic (LQ) stochastic optimal control problem for stochastic differential equations (SDEs) with jump diffusions and random coefficients driven by both the Brownian motion and the (compensated) Poisson process. In our problem setup, the coefficients in the SDE and the objective functional are allowed to be random, and the jump-diffusion part of the SDE depends on the state and control variables. Moreover, the cost parameters in the objective functional need not be (positive) definite matrices. Although the solution to this problem can also be obtained through the stochastic maximum principle or the dynamic programming principle, our approach is simple and direct. In particular, by using the Itô-Wentzell’s formula, together with the integro-type stochastic Riccati differential equation (ISRDE) and the backward SDE (BSDE) with jump diffusions, we obtain the equivalent objective functional that is quadratic in control u under the positive definiteness condition, where the approach is known as the completion of squares method. Then the explicit optimal solution, which is linear in state characterized by the ISRDE and the BSDE jump diffusions, and the associated optimal cost are derived by eliminating the quadratic term of u in the equivalent objective functional. We also verify the optimality of the proposed solution via the verification theorem, which requires solving the stochastic HJB equation, a class of stochastic partial differential equations with jump diffusions.

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Jamba, Nelson T., Patrícia Andreia Filipe, Gonçalo Jacinto, and Carlos A. Braumann. "Stochastic differential equations mixed model for individual growth with the inclusion of genetic characteristics." Statistics, Optimization & Information Computing 12, no. 2 (December 19, 2023): 298–309. http://dx.doi.org/10.19139/soic-2310-5070-1829.

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In early work we have studied a class of stochastic differential equation (SDE) models, for which the Gompertz and the Bertalanffy-Richards stochastic models are particular cases, to describe individual growth in random environments, and applied it to model cattle weight evolution using real data. We have started to work on these type of models considering that the model parameters are fixed, i.e. the same for all the animals. Aiming to incorporate variability among individuals, we consider that the model parameters can be random variables, resulting in SDE mixed models. In additon, here we consider SDE mixed models, allowing the parameters to be random and propose to incorporate each animal's genetic characteristics considering the transformed animal's weight at maturity to be a function of its genetic values. The main objective is to extend the SDE mixed model to the more realistic case where the individual genetic value becomes an important component in the estimated growth curve. For the estimation of the model parameters we have used maximum likelihood estimation theory. Estimates and asymptotic confidence intervals of the parameters are presented. A comparison with SDE non-mixed model and SDE mixed model without the inclusion of genetic characteristics is shown with the conclusion that the incorporation of some genetic characteristics in the model parameters improves the estimation of the animal's growth curve.

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Xie, Hongling. "An efficient and spectral accurate numerical method for computing SDE driven by multivariate Gaussian variables." AIP Advances 12, no. 7 (July 1, 2022): 075306. http://dx.doi.org/10.1063/5.0096285.

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There are many previous studies on designing efficient and high-order numerical methods for stochastic differential equations (SDEs) driven by Gaussian random variables. They mostly focus on proposing numerical methods for SDEs with independent Gaussian random variables and rarely solving SDEs driven by dependent Gaussian random variables. In this paper, we propose a Galerkin spectral method for solving SDEs with dependent Gaussian random variables. Our main techniques are as follows: (1) We design a mapping transformation between multivariate Gaussian random variables and independent Gaussian random variables based on the covariance matrix of multivariate Gaussian random variables. (2) First, we assume the unknown function in the SDE has the generalized polynomial chaos expansion and convert it to be driven by independent Gaussian random variables by the mapping transformation; second, we implement the stochastic Galerkin spectral method for the SDE in the Gaussian measure space; and third, we obtain deterministic differential equations for the coefficients of the expansion. (3) We employ a spectral method solving the deterministic differential equations numerically. We apply the newly proposed numerical method to solve the one-dimensional and two-dimensional stochastic Poisson equations and one-dimensional and two-dimensional stochastic heat equations, respectively. All the presented stochastic equations are driven by two Gaussian random variables, and they are dependent and have multivariate normal distribution of their joint probability density.

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Fan, Yulian. "The PDEs and Numerical Scheme for Derivatives under Uncertainty Volatility." Mathematical Problems in Engineering 2019 (May 29, 2019): 1–7. http://dx.doi.org/10.1155/2019/1268301.

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We use the stochastic differential equations (SDE) driven by G-Brownian motion to describe the basic assets (such as stocks) price processes with volatility uncertainty. We give the estimation method of the SDE’s parameters. Then, by the nonlinear Feynman-Kac formula, we get the partial differential equations satisfied by the derivatives. At last, we give a numerical scheme to solve the nonlinear partial differential equations.

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Yuskovych, V. K. "On asymptotic behavior of solutions of stochastic differential equations in multidimensional space." Theory of Stochastic Processes 27(43), no. 1 (November 16, 2023): 53–66. http://dx.doi.org/10.3842/tsp-9252662178-99.

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Consider the multidimensional SDE dX(t) = a(X(t)) dt + b(X(t)) dW(t). We study the asymptotic behavior of its solution X(t) as t → ∞, namely, we study sufficient conditions of transience of its solution X(t), stabilization of its multidimensional angle X(t)/|X(t)|, and asymptotic equivalence of solutions of the given SDE and the following ODE without noise: dx(t) = a(x(t)) dt.

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Wang, Yongguang, and Shuzhen Yao. "Neural Stochastic Differential Equations with Neural Processes Family Members for Uncertainty Estimation in Deep Learning." Sensors 21, no. 11 (May 26, 2021): 3708. http://dx.doi.org/10.3390/s21113708.

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Existing neural stochastic differential equation models, such as SDE-Net, can quantify the uncertainties of deep neural networks (DNNs) from a dynamical system perspective. SDE-Net is either dominated by its drift net with in-distribution (ID) data to achieve good predictive accuracy, or dominated by its diffusion net with out-of-distribution (OOD) data to generate high diffusion for characterizing model uncertainty. However, it does not consider the general situation in a wider field, such as ID data with noise or high missing rates in practice. In order to effectively deal with noisy ID data for credible uncertainty estimation, we propose a vNPs-SDE model, which firstly applies variants of neural processes (NPs) to deal with the noisy ID data, following which the completed ID data can be processed more effectively by SDE-Net. Experimental results show that the proposed vNPs-SDE model can be implemented with convolutional conditional neural processes (ConvCNPs), which have the property of translation equivariance, and can effectively handle the ID data with missing rates for one-dimensional (1D) regression and two-dimensional (2D) image classification tasks. Alternatively, vNPs-SDE can be implemented with conditional neural processes (CNPs) or attentive neural processes (ANPs), which have the property of permutation invariance, and exceeds vanilla SDE-Net in multidimensional regression tasks.

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Halidias, Nikolaos, and Ioannis S. Stamatiou. "A note on the asymptotic stability of the semi-discrete method for stochastic differential equations." Monte Carlo Methods and Applications 28, no. 1 (February 15, 2022): 13–25. http://dx.doi.org/10.1515/mcma-2022-2102.

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Abstract We study the asymptotic stability of the semi-discrete (SD) numerical method for the approximation of stochastic differential equations. Recently, we examined the order of ℒ 2 {\mathcal{L}^{2}} -convergence of the truncated SD method and showed that it can be arbitrarily close to 1 2 {\frac{1}{2}} ; see [I. S. Stamatiou and N. Halidias, Convergence rates of the semi-discrete method for stochastic differential equations, Theory Stoch. Process. 24 2019, 2, 89–100]. We show that the truncated SD method is able to preserve the asymptotic stability of the underlying SDE. Motivated by a numerical example, we also propose a different SD scheme, using the Lamperti transformation to the original SDE. Numerical simulations support our theoretical findings.

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Ferreiro-Castilla, A., A. E. Kyprianou, and R. Scheichl. "An Euler–Poisson scheme for Lévy driven stochastic differential equations." Journal of Applied Probability 53, no. 1 (March 2016): 262–78. http://dx.doi.org/10.1017/jpr.2015.23.

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Abstract We describe an Euler scheme to approximate solutions of Lévy driven stochastic differential equations (SDEs) where the grid points are given by the arrival times of a Poisson process and thus are random. This result extends the previous work of Ferreiro-Castilla et al. (2014). We provide a complete numerical analysis of the algorithm to approximate the terminal value of the SDE and prove that the mean-square error converges with rate O(n-1/2). The only requirement of the methodology is to have exact samples from the resolvent of the Lévy process driving the SDE. Classical examples, such as stable processes, subclasses of spectrally one-sided Lévy processes, and new families, such as meromorphic Lévy processes (Kuznetsov et al. (2012), are examples for which our algorithm provides an interesting alternative to existing methods, due to its straightforward implementation and its robustness with respect to the jump structure of the driving Lévy process.

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Yang, Jie, and Weidong Zhao. "Convergence of Recent Multistep Schemes for a Forward-Backward Stochastic Differential Equation." East Asian Journal on Applied Mathematics 5, no. 4 (November 2015): 387–404. http://dx.doi.org/10.4208/eajam.280515.211015a.

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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.

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Banchuin, Rawid, and Roungsan Chaisricharoen. "Vector SDE Based Stochastic Analysis of Transformer." ECTI Transactions on Computer and Information Technology (ECTI-CIT) 15, no. 1 (January 5, 2021): 82–107. http://dx.doi.org/10.37936/ecti-cit.2021151.188931.

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In this research, the stochastic behaviours oftransformer have been analysed by using the stochasticdifferential equation approach where both noise in thevoltage source applied to the transformer and the randomvariations in elements and parameters of transformers havebeen considered. The resulting vector stochasticdifferential equations of the transformer have been bothanalytically and numerically solved in the Ito sense wherethe Euler-Maruyama scheme has been adopted fordetermining the numerical solutions which have been theirsample means have been used for verification. With theobtained analytical and numerical solutions, the stochasticproperties of the transformer’s electrical quantities havebeen studied and the influences of noise in the voltagesource and random variations in elements and parametersof transformers to those electrical quantities have beenanalysed. The causes of high and low frequency stochasticvariations of such electrical quantities in both transient andsteady state have been pointed out. Moreover, furtherextension of our stochastic differential equations and therelated mathematical formulations has also been given.

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Redjil, Amel, Zineb Arab, Hanane Ben Gherbal, and Zakaria Boumezbeur. "Temporal regularity of stochastic differential equations driven by G-Brownian motion." Statistics, Optimization & Information Computing 12, no. 4 (March 12, 2024): 1173–83. http://dx.doi.org/10.19139/soic-2310-5070-1898.

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This paper is devoted to study the temporal regularity of the solution of stochastic differential equations driven by G-Brownian motion (G-SDEs) under the global Lipschitz and linear growth conditions. In addition, a numerical simulation of a particular G-SDE is provided.

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Alnafisah, Yousef. "A New Approach to Compare the Strong Convergence of the Milstein Scheme with the Approximate Coupling Method." Fractal and Fractional 6, no. 6 (June 17, 2022): 339. http://dx.doi.org/10.3390/fractalfract6060339.

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Milstein and approximate coupling approaches are compared for the pathwise numerical solutions to stochastic differential equations (SDE) driven by Brownian motion. These methods attain an order one convergence under the nondegeneracy assumption of the diffusion term for the approximate coupling method. We use MATLAB to simulate these methods by applying them to a particular two-dimensional SDE. Then, we analyze the performance of both methods and the amount of time required to obtain the result. This comparison is essential in several areas, such as stochastic analysis, financial mathematics, and some biological applications.

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Kubilius, Kęstutis, and Aidas Medžiūnas. "Pathwise Convergent Approximation for the Fractional SDEs." Mathematics 10, no. 4 (February 21, 2022): 669. http://dx.doi.org/10.3390/math10040669.

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Fractional stochastic differential equation (FSDE)-based random processes are used in a wide spectrum of scientific disciplines. However, in the majority of cases, explicit solutions for these FSDEs do not exist and approximation schemes have to be applied. In this paper, we study one-dimensional stochastic differential equations (SDEs) driven by stochastic process with Hölder continuous paths of order 1/2<γ<1. Using the Lamperti transformation, we construct a backward approximation scheme for the transformed SDE. The inverse transformation provides an approximation scheme for the original SDE which converges at the rate h2γ, where h is a time step size of a uniform partition of the time interval under consideration. This approximation scheme covers wider class of FSDEs and demonstrates higher convergence rate than previous schemes by other authors in the field.

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James, Mirgichan Khobocha, Cyrus Gitonga Ngari, Stephen Karanja, and Robert Muriungi. "Modeling HIV-HBV Co-infection Dynamics: Stochastic Differential Equations and Matlab Simulation with Euler-Maruyama Numerical Method." Asian Research Journal of Mathematics 20, no. 7 (July 11, 2024): 49–69. http://dx.doi.org/10.9734/arjom/2024/v20i7811.

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HIV/AIDS and Hepatitis B co-infection complicates population dynamics and brings forth a wide range of clinical outcomes which makes it a difficult situation for public health. In particular designing treatment plans for the co-infection. A Stochastic Differential Equation (SDE) model is a special class of a stochastic model with continuous parameter space and continuous state space. Deterministic model lacks randomness while an SDE model accounts for randomness and uncertainties. In this study, an SDE model was formulated from an existing deterministic model to examine the variability of dynamic behavior. The analysis and numerical schemes were derived based on Euler-Maruyama SDE algorithms. The model utilized epidemiological insights with current developments in mathematical modeling approaches to represent the interaction between these two viruses. Matlab software was used to obtain SDE numerical results alongside the deterministic solution. Descriptive statistics of the sample paths indicated that the variability of infection outcomes oscillates around the deterministic trajectory. None of the sample paths are absorbed during the time steps. This shows the persistence of the co-infection in the population, in particular The variability of the infections ranges between 1.972 and 202.4, being lowest in AIDS infectives and highest in acute Hepatitis B infectives. An indication that variability cannot be ignored in designing control interventions of co-infections. These results provide new insights into the dynamics of co-infection through in-depth research and simulation, which helps to understand the inherent nature of deterministic model by incorporating the stochastic effects. These understanding will further help the policy makers in health sector to take care of the variability and uncertainty in designing treatment and management strategies.

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Zhao, Weidong, Wei Zhang, and Lili Ju. "A Numerical Method and its Error Estimates for the Decoupled Forward-Backward Stochastic Differential Equations." Communications in Computational Physics 15, no. 3 (March 2014): 618–46. http://dx.doi.org/10.4208/cicp.280113.190813a.

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AbstractIn this paper, a new numerical method for solving the decoupled forward-backward stochastic differential equations (FBSDEs) is proposed based on some specially derived reference equations. We rigorously analyze errors of the proposed method under general situations. Then we present error estimates for each of the specific cases when some classical numerical schemes for solving the forward SDE are taken in the method; in particular, we prove that the proposed method is second-order accurate if used together with the order-2.0 weak Taylor scheme for the SDE. Some examples are also given to numerically demonstrate the accuracy of the proposed method and verify the theoretical results.

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Wang, Tianxiao. "On closed-loop equilibrium strategies for mean-field stochastic linear quadratic problems." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 41. http://dx.doi.org/10.1051/cocv/2019057.

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This article is concerned with linear quadratic optimal control problems of mean-field stochastic differential equations (MF-SDE) with deterministic coefficients. To treat the time inconsistency of the optimal control problems, linear closed-loop equilibrium strategies are introduced and characterized by variational approach. Our developed methodology drops the delicate convergence procedures in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. When the MF-SDE reduces to SDE, our Riccati system coincides with the analogue in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. However, these two systems are in general different from each other due to the conditional mean-field terms in the MF-SDE. Eventually, the comparisons with pre-committed optimal strategies, open-loop equilibrium strategies are given in details.

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Liu, Shuaiqiang, Lech A. Grzelak, and Cornelis W. Oosterlee. "The Seven-League Scheme: Deep Learning for Large Time Step Monte Carlo Simulations of Stochastic Differential Equations." Risks 10, no. 3 (February 23, 2022): 47. http://dx.doi.org/10.3390/risks10030047.

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We propose an accurate data-driven numerical scheme to solve stochastic differential equations (SDEs), by taking large time steps. The SDE discretization is built up by means of the polynomial chaos expansion method, on the basis of accurately determined stochastic collocation (SC) points. By employing an artificial neural network to learn these SC points, we can perform Monte Carlo simulations with large time steps. Basic error analysis indicates that this data-driven scheme results in accurate SDE solutions in the sense of strong convergence, provided the learning methodology is robust and accurate. With a method variant called the compression–decompression collocation and interpolation technique, we can drastically reduce the number of neural network functions that have to be learned, so that computational speed is enhanced. As a proof of concept, 1D numerical experiments confirm a high-quality strong convergence error when using large time steps, and the novel scheme outperforms some classical numerical SDE discretizations. Some applications, here in financial option valuation, are also presented.

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BELFADLI, R., S. HAMADÈNE, and Y. OUKNINE. "ON ONE-DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS INVOLVING THE MAXIMUM PROCESS." Stochastics and Dynamics 09, no. 02 (June 2009): 277–92. http://dx.doi.org/10.1142/s0219493709002671.

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We prove existence and pathwise uniqueness results for four different types of stochastic differential equations (SDEs) perturbed by the past maximum process and/or the local time at zero. Along the first three studies, the coefficients are no longer Lipschitz. The first type is the equation [Formula: see text] The second type is the equation [Formula: see text] The third type is the equation [Formula: see text] We end the paper by establishing the existence of strong solution and pathwise uniqueness, under Lipschitz condition, for the SDE [Formula: see text]

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Narmontas, Martynas, Petras Rupšys, and Edmundas Petrauskas. "Models for Tree Taper Form: The Gompertz and Vasicek Diffusion Processes Framework." Symmetry 12, no. 1 (January 2, 2020): 80. http://dx.doi.org/10.3390/sym12010080.

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In this work, we employ stochastic differential equations (SDEs) to model tree stem taper. SDE stem taper models have some theoretical advantages over the commonly employed regression-based stem taper modeling techniques, as SDE models have both simple analytic forms and a high level of accuracy. We perform fixed- and mixed-effect parameters estimation for the stem taper models by developing an approximated maximum likelihood procedure and using a data set of longitudinal measurements from 319 mountain pine trees. The symmetric Vasicek- and asymmetric Gompertz-type diffusion processes used adequately describe stem taper evolution. The proposed SDE stem taper models are compared to four regression stem taper equations and four volume equations. Overall, the best goodness-of-fit statistics are produced by the mixed-effect parameters SDEs stem taper models. All results are obtained in the Maple computer algebra system.

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Geiser, Jürgen. "Numerical Picard Iteration Methods for Simulation of Non-Lipschitz Stochastic Differential Equations." Symmetry 12, no. 3 (March 3, 2020): 383. http://dx.doi.org/10.3390/sym12030383.

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In this paper, we present splitting approaches for stochastic/deterministic coupled differential equations, which play an important role in many applications for modelling stochastic phenomena, e.g., finance, dynamics in physical applications, population dynamics, biology and mechanics. We are motivated to deal with non-Lipschitz stochastic differential equations, which have functions of growth at infinity and satisfy the one-sided Lipschitz condition. Such problems studied for example in stochastic lubrication equations, while we deal with rational or polynomial functions. Numerically, we propose an approximation, which is based on Picard iterations and applies the Doléans-Dade exponential formula. Such a method allows us to approximate the non-Lipschitzian SDEs with iterative exponential methods. Further, we could apply symmetries with respect to decomposition of the related matrix-operators to reduce the computational time. We discuss the different operator splitting approaches for a nonlinear SDE with multiplicative noise and compare this to standard numerical methods.

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Sun, Yabing, Jie Yang, and Weidong Zhao. "Itô-Taylor Schemes for Solving Mean-Field Stochastic Differential Equations." Numerical Mathematics: Theory, Methods and Applications 10, no. 4 (September 12, 2017): 798–828. http://dx.doi.org/10.4208/nmtma.2017.0007.

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AbstractThis paper is devoted to numerical methods for mean-field stochastic differential equations (MSDEs). We first develop the mean-field Itô formula and mean-field Itô-Taylor expansion. Then based on the new formula and expansion, we propose the Itô-Taylor schemes of strong order γ and weak order η for MSDEs, and theoretically obtain the convergence rate γ of the strong Itô-Taylor scheme, which can be seen as an extension of the well-known fundamental strong convergence theorem to the mean-field SDE setting. Finally some numerical examples are given to verify our theoretical results.

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Thiruthummal, Abhiram Anand, and Eun-jin Kim. "Monte Carlo Simulation of Stochastic Differential Equation to Study Information Geometry." Entropy 24, no. 8 (August 12, 2022): 1113. http://dx.doi.org/10.3390/e24081113.

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Information Geometry is a useful tool to study and compare the solutions of a Stochastic Differential Equations (SDEs) for non-equilibrium systems. As an alternative method to solving the Fokker–Planck equation, we propose a new method to calculate time-dependent probability density functions (PDFs) and to study Information Geometry using Monte Carlo (MC) simulation of SDEs. Specifically, we develop a new MC SDE method to overcome the challenges in calculating a time-dependent PDF and information geometric diagnostics and to speed up simulations by utilizing GPU computing. Using MC SDE simulations, we reproduce Information Geometric scaling relations found from the Fokker–Planck method for the case of a stochastic process with linear and cubic damping terms. We showcase the advantage of MC SDE simulation over FPE solvers by calculating unequal time joint PDFs. For the linear process with a linear damping force, joint PDF is found to be a Gaussian. In contrast, for the cubic process with a cubic damping force, joint PDF exhibits a bimodal structure, even in a stationary state. This suggests a finite memory time induced by a nonlinear force. Furthermore, several power-law scalings in the characteristics of bimodal PDFs are identified and investigated.

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Jamba, Nelson T., Gonçalo Jacinto, Patrícia A. Filipe, and Carlos A. Braumann. "Estimation for stochastic differential equation mixed models using approximation methods." AIMS Mathematics 9, no. 4 (2024): 7866–94. http://dx.doi.org/10.3934/math.2024383.

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<abstract><p>We used a class of stochastic differential equations (SDE) to model the evolution of cattle weight that, by an appropriate transformation of the weight, resulted in a variant of the Ornstein-Uhlenbeck model. In previous works, we have dealt with estimation, prediction, and optimization issues for this class of models. However, to incorporate individual characteristics of the animals, the average transformed size at maturity parameter $ \alpha $ and/or the growth parameter $ \beta $ may vary randomly from animal to animal, which results in SDE mixed models. Obtaining a closed-form expression for the likelihood function to apply the maximum likelihood estimation method is a difficult, sometimes impossible, task. We compared the known Laplace approximation method with the delta method to approximate the integrals involved in the likelihood function. These approaches were adapted to allow the estimation of the parameters even when the requirement of most existing methods, namely having the same age vector of observations for all trajectories, fails, as it did in our real data example. Simulation studies were also performed to assess the performance of these approximation methods. The results show that the approximation methods under study are a very good alternative for the estimation of SDE mixed models.</p></abstract>

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Herzog, Bodo. "Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion." Mathematics 10, no. 3 (January 23, 2022): 340. http://dx.doi.org/10.3390/math10030340.

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The aim of this work is to establish and generalize a relationship between fractional partial differential equations (fPDEs) and stochastic differential equations (SDEs) to a wider class of stochastic processes, including fractional Brownian motions {BtH,t≥0} and sub-fractional Brownian motions {ξtH,t≥0} with Hurst parameter H∈(12,1). We start by establishing the connection between a fPDE and SDE via the Feynman–Kac Theorem, which provides a stochastic representation of a general Cauchy problem. In hindsight, we extend this connection by assuming SDEs with fractional- and sub-fractional Brownian motions and prove the generalized Feynman–Kac formulas under a (sub-)fractional Brownian motion. An application of the theorem demonstrates, as a by-product, the solution of a fractional integral, which has relevance in probability theory.

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İnce, Nihal, and Aladdin Shamilov. "An Application of New Method to Obtain Probability Density Function of Solution of Stochastic Differential Equations." Applied Mathematics and Nonlinear Sciences 5, no. 1 (March 31, 2020): 337–48. http://dx.doi.org/10.2478/amns.2020.1.00031.

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AbstractIn this study, a new method to obtain approximate probability density function (pdf) of random variable of solution of stochastic differential equations (SDEs) by using generalized entropy optimization methods (GEOM) is developed. By starting given statistical data and Euler–Maruyama (EM) method approximating SDE are constructed several trajectories of SDEs. The constructed trajectories allow to obtain random variable according to the fixed time. An application of the newly developed method includes SDE model fitting on weekly closing prices of Honda Motor Company stock data between 02 July 2018 and 25 March 2019.

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Hu, Yaozhong, and Qun Shi. "Strong solution of stochastic differential equations with discontinuous and unbounded coefficients." Transactions of the American Mathematical Society, Series B 11, no. 44 (December 18, 2024): 1509–55. https://doi.org/10.1090/btran/213.

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In this paper we study the existence and uniqueness of strong solution of the following d d -dimensional stochastic differential equation (SDE) driven by Brownian motion: d X t = b ( t , X t ) d t + σ ( t , X t ) d B t , X 0 = x , \begin{equation*} dX_t=b(t,X_t)dt+\sigma (t,X_t)dB_t, X_0=x, \end{equation*} where B B is a d d -dimensional standard Brownian motion; the diffusion coefficient σ \sigma is a Hölder continuous and uniformly nondegenerate d × d d\times d matrix-valued function and the drift coefficient b b may be discontinuous and unbounded, not necessarily in L p q \mathbb {L}_p^q , extending the previous works to discontinuous and unbounded drift coefficient situation. The idea is to combine the Zvonkin’s transformation with the Lyapunov function approach. Zvonkin’s transformation is a one-to-one (and quasi-isometric) transformation of a phase space that allows us to pass from a diffusion process with nonzero drift coefficient to a process without drift. To this end, we need to establish a local version of the connection between the solutions of the SDE up to the exit time of a bounded connected open set D D and the associated partial differential equation on this domain. As an interesting by-product, we establish a localized version of the Krylov estimates and a localized version of the stability result of the SDEs of discontinuous coefficients.

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Oladayo, ODUSELU-HASSAN, Emmanuel. "Advancing Hybrid Numerical Methods for Nonlinear Stochastic Differential Equations: Applications in Complex Systems." Asian Journal of Research in Computer Science 18, no. 1 (January 14, 2025): 124–32. https://doi.org/10.9734/ajrcos/2025/v18i1553.

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The focus of this work is to consider composite numerical techniques for the approximation of SDEs with nonlinear coefficients in the drift and diffusion terms. SDEs, crucial for modeling systems with stochastic components, contain nonlinear terms that cause analytical solvability, numerical stiffness, and sensitivity to noise. These difficulties pose a problem for traditional techniques such as Euler-Maruyama or Milstein schemes, specifically in stiff or very nonlinear systems. Accompanying exact methods are numerical methods that include a deterministic synthesis of drift terms and a stochastic interpolation of diffusion terms with the purpose of increasing precision and stability and optimizing used computing time. Discussed approaches include implicit-explicit (IMEX) schemes, spectral collocation methods, and machine learning-assisted techniques. IMEX methods handle stiffness in nonlinear drift terms implicitly, while explicitly handling stochastic diffusion. Spectral-collocation methods utilize high-order polynomial approximations for accuracy in discretization where solutions are smooth and defined in a bounded domain. The combination of these techniques and machine learning extends SDE analysis and concentrates on SDE nonlinearities as well as adaptive solution strategies. They find use in every area of discipline, such as stochastic volatility models in finance, population dynamics in biology, and turbulent fluid flows in engineering. Simulation results show that hybrid schemes outperform other methods in terms of accuracy, stability, and computational expense. This work outlines how the integration of the suggested methods can overcome the shortcomings of the classic approaches so as to enable progression in solving complex, high-dimensional, and nonlinear stochastic problems. Subsequent studies will continue to investigate additional adaptive frameworks and more domain-specific and machine learning-based improvements to expand the spectrum of hybrid use.

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Journal articles on the topic 'Stochastic Differential Equations (SDE)'

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