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Journal articles on the topic 'Mean-field stochastic differential equations (SDE)'

Author: Grafiati

Published: 4 May 2024

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1

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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2

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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3

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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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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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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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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7

Higham, Desmond J., Xuerong Mao, and Andrew M. Stuart. "Exponential Mean-Square Stability of Numerical Solutions to Stochastic Differential Equations." LMS Journal of Computation and Mathematics 6 (2003): 297–313. http://dx.doi.org/10.1112/s1461157000000462.

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AbstractPositive results are proved here about the ability of numerical simulations to reproduce the exponential mean-square stability of stochastic differential equations (SDEs). The first set of results applies under finite-time convergence conditions on the numerical method. Under these conditions, the exponential mean-square stability of the SDE and that of the method (for sufficiently small step sizes) are shown to be equivalent, and the corresponding second-moment Lyapunov exponent bounds can be taken to be arbitrarily close. The required finite-time convergence conditions hold for the class of stochastic theta methods on globally Lipschitz problems. It is then shown that exponential mean-square stability for non-globally Lipschitz SDEs is not inherited, in general, by numerical methods. However, for a class of SDEs that satisfy a one-sided Lipschitz condition, positive results are obtained for two implicit methods. These results highlight the fact that for long-time simulation on nonlinear SDEs, the choice of numerical method can be crucial.

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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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Rupšys, Petras. "Modeling Dynamics of Structural Components of Forest Stands Based on Trivariate Stochastic Differential Equation." Forests 10, no. 6 (June 14, 2019): 506. http://dx.doi.org/10.3390/f10060506.

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Research Highlights: Today’s approaches to modeling of forest stands are in most cases based on that the regression models and they are constructed as static sub-models describing individual stands variables. The disadvantages of this method; it is laborious because too many different equations need to be assessed and empirical choices of candidate equations make the results subjective; it does not relate to the stand variables dynamics against the age dimension (time); and does not consider the underlying covariance structure driving changes in the stand variables. In this study, the dynamical model defined by a fixed-and mixed effect parameters trivariate stochastic differential equation (SDE) is introduced and described how such a model can be used to model quadratic mean diameter, mean height, number of trees per hectare, self-thinning line, stand basal area, stand volume per hectare and much more. Background and Objectives: New developed marginal and conditional trivariate probability density functions, combining information generated from an age-dependent variance-covariance matrix of quadratic mean diameter, mean height and number of trees per hectare, improve stand growth prediction, and forecast (in forecast the future is completely unavailable and must only be estimated from historical patterns) accuracies. Materials and Methods: Fixed-and mixed effect parameters SDE models were harmonized to predict and forecast the dynamics of quadratic mean diameter, mean height, number of trees per hectare, basal area, stand volume per hectare, and their current and mean increments. The results and experience from applying the SDE concepts and techniques in an extensive whole stand growth and yield analysis are described using a Scots pine (Pinus sylvestris L.) experimental dataset in Lithuania. Results: The mixed effects scenario SDE model showed high accuracy, the percentage root mean square error values for quadratic mean diameter, mean height, number of trees per hectare, stand basal area and stand volume per hectare predictions (forecasts) were 3.37% (10.44%), 1.82% (2.07%), 1.76% (2.93%), 6.65% (10.41%) and 6.50% (8.93%), respectively. In the same way, the quadratic mean diameter, mean height, number of trees per hectare, stand basal area and stand volume per hectare prediction (forecast) relationships had high values of the coefficient of determination, R2, 0.998 (0.987), 0.997 (0.992), 0.997 (0.988), 0.968 (0.984) and 0.966 (0.980), respectively. Conclusions: The approach presented in this paper can be used for developing a new generation stand growth and yield models.

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10

Jaworski, Piotr. "On Copula-Itô processes." Dependence Modeling 7, no. 1 (November 1, 2019): 322–47. http://dx.doi.org/10.1515/demo-2019-0017.

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AbstractWe study the dynamics of the family of copulas {Ct}t≥0 of a pair of stochastic processes given by stochastic differential equations (SDE). We associate to it a parabolic partial differential equation (PDE). Having embedded the set of bivariate copulas in a dual of a Sobolev Hilbert space H1 (ℝ2)* we calculate the derivative with respect to t and the *weak topology i.e. the tangent vector field to the image of the curve t → Ct. Furthermore we show that the family {Ct}t≥0 is an orbit of a strongly continuous semigroup of transformations and provide the infinitesimal generator of this semigroup.

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Mykhailenko, Viacheslav, and Pavol Bobik. "Statistical Error for Cosmic Rays Modulation Evaluated by SDE Backward in Time Method for 1D Model." Fluids 7, no. 2 (January 19, 2022): 46. http://dx.doi.org/10.3390/fluids7020046.

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The propagation of cosmic rays through the heliosphere has been solved for more than half a century by stochastic methods based on Ito’s lemma. This work presents the estimation of statistical error of solution of Fokker–Planck equation by the 1D backward in time stochastic differential equations method. The error dependence on simulation statistics and energy is presented for different combinations of input parameters. The 1% precision criterion in mean value units of intensity standard deviation is defined as a function of solar wind velocity and diffusion coefficient value.

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Averina, Tatyana. "Conditional Optimization of Algorithms for Estimating Distributions of Solutions to Stochastic Differential Equations." Mathematics 12, no. 4 (February 16, 2024): 586. http://dx.doi.org/10.3390/math12040586.

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This article discusses an alternative method for estimating marginal probability densities of the solution to stochastic differential equations (SDEs). Two algorithms for calculating the numerical–statistical projection estimate for distributions of solutions to SDEs using Legendre polynomials are proposed. The root-mean-square error of this estimate is studied as a function of the projection expansion length, while the step of a numerical method for solving SDE and the sample size for expansion coefficients are fixed. The proposed technique is successfully verified on three one-dimensional SDEs that have stationary solutions with given one-dimensional distributions and exponential correlation functions. A comparative analysis of the proposed method for calculating the numerical–statistical projection estimate and the method for constructing the histogram is carried out.

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13

Hutzenthaler, Martin, Arnulf Jentzen, and Peter E. Kloeden. "Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467, no. 2130 (December 15, 2010): 1563–76. http://dx.doi.org/10.1098/rspa.2010.0348.

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The stochastic Euler scheme is known to converge to the exact solution of a stochastic differential equation (SDE) with globally Lipschitz continuous drift and diffusion coefficients. Recent results extend this convergence to coefficients that grow, at most, linearly. For superlinearly growing coefficients, finite-time convergence in the strong mean-square sense remains. In this article, we answer this question to the negative and prove, for a large class of SDEs with non-globally Lipschitz continuous coefficients, that Euler’s approximation converges neither in the strong mean-square sense nor in the numerically weak sense to the exact solution at a finite time point. Even worse, the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean-square sense and in the numerically weak sense.

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Esquível, Manuel L., Paula Patrício, and Gracinda R. Guerreiro. "From ODE to Open Markov Chains, via SDE: an application to models for infections in individuals and populations." Computational and Mathematical Biophysics 8, no. 1 (December 17, 2020): 180–97. http://dx.doi.org/10.1515/cmb-2020-0110.

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AbstractWe present a methodology to connect an ordinary differential equation (ODE) model of interacting entities at the individual level, to an open Markov chain (OMC) model of a population of such individuals, via a stochastic differential equation (SDE) intermediate model. The ODE model here presented is formulated as a dynamic change between two regimes; one regime is of mean reverting type and the other is of inverse logistic type. For the general purpose of defining an OMC model for a population of individuals, we associate an Ito processes, in the form of SDE to ODE system of equations, by means of the addition of Gaussian noise terms which may be thought to model non essential characteristics of the phenomena with small and undifferentiated influences. The next step consists on discretizing the SDE and using the discretized trajectories computed by simulation to define transitions of a finite valued Markov chain; for that, the state space of the Ito processes is partitioned according to some rule. For the example proposed for illustration, the state space of the ODE system referred – corresponding to a model of a viral infection – is partitioned into six infection classes determined by some of the critical points of the ODE system; we detail the evolution of some infected population in these infection classes.

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Rupšys, Petras. "Generalized fixed-effects and mixed-effects parameters height–diameter models with diffusion processes." International Journal of Biomathematics 08, no. 05 (August 13, 2015): 1550060. http://dx.doi.org/10.1142/s1793524515500606.

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Statistical models using stochastic differential equations (SDEs) to describe dynamically evolving natural systems are appearing in the scientific literature with some regularity in recent years. In this study, the SDE mixed-effects parameter models based on a Vasicek non-homogeneous diffusion process are formulated. The breast height diameter-dependent drift function additionally depends on deterministic function that describes the dynamic of certain exogenous stand variables (crown height, c h , crown width, c w , mean breast height diameter, d0, mean height, h0, age, A, soil fertility index, SFI, stocking level, S) versus breast height diameter. The mixed-effects parameters SDE models included a random parameter that affected the models asymptote. The parameter estimators are evaluated by maximum likelihood procedure. The objective of the research was to develop a generalized mixed-effects parameters SDE height–diameter models and to illustrate issues using dataset of Scots pine trees (Pinus sylvestris L.) in Lithuania with the breast height diameter outside the bark larger than 0 cm. The parameters of all used models were estimated using an estimation dataset and were evaluated using a validation dataset. The new developed height–diameter models are an improvement over exogenous stand variables, in that it can be calibrated to a new stand with observed height–diameter pairs, thus improving height prediction.

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Fagin, Joshua, Ji Won Park, Henry Best, James H. H. Chan, K. E. Saavik Ford, Matthew J. Graham, V. Ashley Villar, Shirley Ho, and Matthew O’Dowd. "Latent Stochastic Differential Equations for Modeling Quasar Variability and Inferring Black Hole Properties." Astrophysical Journal 965, no. 2 (April 1, 2024): 104. http://dx.doi.org/10.3847/1538-4357/ad2988.

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Abstract Quasars are bright and unobscured active galactic nuclei (AGN) thought to be powered by the accretion of matter around supermassive black holes at the centers of galaxies. The temporal variability of a quasar’s brightness contains valuable information about its physical properties. The UV/optical variability is thought to be a stochastic process, often represented as a damped random walk described by a stochastic differential equation (SDE). Upcoming wide-field telescopes such as the Rubin Observatory Legacy Survey of Space and Time (LSST) are expected to observe tens of millions of AGN in multiple filters over a ten year period, so there is a need for efficient and automated modeling techniques that can handle the large volume of data. Latent SDEs are machine learning models well suited for modeling quasar variability, as they can explicitly capture the underlying stochastic dynamics. In this work, we adapt latent SDEs to jointly reconstruct multivariate quasar light curves and infer their physical properties such as the black hole mass, inclination angle, and temperature slope. Our model is trained on realistic simulations of LSST ten year quasar light curves, and we demonstrate its ability to reconstruct quasar light curves even in the presence of long seasonal gaps and irregular sampling across different bands, outperforming a multioutput Gaussian process regression baseline. Our method has the potential to provide a deeper understanding of the physical properties of quasars and is applicable to a wide range of other multivariate time series with missing data and irregular sampling.

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Giles, Michael B., Mateusz B. Majka, Lukasz Szpruch, Sebastian J. Vollmer, and Konstantinos C. Zygalakis. "Multi-level Monte Carlo methods for the approximation of invariant measures of stochastic differential equations." Statistics and Computing 30, no. 3 (September 10, 2019): 507–24. http://dx.doi.org/10.1007/s11222-019-09890-0.

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Abstract We develop a framework that allows the use of the multi-level Monte Carlo (MLMC) methodology (Giles in Acta Numer. 24:259–328, 2015. 10.1017/S096249291500001X) to calculate expectations with respect to the invariant measure of an ergodic SDE. In that context, we study the (over-damped) Langevin equations with a strongly concave potential. We show that when appropriate contracting couplings for the numerical integrators are available, one can obtain a uniform-in-time estimate of the MLMC variance in contrast to the majority of the results in the MLMC literature. As a consequence, a root mean square error of $$\mathcal {O}(\varepsilon )$$O(ε) is achieved with $$\mathcal {O}(\varepsilon ^{-2})$$O(ε-2) complexity on par with Markov Chain Monte Carlo (MCMC) methods, which, however, can be computationally intensive when applied to large datasets. Finally, we present a multi-level version of the recently introduced stochastic gradient Langevin dynamics method (Welling and Teh, in: Proceedings of the 28th ICML, 2011) built for large datasets applications. We show that this is the first stochastic gradient MCMC method with complexity $$\mathcal {O}(\varepsilon ^{-2}|\log {\varepsilon }|^{3})$$O(ε-2|logε|3), in contrast to the complexity $$\mathcal {O}(\varepsilon ^{-3})$$O(ε-3) of currently available methods. Numerical experiments confirm our theoretical findings.

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Sharma, Shambhu N., and H. Parthasarathy. "Dynamics of a stochastically perturbed two-body problem." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2080 (January 16, 2007): 979–1003. http://dx.doi.org/10.1098/rspa.2006.1801.

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In classical mechanics, the two-body problem has been well studied. The governing equations form a system of two-coupled second-order nonlinear differential equations for the radial and angular coordinates. The perturbation induced by the astronomical disturbance like ‘dust’ is normally not considered in the orbit dynamics. Distributed dust produces an additional random force on the orbiting particle, which can be modelled as a random force having ‘Gaussian statistics’. The estimation of accurate positioning of the orbiting particle is not possible without accounting for the stochastic perturbation felt by the orbiting particle. The objective of this paper is to use the stochastic differential equation (SDE) formalism to study the effect of such disturbances on the orbiting body. Specifically, in this paper, we linearize SDEs about the mean of the state vector. The linearization operation performed above, transforms the system of SDEs into another system of SDEs that resembles a bilinear system, as described in signal processing and control literature. However, the mean trajectory of the resulting bilinear stochastic differential model does not preserve the perturbation effect felt by the orbiting particle; only the variance trajectory includes the perturbation effect. For this reason, the effectiveness of the dust-perturbed model is examined on the basis of the bilinear and second-order approximations of the system nonlinearity . The bilinear and second-order approximations of the system nonlinearity allow substantial simplifications for the numerical implementation and preserve some of the properties of the original stochastically perturbed model. Most notably, this paper reveals that the Brownian motion process is accurate to model and study the effect of dust perturbation on the orbiting particle. In addition, analytical findings are supported with finite difference method-based numerical simulations.

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Lahiri, Abhirup, and Tarun Kumar Rawat. "Noise analysis of single stage fractional-order low-pass filter using stochastic and fractional Calculus." ECTI Transactions on Electrical Engineering, Electronics, and Communications 7, no. 2 (September 5, 2008): 47–54. http://dx.doi.org/10.37936/ecti-eec.200972.171889.

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In this paper, we present the noise analysis of a simple single stage low-pass ¯lter (SSLPF) with the fractional-order capacitor, using stochastic differential equations (SDE). The input noise is considered to be white and various solution statistics of output namely mean, variance, auto-correlation and power spectral density (PSD) are obtained using tools from both stochastic and fractional calculus. We investigate the change in these statistics with the change in the capacitor order. The closed form solutions of the step response of the fractional filter are also provided and it has been found that filters with capacitor order greater than unity have a faster step response but suffer from higher output noise and although, the filters with capacitor order less than unity enjoy advantage of less output noise, but they have a sluggish step response. And hence, an appropriate fractional capacitor can be chosen for the desired circuit behavior. A brief study of more generic class of single stage fractional-order high-pass and all-pass ¯ltering functions has been included. The idea can be extended to more complex and practical fractional order circuits.

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Xue, Xirui, Shucai Huang, Daozhi Wei, and Jiahao Xie. "Multiradar Joint Tracking of Cluster Targets Based on Graph-LSTMs." Journal of Sensors 2022 (November 14, 2022): 1–20. http://dx.doi.org/10.1155/2022/8556477.

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The cluster target brings a serious challenge to the traditional multisensor multitarget tracking algorithm because of its large number of members and the cooperative interaction between members. Using multiradar joint tracking cluster target is an alternative method to solve the problem of cluster target tracking, but it inevitably brings the problem of radar-target assignment and tracking information fusion. Aiming at the problem of radar-target assignment and tracking information fusion, a joint tracking method based on graph-long short-term memory neural nets (Graph-LSTMs) is proposed. Firstly, we use multivariable stochastic differential equations (SDE) to model the cooperative interaction of cluster members and transform the derived state space model of cluster members into the same form as the constant velocity (CV) motion model, and the target state equation of cluster which can be used for Bayesian filtering iteration is established. Secondly, based on the detection relationship between radars and cluster members, we introduce the detection confirmation matrix and propose a radar-target assignment method to achieve multiple measurements of single member and detection coverage of all cluster members. Then, each radar uses δ-GLMB filter to estimate the motion state of the assigned targets. Finally, on the basis of spatial discretization, the labels of multiple estimates of cluster member states are obtained. We use the designed Graph-LSTMs to learn the cooperative relationship between target states to fuse the labels and obtain better tracking effect. The experimental results show that the proposed method effectively simulates the cluster motion and realizes the joint estimation of cluster target motion state by multiradar. Our method makes up for the defect that a single radar cannot stably track adjacent multiple targets and achieves better estimation fusion effect than the expectation-maximization (EM) algorithm and mean method.

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Zhu, Jie. "The Mean Field Forward Backward Stochastic Differential Equations and Stochastic Partial Differential Equations." Pure and Applied Mathematics Journal 4, no. 3 (2015): 120. http://dx.doi.org/10.11648/j.pamj.20150403.20.

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Li, Zhi, and Jiaowan Luo. "Mean-field reflected backward stochastic differential equations." Statistics & Probability Letters 82, no. 11 (November 2012): 1961–68. http://dx.doi.org/10.1016/j.spl.2012.06.018.

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Buckdahn, Rainer, Juan Li, and Shige Peng. "Mean-field backward stochastic differential equations and related partial differential equations." Stochastic Processes and their Applications 119, no. 10 (October 2009): 3133–54. http://dx.doi.org/10.1016/j.spa.2009.05.002.

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Agram, Nacira, Yaozhong Hu, and Bernt Øksendal. "Mean-field backward stochastic differential equations and applications." Systems & Control Letters 162 (April 2022): 105196. http://dx.doi.org/10.1016/j.sysconle.2022.105196.

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Li, Juan, and Hui Min. "Weak solutions of mean-field stochastic differential equations." Stochastic Analysis and Applications 35, no. 3 (February 15, 2017): 542–68. http://dx.doi.org/10.1080/07362994.2017.1278706.

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Zong, Gaofeng, and Zengjing Chen. "Harnack inequality for mean-field stochastic differential equations." Statistics & Probability Letters 83, no. 5 (May 2013): 1424–32. http://dx.doi.org/10.1016/j.spl.2013.01.035.

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Buckdahn, Rainer, Juan Li, Shige Peng, and Catherine Rainer. "Mean-field stochastic differential equations and associated PDEs." Annals of Probability 45, no. 2 (March 2017): 824–78. http://dx.doi.org/10.1214/15-aop1076.

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Zhu, Qingfeng, and Yufeng Shi. "Mean-Field Forward-Backward Doubly Stochastic Differential Equations and Related Nonlocal Stochastic Partial Differential Equations." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/194341.

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Mean-field forward-backward doubly stochastic differential equations (MF-FBDSDEs) are studied, which extend many important equations well studied before. Under some suitable monotonicity assumptions, the existence and uniqueness results for measurable solutions are established by means of a method of continuation. Furthermore, the probabilistic interpretation for the solutions to a class of nonlocal stochastic partial differential equations (SPDEs) combined with algebra equations is given.

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Dumitrescu, Roxana, Bernt Øksendal, and Agnès Sulem. "Stochastic Control for Mean-Field Stochastic Partial Differential Equations with Jumps." Journal of Optimization Theory and Applications 176, no. 3 (February 20, 2018): 559–84. http://dx.doi.org/10.1007/s10957-018-1243-3.

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Elbarrimi, Oussama, and Youssef Ouknine. "Approximation of solutions of mean-field stochastic differential equations." Stochastics and Dynamics 21, no. 01 (March 11, 2020): 2150003. http://dx.doi.org/10.1142/s0219493721500039.

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Our aim in this paper is to establish some strong stability properties of solutions of mean-field stochastic differential equations. These latter are stochastic differential equations where the coefficients depend not only on the state of the unknown process but also on its probability distribution. The results are obtained assuming that the pathwise uniqueness property holds and using Skorokhod’s selection theorem.

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Lu, Wen, and Yong Ren. "MEAN-FIELD BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS ON MARKOV CHAINS." Bulletin of the Korean Mathematical Society 54, no. 1 (January 31, 2017): 17–28. http://dx.doi.org/10.4134/bkms.b150007.

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32

Hao, Tao. "Anticipated mean-field backward stochastic differential equations with jumps∗." Lithuanian Mathematical Journal 60, no. 3 (May 31, 2020): 359–75. http://dx.doi.org/10.1007/s10986-020-09484-8.

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33

Buckdahn, Rainer, Boualem Djehiche, Juan Li, and Shige Peng. "Mean-field backward stochastic differential equations: A limit approach." Annals of Probability 37, no. 4 (July 2009): 1524–65. http://dx.doi.org/10.1214/08-aop442.

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34

Min, Hui, Ying Peng, and Yongli Qin. "Fully Coupled Mean-Field Forward-Backward Stochastic Differential Equations and Stochastic Maximum Principle." Abstract and Applied Analysis 2014 (2014): 1–15. http://dx.doi.org/10.1155/2014/839467.

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Abstract:

We discuss a new type of fully coupled forward-backward stochastic differential equations (FBSDEs) whose coefficients depend on the states of the solution processes as well as their expected values, and we call them fully coupled mean-field forward-backward stochastic differential equations (mean-field FBSDEs). We first prove the existence and the uniqueness theorem of such mean-field FBSDEs under some certain monotonicity conditions and show the continuity property of the solutions with respect to the parameters. Then we discuss the stochastic optimal control problems of mean-field FBSDEs. The stochastic maximum principles are derived and the related mean-field linear quadratic optimal control problems are also discussed.

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35

Zhu, Qingfeng, Lijiao Su, Fuguo Liu, Yufeng Shi, Yong’ao Shen, and Shuyang Wang. "Mean-field type forward-backward doubly stochastic differential equations and related stochastic differential games." Frontiers of Mathematics in China 15, no. 6 (December 2020): 1307–26. http://dx.doi.org/10.1007/s11464-020-0889-y.

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36

Ma, Limin, Weihai Zhang, and Zhenbin Liu. "Relationship between Nash Equilibrium Strategies and H2/H∞ Control of Mean-Field Stochastic Differential Equations with Multiplicative Noise." Processes 11, no. 11 (November 4, 2023): 3154. http://dx.doi.org/10.3390/pr11113154.

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The relationship between finite-horizon mean-field stochastic H2/H∞ control and Nash equilibrium strategies is investigated in this technical note. First, the finite-horizon mean-field stochastic bounded real lemma (SBRL) is established, which is key to developing the H∞ theory. Second, for mean-field stochastic differential equations (MF-SDEs) with control- and state-dependent noises, it is revealed that the existence of Nash equilibrium strategies is equivalent to the solvability of generalized differential Riccati equations (GDREs). Furthermore, the existence of Nash equilibrium strategies is equivalent to the solvability of H2/H∞ control for MF-SDEs with control- and state-dependent noises. However, for mean-field stochastic systems with disturbance-dependent noises, these two problems are not equivalent. Finally, a sufficient and necessary condition is presented via coupled matrix-valued equations for the finite-horizon H2/H∞ control of mean-field stochastic differential equations with disturbance-dependent noises.

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37

Sun, Yabing, and Weidong Zhao. "Numerical methods for mean-field stochastic differential equations with jumps." Numerical Algorithms 88, no. 2 (February 4, 2021): 903–37. http://dx.doi.org/10.1007/s11075-020-01062-w.

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38

Xiaocui, Ma, and Xi Fubao. "Moderate deviations for mean-field stochastic differential equations with jumps." SCIENTIA SINICA Mathematica 50, no. 1 (August 5, 2019): 87. http://dx.doi.org/10.1360/n012018-00192.

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39

Hancheng, Guo, and Ren Xiuyun. "Mean-field backward stochastic differential equations with uniformly continuous generators." Journal of Control and Decision 2, no. 2 (April 3, 2015): 142–54. http://dx.doi.org/10.1080/23307706.2015.1027796.

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40

Cai, Yujie, Jianhui Huang, and Vasileios Maroulas. "Large deviations of mean-field stochastic differential equations with jumps." Statistics & Probability Letters 96 (January 2015): 1–9. http://dx.doi.org/10.1016/j.spl.2014.08.010.

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41

Sun, Yabing, Weidong Zhao, and Tao Zhou. "Explicit theta-Schemes for Mean-Field Backward Stochastic Differential Equations." SIAM Journal on Numerical Analysis 56, no. 4 (January 2018): 2672–97. http://dx.doi.org/10.1137/17m1161944.

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42

Lu, Wen, Yong Ren, and Lanying Hu. "Mean-field backward stochastic differential equations in general probability spaces." Applied Mathematics and Computation 263 (July 2015): 1–11. http://dx.doi.org/10.1016/j.amc.2015.04.014.

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43

Nykänen, Jani. "Mean-field stochastic differential equations with a discontinuous diffusion coefficient." Probability, Uncertainty and Quantitative Risk 8, no. 3 (2023): 351–72. http://dx.doi.org/10.3934/puqr.2023016.

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44

Li, Junsong, Chao Mi, Chuanzhi Xing, and Dehao Zhao. "General Coupled Mean-Field Reflected Forward-Backward Stochastic Differential Equations." Acta Mathematica Scientia 43, no. 5 (July 12, 2023): 2234–62. http://dx.doi.org/10.1007/s10473-023-0518-4.

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45

Li, Xun, Jingtao Shi, and Jiongmin Yong. "Mean-field linear-quadratic stochastic differential games in an infinite horizon." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 81. http://dx.doi.org/10.1051/cocv/2021078.

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This paper is concerned with two-person mean-field linear-quadratic non-zero sum stochastic differential games in an infinite horizon. Both open-loop and closed-loop Nash equilibria are introduced. The existence of an open-loop Nash equilibrium is characterized by the solvability of a system of mean-field forward-backward stochastic differential equations in an infinite horizon and the convexity of the cost functionals, and the closed-loop representation of an open-loop Nash equilibrium is given through the solution to a system of two coupled non-symmetric algebraic Riccati equations. The existence of a closed-loop Nash equilibrium is characterized by the solvability of a system of two coupled symmetric algebraic Riccati equations. Two-person mean-field linear-quadratic zero-sum stochastic differential games in an infinite horizon are also considered. Both the existence of open-loop and closed-loop saddle points are characterized by the solvability of a system of two coupled generalized algebraic Riccati equations with static stabilizing solutions. Mean-field linear-quadratic stochastic optimal control problems in an infinite horizon are discussed as well, for which it is proved that the open-loop solvability and closed-loop solvability are equivalent.

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46

Li, Juan, and Hui Min. "Weak Solutions of Mean-Field Stochastic Differential Equations and Application to Zero-Sum Stochastic Differential Games." SIAM Journal on Control and Optimization 54, no. 3 (January 2016): 1826–58. http://dx.doi.org/10.1137/15m1015583.

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47

Zhao, Nana, Jinghan Wang, Yufeng Shi, and Qingfeng Zhu. "General Time-Symmetric Mean-Field Forward-Backward Doubly Stochastic Differential Equations." Symmetry 15, no. 6 (May 24, 2023): 1143. http://dx.doi.org/10.3390/sym15061143.

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In this paper, a general class of time-symmetric mean-field stochastic systems, namely the so-called mean-field forward-backward doubly stochastic differential equations (mean-field FBDSDEs, in short) are studied, where coefficients depend not only on the solution processes but also on their law. We first verify the existence and uniqueness of solutions for the forward equation of general mean-field FBDSDEs under Lipschitz conditions, and we obtain the associated comparison theorem; similarly, we also verify those results about the backward equation. As the above two comparison theorems’ application, we prove the existence of the maximal solution for general mean-field FBDSDEs under some much weaker monotone continuity conditions. Furthermore, under appropriate assumptions we prove the uniqueness of the solution for the equations. Finally, we also obtain a comparison theorem for coupled general mean-field FBDSDEs.

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48

Sun, Shengqiu. "Mean‐field backward stochastic differential equations driven by G ‐Brownian motion and related partial differential equations." Mathematical Methods in the Applied Sciences 43, no. 12 (May 31, 2020): 7484–505. http://dx.doi.org/10.1002/mma.6573.

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49

Du, Kai, and Zhen Wu. "Linear-Quadratic Stackelberg Game for Mean-Field Backward Stochastic Differential System and Application." Mathematical Problems in Engineering 2019 (February 21, 2019): 1–17. http://dx.doi.org/10.1155/2019/1798585.

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This paper is concerned with a new kind of Stackelberg differential game of mean-field backward stochastic differential equations (MF-BSDEs). By means of four Riccati equations (REs), the follower first solves a backward mean-field stochastic LQ optimal control problem and gets the corresponding open-loop optimal control with the feedback representation. Then the leader turns to solve an optimization problem for a 1×2 mean-field forward-backward stochastic differential system. In virtue of some high-dimensional and complicated REs, we obtain the open-loop Stackelberg equilibrium, and it admits a state feedback representation. Finally, as applications, a class of stochastic pension fund optimization problems which can be viewed as a special case of our formulation is studied and the open-loop Stackelberg strategy is obtained.

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50

Shi, Yu Feng, Jia Qiang Wen, and Jie Xiong. "Mean-Field Backward Stochastic Differential Equations Driven by Fractional Brownian Motion." Acta Mathematica Sinica, English Series 37, no. 7 (July 2021): 1156–70. http://dx.doi.org/10.1007/s10114-021-0002-9.

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