Search Results (120)

We investigate a class of quadratic backward stochastic differential equations (BSDEs) with generators that are singular in y. First, we establish the existence of solutions and a comparison theorem, thereby extending the existing results in the literature. Furthermore, we analyze the stability properties, derive the Feynman–Kac formula, and prove the uniqueness of viscosity solutions for the corresponding singular semi-linear partial differential equations (PDEs). Finally, we demonstrate applications in the context of robust control linked to stochastic differential utility and the certainty equivalent based on g-expectation. In these applications, the quadratic coefficients in the generators, respectively, quantify ambiguity aversion and absolute risk aversion. Full article

A well-known model of infectious diseases, described by a nonlinear system of delay differential equations, is investigated under the influence of stochastic perturbations. Using the general method of Lyapunov functional construction combined with the linear matrix inequality (LMI) approach, we derive sufficient conditions for the stability of the equilibria of the considered system. Numerical simulations illustrating the system’s behavior under stochastic perturbations are provided to support the thoretical findings. The proposed method for stability analysis is broadly applicable to other systems of nonlinear stochastic differential equations across various fields. Full article

We investigate the bridge problem for stochastic processes, that is, we analyze the statistical properties of trajectories constrained to begin and terminate at a fixed position within a time interval $\tau$. Our primary focus is the time-reversal symmetry of these trajectories: under which conditions do the statistical properties remain invariant under the transformation $t\to \tau -t$? To address this question, we compare the stochastic differential equation describing the bridge, derived equivalently via Doob’s transform or stochastic optimal control, with the corresponding equation for the time-reversed bridge. We aim to provide a concise overview of these well-established derivation techniques and subsequently obtain a local condition for the time-reversal asymmetry that is specifically valid for the bridge. We are specifically interested in cases in which detailed balance is not satisfied and aim to eventually quantify the bridge asymmetry and understand how to use it to derive useful information about the underlying out-of-equilibrium dynamics. To this end, we derived a necessary condition for time-reversal symmetry, expressed in terms of the current velocity of the original stochastic process and a quantity linked to detailed balance. As expected, this formulation demonstrates that the bridge is symmetric when detailed balance holds, a sufficient condition that was already known. However, it also suggests that a bridge can exhibit symmetry even when the underlying process violates detailed balance. While we did not identify a specific instance of complete symmetry under broken detailed balance, we present an example of partial symmetry. In this case, some, but not all, components of the bridge display time-reversal symmetry. This example is drawn from a minimal non-equilibrium model, namely Brownian Gyrators, that are linear stochastic processes. We examined non-equilibrium systems driven by a "mechanical” force, specifically those in which the linear drift cannot be expressed as the gradient of a potential. While Gaussian processes like Brownian Gyrators offer valuable insights, it is known that they can be overly simplistic, even in their time-reversal properties. Therefore, we transformed the model into polar coordinates, obtaining a non-Gaussian process representing the squared modulus of the original process. Despite this increased complexity and the violation of detailed balance in the full process, we demonstrate through exact calculations that the bridge of the squared modulus in the isotropic case, constrained to start and end at the origin, exhibits perfect time-reversal symmetry. Full article

This paper addresses the problem of optimal $H_2$-filtering for a class of continuous-time linear McKean–Vlasov stochastic differential equations under sampled measurements. The main tool used to solve the filtering problem is a forward jump matrix linear differential equation with a Riccati-type jumping operator. More specifically, the stabilizing solution of such a jump Riccati-type equation plays a key role. Full article

The white-noise-driven KdV-type Boussinesq system is a class of stochastic partial differential equations (SPDEs) that describe nonlinear wave propagation under the influence of random noise—specifically white noise—and generalize features from both the Korteweg–de Vries (KdV) and Boussinesq equations. We consider a Cauchy problem for two stochastic systems based on the KdV-type Boussinesq equations. For these systems, we determine sufficient conditions to ensure that this problem is locally and globally well posed for initial data in Sobolev spaces by the linear and bilinear estimates and their modification together with the Banach fixed point. Full article

This paper presents a modified Euler—Maruyama (EM) method for mixed stochastic fractional integro—differential equations (mSFIEs) with Caputo—type fractional derivatives whose coefficients satisfy local Lipschitz and linear growth conditions. First, we transform the mSFIEs into an equivalent mixed stochastic Volterra integral equations (mSVIEs) using a fractional calculus technique. Then, we establish the well—posedness of the analytical solutions of the mSVIEs. After that, a modified EM scheme is formulated to approximate the numerical solutions of the mSVIEs, and its strong convergence is proven based on local Lipschitz and linear growth conditions. Furthermore, we derive the modified EM scheme under the same conditions in the $L^2$ sense, which is consistent with the strong convergence result of the corresponding EM scheme. Notably, the strong convergence order under local Lipschitz conditions is inherently lower than the corresponding order under global Lipschitz conditions. Finally, numerical experiments are presented to demonstrate that our approach not only circumvents the restrictive integrability conditions imposed by singular kernels, but also achieves a rigorous convergence order in the $L^2$ sense. Full article

This study investigates the computation of fractional and higher integer-order moments for a stochastic process governed by a one-dimensional, non-homogeneous linear stochastic differential equation (SDE) driven by fractional Brownian motion (fBm). Unlike conventional approaches relying on moment-generating functions or Fokker–Planck equations, which often yield intractable expressions, we derive explicit closed-form formulas for these moments. Our methodology leverages the Wick–Itô calculus (fractional Itô formula) and the properties of Hermite polynomials to express moments efficiently. Additionally, we establish a recurrence relation for moment computation and propose an alternative approach based on generalized binomial expansions. To validate our findings, Monte Carlo simulations are performed, demonstrating a high degree of accuracy between theoretical and empirical results. The proposed framework provides novel insights into stochastic processes with long-memory properties, with potential applications in statistical inference, mathematical finance, and physical modeling of anomalous diffusion. Full article

This paper investigates a backward stochastic linear quadratic control problem with an expected-type equality constraint on the initial state. By using the Lagrange multiplier method, the problem with a uniformly convex cost functional is first transformed into an equivalent unconstrained parameterized backward stochastic linear quadratic control problem. Then, under the surjectivity of the linear constraint, the equivalence between the original problem and the dual problem is proven by Lagrange duality theory. Subsequently, with the help of the maximum principle, an explicit solution of the optimal control for the unconstrained problem is obtained. This solution is feedback-based and determined by an adjoint stochastic differential equation, a Riccati-type ordinary differential equation, a backward stochastic differential equation, and an equality, thereby yielding the optimal control for the original problem. Finally, an optimal control for an investment portfolio problem with an expected-type equality constraint on the initial state is explicitly provided. Full article

This paper extends classical Markov switching models. We introduce a generalized semi-Markov switching framework in which the system dynamics are governed by an Itô stochastic differential equation. Of note, the optimal control synthesis problem is formulated for stochastic dynamic systems with semi-Markov parameters. Further, a system of ordinary differential equations is derived to characterize the Bellman functional and the corresponding optimal control. We investigate the case of linear dynamics in detail, and propose a closed-form solution for the optimal control law. A numerical example is presented to illustrate the theoretical results. Full article

Brownian motion has been studied since 1827, leading to numerous important advances in many branches of science and to it being studied primarily as a stochastic dynamical system. In this paper, we present a deterministic model for the Brownian motion for a particle in a constant force field based on the Ornstein–Uhlenbeck model. By adding one degree of freedom, the system evolves into three differential equations. This change in the model is based on the Jerk equation with commutation surfaces and is analyzed in three cases: overdamped, critically damped, and underdamped. The dynamics of the proposed model are compared with classical results using a random process with normal distribution, where despite the absence of a stochastic component, the model preserves key Brownian motion characteristics, which are lost in stochastic models, giving a new perspective to the study of particle dynamics under different force fields. This is validated by a linear average square displacement and a Gaussian distribution of particle displacement in all cases. Furthermore, the correlation properties are examined using detrended fluctuation analysis (DFA) for compared cases, which confirms that the model effectively replicates the essential behaviors of Brownian motion that the classic models lose. Full article

This paper focuses on the quasi-sure exponential stability of the stochastic differential delay equation driven by G-Brownian motion (SDDE-GBM): $d\xi \left(t\right)=f(t,\xi (t-{\kappa }_1\left(t\right)))dt+g(t,\xi (t-{\kappa }_2\left(t\right)))d\mathcal{Z}\left(t\right),$ where ${\kappa }_1(·),{\kappa }_2(·):{\mathbb{R}}_{+}\to [0,\tau ]$ denote variable delays, and $\mathcal{Z}\left(t\right)$ denotes scalar G-Brownian motion, which has a symmetry distribution. It is shown that the SDDE-GBM is quasi-surely exponentially stable for each $\tau >0$ bounded by ${\tau }^{*}$, where the corresponding (non-delay) stochastic differential equation driven by G-Bronwian motion (SDE-GBM), $d\eta \left(t\right)=f(t,\eta (t\left)\right)dt+g(t,\eta (t\left)\right)d\mathcal{Z}\left(t\right)$, is quasi-surely exponentially stable. Moreover, by solving the non-linear equation on $\tau$, we can obtain the implicit lower bound ${\tau }^{*}$. Finally, illustrating examples are provided. Full article

A class of stochastic linear-quadratic (LQ) dynamic optimization problems involving a large population is investigated in this work. Here, the agents cooperate with each other to minimize certain social costs. Furthermore, differently from the classic social optima literature, the dynamics in this framework are driven by anticipated backward stochastic differential equations (ABSDE) in which the terminal instead of the initial condition is specified and the anticipated terms are involved. The individual admissible controls are constrained in closed convex subsets, and the common noise is considered. As a result, the related social cost is represented by a recursive functional in which the initial state is involved. By virtue of the so-called anticipated person-by-person optimality principle, a decentralized strategy can be derived. This is based on a class of new consistency condition systems, which are mean-field-type anticipated forward-backward stochastic differential delay equations (AFBSDDEs). The well-posedness of such a consistency condition system is obtained through a discounting decoupling method. Finally, the corresponding asymptotic social optimality is proved. Full article

This study introduces a novel hybrid model combining Bayesian Stochastic Partial Differential Equations (SPDE) with deep learning, specifically Convolutional Neural Networks (CNN) and Deep Feedforward Neural Networks (DFFNN), to predict PM_2.5 concentrations. Traditional models often fail to account for non-linear relationships and complex spatial dependencies, critical in urban settings. By integrating SPDE’s spatial-temporal structure with neural networks’ capacity for non-linearity, our model significantly outperforms standalone methods. Accurately predicting air pollution supports sustainable public health strategies and targeted interventions, which are critical for mitigating the adverse health effects of PM_2.5, particularly in urban areas heavily impacted by climate change. The hybrid model was applied to the Pleasant Run Airshed in Indianapolis, Indiana, utilizing a comprehensive dataset that included PM_2.5 sensor data, meteorological variables, and land-use information. By combining SPDE’s ability to model spatial-temporal structures with the adaptive power of neural networks, the model achieved a high level of predictive accuracy, significantly outperforming standalone methods. Additionally, the model’s interpretability was enhanced through the use of SHAP (Shapley Additive Explanations) values, which provided insights into the contribution of each variable to the model’s predictions. This framework holds the potential for improving air quality monitoring and supports more targeted public health interventions and policy-making efforts. Full article

This study presents an integrated approach that combines the Levenberg–Marquardt (LM) and Luus–Jaakola (LJ) algorithms to enhance parameter estimation for various applications. The LM algorithm, known for its precision in solving non-linear least squares problems, is effectively paired with the LJ algorithm, a robust stochastic optimization method, to improve accuracy and computational efficiency. This hybrid LM-LJ methodology is demonstrated through several case studies, including the optimization of MESH equations in distillation processes, modeling controlled diffusion in biopolymer films, and analyzing heat and mass transfer during the drying of cylindrical quince slices. By overcoming the convergence issues typical of gradient-based methods and performing global searches without initial parameter bounds, this approach effectively handles complex models and closely aligns simulation results with experimental data. These capabilities highlight the versatility of this approach in engineering and environmental modeling, significantly enhancing parameter estimation in systems governed by differential equations. Full article

With the load center’s continuous expansion and development of the AC power grid’s scale and construction, the recipient grid under the multi-feed DC environment is facing severe challenges of DC commutation failure and bipolar blocking due to the high strength of AC-DC coupling and the low level of system inertia, which brings many complexities and uncertainties to economic scheduling. In addition, the large-scale grid integration of wind power, photovoltaic, and other intermittent energy sources makes the ultra-high-voltage (UHV) DC channel operation state randomized. The deterministic scenario-based timing power simulation is no longer suitable for the current complex and changeable grid operation state. In this paper, we first start with the description and analysis of the uncertainty in renewable energy (RE) sources, such as wind and solar, and reconstruct the time-sequence power model by using the stochastic differential equation model. Then, a carbon emission trading cost (CET) model is constructed based on the CET mechanism, and the two-stage locating and capacity optimization model for the UHV DC receiving end is proposed under the constraint of dispatch safety and stability. Among them, the first stage starts with the objective of maximizing the carrying capacity of the UHV DC receiving end grid; the second stage checks its dynamic safety under the basic and fault modes according to the results of the first stage and corrects the drop point and capacity of the UHV DC line with the objective of achieving safe and stable UHV DC operation at the lowest economic investment. In addition, the two-stage model innovatively proposes UHV DC relative inertia constraints, peak adjustment margin constraints, transient voltage support constraints under commutation failure conditions, and frequency support constraints under a DC blocking state. In addition, to address the problem that the probabilistic constraints of the scheduling model are difficult to solve, the discrete step-size transformation and convolution sequence operation methods are proposed to transform the chance-constrained planning into mixed-integer linear planning for solving. Finally, the proposed model is validated with a UHV DC channel in 2023, and the results confirm the feasibility and effectiveness of the model. Full article