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In [SIAM J. Appl. Dyn. Syst., 12 (2013), pp. 2068--2092], Widiasih proposed and analyzed a deterministic one-dimensional Budyko--Sellers energy-balance model with a moving ice line. In this paper, we extend this model to the stochastic setting and analyze it within the framework of stochastic slow-fast systems. We derive the dynamics for the ice line in the limit of a small parameter as a solution to a stochastic differential equation. The stochastic approach enables the study of co-existing (metastable) climate states as well as the transition dynamics between them. [ABSTRACT FROM AUTHOR]

The G-Brownian-motion-driven stochastic differential equations (G-SDEs) as well as the G-expectation, which were seminally proposed by Peng and his colleagues, have been extensively applied to describing a particular kind of uncertainty arising in real-world systems modeling. Mathematically depicting long-time and limit behaviors of the solution produced by G-SDEs is beneficial to understanding the mechanisms of system's evolution. Here, we develop a new G-semimartingale convergence theorem and further establish a new invariance principle for investigating the long-time behaviors emergent in G-SDEs. We also validate the uniqueness and the global existence of the solution of G-SDEs whose vector fields are only locally Lipschitzian with a linear upper bound. To demonstrate the broad applicability of our analytically established results, we investigate its application to achieving G-stochastic control in a few representative dynamical systems. [ABSTRACT FROM AUTHOR]

We introduce the notion of mean viability for controlled stochastic differential equations and establish counterparts of Nagumo's classical viability theorems (necessary and sufficient conditions for mean viability). As an application, we provide a purely probabilistic proof of a comparison principle and of existence for contingent and viscosity solutions of second-order fully nonlinear path-dependent Hamilton--Jacobi--Bellman equations. We do not use compactness and optimal stopping arguments, which are usually employed in the literature on viscosity solutions for second-order path-dependent PDEs. [ABSTRACT FROM AUTHOR]

In this manuscript we consider a class of optimal control problems of stochastic differential delay equations. First, we rewrite the problem in a suitable infinite-dimensional Hilbert space. Then, using the dynamic programming approach, we characterize the value function of the problem as the unique viscosity solution of the associated infinite-dimensional Hamilton-Jacobi-Bellman equation. Finally, we prove a C1,α -partial regularity of the value function. We apply these results to path dependent financial and economic problems (Merton-like portfolio problem and optimal advertising). [ABSTRACT FROM AUTHOR]

The multilevel Monte Carlo (MLMC) method is highly efficient for estimating expectations of a functional of a solution to a stochastic differential equation (SDE). However, MLMC estimators may be unstable and have a poor (noncanonical) complexity in the case of low regularity of the functional. To overcome this issue, we extend our previously introduced idea of numerical smoothing in [Quant. Finance, 23 (2023), pp. 209-227], in the context of deterministic quadrature methods to the MLMC setting. The numerical smoothing technique is based on root-finding methods combined with one-dimensional numerical integration with respect to a single well-chosen variable. This study is motivated by the computation of probabilities of events, pricing options with a discontinuous payoff, and density estimation problems for dynamics where the discretization of the underlying stochastic processes is necessary. The analysis and numerical experiments reveal that the numerical smoothing significantly improves the strong convergence and, consequently, the complexity and robustness (by making the kurtosis at deep levels bounded) of the MLMC method. In particular, we show that numerical smoothing enables recovering the MLMC complexities obtained for Lipschitz functionals due to the optimal variance decay rate when using the Euler--Maruyama scheme. For the Milstein scheme, numerical smoothing recovers the canonical MLMC complexity, even for the nonsmooth integrand mentioned above. Finally, our approach efficiently estimates univariate and multivariate density functions. [ABSTRACT FROM AUTHOR]

First hitting times (FHTs) describe the time it takes a random "searcher" to find a "target" and are used to study timescales in many applications. FHTs have been well-studied for diffusive search, especially for small targets, which is called the narrow capture or narrow escape problem. In this paper, we study the FHT to small targets for a one-dimensional superdiffusive search described by a Lévy flight. By applying the method of matched asymptotic expansions to a fractional differential equation we obtain an explicit asymptotic expansion for the mean FHT (MFHT). For fractional order s ε (0, 1) (describing a (2s)-stable Lévy flight whose squared displacement scales as t1/s in time t) and targets of radius \varepsilon \ll 1, we show that the MFHT is order one for s ε (1/2, 1) and diverges as log(1/\varepsilon) for s = 1/2 and \varepsilon 2s 1 for s ε (0, 1/2). We then use our asymptotic results to identify the value of s ε (0, 1] which minimizes the average MFHT and find that (a) this optimal value of s vanishes for sparse targets and (b) the value s = 1/2 (corresponding to an inverse square Lévy search) is optimal in only very specific circumstances. We confirm our results by comparison to both deterministic numerical solutions of the associated fractional differential equation and stochastic simulations. [ABSTRACT FROM AUTHOR]

In this paper, we study optimal stochastic control problems for stochastic systems driven by non-Markov subdiffusion BLt, which have mixed features of deterministic and stochastic controls. Here Bt is the standard Brownian motion on R, and Lt := inf { r > 0 : Sr > t}, t ≥ 0, is the inverse of a subordinator St with drift κ > 0 that is independent of Bt. We obtain stochastic maximum principles (SMPs) for these systems using both convex and spiking variational methods, depending on whether or not the domain is convex. To derive SMPs, we first establish a martingale representation theorem for subdiffusions BLt , and then use it to derive the existence and uniqueness result for the solutions of backward stochastic differential equations (BSDEs) driven by subdiffusions, which may be of independent interest. We also derive sufficient SMPs. Application to a linear quadratic system is given to illustrate the main results of this paper. [ABSTRACT FROM AUTHOR]

This paper provides some sufficient conditions for the existence and uniqueness and the stochastic stability of the global solution for nonlinear neutral stochastic functional differential equations. When the drift term and the diffusion term satisfy a locally Lipschitz condition, and the Lyapunov monotonicity condition has a sign-changed time-varying coefficient, the existence and uniqueness of the global solution for such equations will be studied by using the Lyapunov--Krasovskii function approach and the theory of stochastic analysis. The stability in pth-moment, the asymptotical stability in pth-moment, and the exponential stability in pth-moment will be investigated. Different characterizations for these three kinds of stochastic stability in moment will be established, which are presented with respect to integration conditions. These results have seldom been reported in the existing literature. The almost surely exponential stability for the global solution of such equations is also discussed. Some discussions and comparisons are provided. Two examples are given to check the effectiveness of the theoretical results obtained. [ABSTRACT FROM AUTHOR]

We introduce a predictor-corrector discretization scheme for the numerical integration of a class of stochastic differential equations and prove that it converges with weak order 1.0. The key feature of the new scheme is that it builds up sequentially (and recursively) in the dimension of the state space of the solution, hence making it suitable for approximations of high-dimensional state space models. We show, using the stochastic Lorenz 96 system as a test model, that the proposed method can operate with larger time steps than the standard Euler--Maruyama scheme and, therefore, generate valid approximations with a smaller computational cost. We also introduce the theoretical analysis of the error incurred by the new predictor-corrector scheme when used as a building block for discrete-time Bayesian filters for continuous-time systems. Finally, we assess the performance of several ensemble Kalman filters that incorporate the proposed sequential predictor-corrector Euler scheme and the standard Euler--Maruyama method. The numerical experiments show that the filters employing the new sequential scheme can operate with larger time steps, smaller Monte Carlo ensembles, and noisier systems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to carry out convergence analysis and algorithm implementation of a novel sample-wise backpropagation method for training a class of stochastic neural networks (SNNs). The preliminary discussion on such an SNN framework was first introduced in [Archibald et al., Discrete Contin. Dyn. Syst. Ser. S, 15 (2022), pp. 2807--2835]. The structure of the SNN is formulated as a discretization of a stochastic differential equation (SDE). A stochastic optimal control framework is introduced to model the training procedure, and a sample-wise approximation scheme for the adjoint backward SDE is applied to improve the efficiency of the stochastic optimal control solver, which is equivalent to the backpropagation for training the SNN. The convergence analysis is derived by introducing a novel joint conditional expectation for the gradient process. Under the convexity assumption, our result indicates that the number of SNN training steps should be proportional to the square of the number of layers in the convex optimization case. In the implementation of the sample-based SNN algorithm with the benchmark MNIST dataset, we adopt the convolution neural network (CNN) architecture and demonstrate that our sample-based SNN algorithm is more robust than the conventional CNN. [ABSTRACT FROM AUTHOR]

This paper is concerned with an optimal control problem for a mean-field linear stochastic differential equation with a quadratic functional in the infinite time horizon. Under suitable conditions, including the stabilizability, the (strong) exponential, integral, and mean-square turnpike properties for the optimal pair are established. The keys are to correctly formulate the corresponding static optimization problem and find the equations determining the correction processes. These have revealed the main feature of the stochastic problems which are significantly different from the deterministic version of the theory. [ABSTRACT FROM AUTHOR]

In this paper we study the stochastic control problem of a partially observed (multidimensional) stochastic system driven by both Brownian motions and fractional Brownian motions. In the absence of the powerful tool of Girsanov transformation, we introduce and study new stochastic processes which are used to transform the original problem to a "classical one"". The adjoint backward stochastic differential equations and the necessary condition satisfied by the optimal control (maximum principle) are obtained. [ABSTRACT FROM AUTHOR]

We develop two new sets of stable, rank-adaptive dynamically orthogonal Runge--Kutta (DORK) schemes that capture the high-order curvature of the nonlinear low-rank manifold. The DORK schemes asymptotically approximate the truncated singular value decomposition at a greatly reduced cost while preserving mode continuity using newly derived retractions. We show that arbitrarily high-order optimal perturbative retractions can be obtained, and we prove that these new retractions are stable. In addition, we demonstrate that repeatedly applying retractions yields a gradient-descent algorithm on the low-rank manifold that converges superlinearly when approximating a low-rank matrix. When approximating a higher-rank matrix, iterations converge linearly to the best low-rank approximation. We then develop a rank-adaptive retraction that is robust to overapproximation. Building off of these retractions, we derive two rank-adaptive integration schemes that dynamically update the subspace upon which the system dynamics are projected within each time step: the stable, optimal dynamically orthogonal Runge--Kutta (so-DORK) and gradient-descent dynamically orthogonal Runge--Kutta (gd-DORK) schemes. These integration schemes are numerically evaluated and compared on an ill-conditioned matrix differential equation, an advection-diffusion partial differential equation, and a nonlinear, stochastic reaction-diffusion partial differential equation. Results show a reduced error accumulation rate with the new stable, optimal and gradient-descent integrators. In addition, we find that rank adaptation allows for highly accurate solutions while preserving computational efficiency. [ABSTRACT FROM AUTHOR]

We consider the problem of finding nearly optimal solutions of optimization problems with random objective functions. Such problems arise widely in the theory of random graphs, theoretical computer science, and statistical physics. Two concrete problems we consider are (a) optimizing the Hamiltonian of a spherical or Ising p-spin glass model and (b) finding a large independent set in a sparse Erd\H os-Renyi graph. The following families of algorithms are considered: (a) low-degree polynomials of the input-a general framework that captures many prior algorithms; (b) low-depth Boolean circuits; (c) the Langevin dynamics algorithm, a canonical Monte Carlo analogue of the gradient descent algorithm. We show that these families of algorithms cannot have high success probability. For the case of Boolean circuits, our results improve the state-of-the-art bounds known in circuit complexity theory (although we consider the search problem as opposed to the decision problem). Our proof uses the fact that these models are known to exhibit a variant of the overlap gap property (OGP) of near-optimal solutions. Specifically, for both models, every two solutions whose objectives are above a certain threshold are either close to or far from each other. The crux of our proof is that the classes of algorithms we consider exhibit a form of stability (noise-insensitivity): a small perturbation of the input induces a small perturbation of the output. We show by an interpolation argument that stable algorithms cannot overcome the OGP barrier. The stability of Langevin dynamics is an immediate consequence of the well-posedness of stochastic differential equations. The stability of low-degree polynomials and Boolean circuits is established using tools from Gaussian and Boolean analysis-namely hypercontractivity and total influence, as well as a novel lower bound for random walks avoiding certain subsets, which we expect to be of independent interest. In the case of Boolean circuits, the result also makes use of Linial--Mansour--Nisan's classical theorem. Our techniques apply more broadly to low influence functions, and we expect that they may apply more generally. [ABSTRACT FROM AUTHOR]

One of the core problems in mean-field control and mean-field games is to solve the corresponding McKean--Vlasov forward-backward stochastic differential equations (MV-FBSDEs). Most existing methods are tailored to special cases in which the mean-field interaction only depends on expectation or other moments and thus are inadequate to solve problems when the mean-field interaction has full distribution dependence. In this paper, we propose a novel deep learning method for computing MV-FBSDEs with a general form of mean-field interactions. Specifically, built on fictitious play, we recast the problem into repeatedly solving standard FBSDEs with explicit coefficient functions. These coefficient functions are used to approximate the MV-FBSDEs' model coefficients with full distribution dependence, and are updated by solving another supervising learning problem using training data simulated from the last iteration's FBSDE solutions. We use deep neural networks to solve standard BSDEs and approximate coefficient functions in order to solve high-dimensional MV-FBSDEs. Under proper assumptions on the learned functions, we prove that the convergence of the proposed method is free of the curse of dimensionality (CoD) by using a class of integral probability metrics previously developed in [J. Han, R. Hu, and J. Long, Stochastic Process. Appl., 164 (2023), pp. 242--287]. The proved theorem shows the advantage of the method in high dimensions. We present the numerical performance in high-dimensional MV-FBSDE problems, including a mean-field game example of the well-known Cucker--Smale model, the cost of which depends on the full distribution of the forward process. [ABSTRACT FROM AUTHOR]

In this paper, we consider the partial controllability of linear stochastic control systems with terminal constraints. Some necessary and sufficient conditions for this controllability are obtained with the help of backward stochastic differential equations (BSDEs). We establish the equivalence between the controllability of a game-based control system (GBCS), the controllability of a forward backward stochastic differential equation (FBSDE), and the partial controllability of a related stochastic differential equation (SDE) with terminal constraints. By applying our results, we obtain some necessary and sufficient conditions for the controllability of GBCSs with jumps. Then we embed the GBCSs driven only by Brownian motion and deterministic GBCSs into our framework with jumps. Previous results of Zhang and Guo are covered and extended. [ABSTRACT FROM AUTHOR]

The spike variation technique plays a crucial role in deriving Pontryagin's type maximum principle of optimal controls for ordinary differential equations (ODEs), partial differential equations (PDEs), stochastic differential equations (SDEs), and (deterministic forward) Volterra integral equations (FVIEs), when the control domains are not assumed to be convex. It is natural to expect that such a technique could be extended to the case of (forward) stochastic Volterra integral equations (FSVIEs). However, by mimicking the case of SDEs, one encounters an essential difficulty of handling an involved quadratic term. To overcome this difficulty, we introduce an auxiliary process for which one can use It's formula, and develop new technologies inspired by stochastic linear-quadratic optimal control problems. Then the suitable representation of the above-mentioned quadratic form is obtained, and the second-order adjoint equations are derived. Consequently, the maximum principle of Pontryagin type is established. Some relevant extensions are investigated as well. [ABSTRACT FROM AUTHOR]

Despite its popularity in the reinforcement learning community, a provably convergent policy gradient method for continuous space-time control problems with nonlinear state dynamics has been elusive. This paper proposes proximal gradient algorithms for feedback controls of finite-time horizon stochastic control problems. The state dynamics are nonlinear diffusions with control-affine drift, and the cost functions are nonconvex in the state and nonsmooth in the control. The system noise can degenerate, which allows for deterministic control problems as special cases. We prove under suitable conditions that the algorithm converges linearly to a stationary point of the control problem and is stable with respect to policy updates by approximate gradient steps. The convergence result justifies the recent reinforcement learning heuristics that adding entropy regularization or a fictitious discount factor to the optimization objective accelerates the convergence of policy gradient methods. The proof exploits careful regularity estimates of backward stochastic differential equations. [ABSTRACT FROM AUTHOR]

We consider the optimal control problem of stochastic evolution equations in a Hilbert space under a recursive utility, which is described as the solution of a backward stochastic differential equation (BSDE). A very general maximum principle is given for the optimal control, allowing the control domain to not be convex and the generator of the BSDE to vary with the second unknown variable z. The associated second-order adjoint process is characterized as a unique solution of a conditionally expected operator-valued backward stochastic integral equation. [ABSTRACT FROM AUTHOR]

This paper derives a free analogue of the Euler-Maruyama method (fEMM) to numerically approximate solutions of free stochastic differential equations (fSDEs). Simply speaking, fSDEs are SDEs in the context of noncommutative random variables (e.g., large random matrices). By applying the theory of multiple operator integrals, we derive a free Itô formula from Taylor expansion of operator-valued functions. Iterating the free Itô formula allows us to motivate and define fEMM. Then we consider weak and strong convergence in the fSDE setting and prove strong convergence order of 1/2 and weak convergence order of 1. Numerical examples support the theoretical results and show solutions for equations where no analytical solution is known. [ABSTRACT FROM AUTHOR]

This work presents a probabilistic scheme for solving semilinear nonlocal diffusion equations with volume constraints and integrable kernels. The nonlocal model of interest is defined by a time-dependent semilinear partial integro-differential equation (PIDE), in which the integrodifferential operator consists of both local convection-diffusion and nonlocal diffusion operators. Our numerical scheme is based on the direct approximation of the nonlinear Feynman-Kac formula that establishes a link between nonlinear PIDEs and stochastic differential equations. The exploitation of the Feynman-Kac representation avoids solving dense linear systems arising from nonlocal operators. Compared with existing stochastic approaches, our method can achieve first-order convergence after balancing the temporal and spatial discretization errors, which is a significant improvement of existing probabilistic/stochastic methods for nonlocal diffusion problems. Error analysis of our numerical scheme is established. The effectiveness of our approach is shown in two numerical examples. The first example considers a three-dimensional nonlocal diffusion equation to numerically verify the error analysis results. The second example presents a physics problem motivated by the study of heat transport in magnetically confined fusion plasmas. [ABSTRACT FROM AUTHOR]

This paper studies a class of partial information linear-quadratic mean-field game problems. A general stochastic large population system is considered, where the diffusion term of the dynamic of each agent can depend on the state and control. We study both the control constrained case and the unconstrained case. In the control constrained case, by using a Hamiltonian approach and convex analysis, the explicit decentralized strategies can be obtained through a projection operator. The corresponding Hamiltonian type consistency condition system is derived, which turns out to be a nonlinear mean-field forward-backward stochastic differential equation with projection operator. The well-posedness of such equations is proved by using a discounting method. Moreover, the corresponding E-varepsilon -Nash equilibrium property is verified. In the control unconstrained case, the decentralized strategies can be further represented explicitly as the feedback of the filtered state through a Riccati approach. The existence and uniqueness of a solution to a new Riccati type consistency condition system is also discussed. As an application, a general interbank borrowing and lending problem is studied to illustrate that the effect of partial information cannot be ignored. [ABSTRACT FROM AUTHOR]

We study the optimal control formulation for stochastic nonlinear Schrödinger equation (SNLSE) on a finite graph. By viewing the SNLSE as a stochastic Wasserstein Hamiltonian flow on density manifold, we show the global existence of a unique strong solution for SNLSE with a linear drift control or a linear diffusion control on graph. Furthermore, we provide the gradient formula, the existence of the optimal control and a description on the optimal condition via the forward and backward stochastic differential equations. [ABSTRACT FROM AUTHOR]

First-order methods are often analyzed via their continuous-time models, where their worst-case convergence properties are usually approached via Lyapunov functions. In this work, we provide a systematic and principled approach to finding and verifying Lyapunov functions for classes of ordinary and stochastic differential equations. More precisely, we extend the performance estimation framework, originally proposed by Drori and Teboulle [Math. Program., 145 (2014), pp. 451-482], to continuous-time models. We retrieve convergence results comparable to those of discrete-time methods using fewer assumptions and inequalities and provide new results for a family of stochastic accelerated gradient flows. [ABSTRACT FROM AUTHOR]

Several differential equation models have been proposed to explain the formation of patterns characteristic of the grid cell network. Understanding the robustness of these patterns with respect to noise is one of the key open questions in computational neuroscience. In the present work, we analyze a family of stochastic differential systems modeling grid cell networks. Furthermore, the well-posedness of the associated McKean--Vlasov and Fokker--Planck equations, describing the average behavior of the networks, is established. Finally, we rigorously prove the mean field limit of these systems and provide a sharp rate of convergence for their empirical measures. [ABSTRACT FROM AUTHOR]

This paper studies a new and highly efficient Markov chain Monte Carlo (MCMC) methodology to perform Bayesian inference in low-photon imaging problems, with particular attention given to situations involving observation noise processes that deviate significantly from Gaussian noise, such as binomial, geometric, and low-intensity Poisson noise. These problems are challenging for many reasons. From an inferential viewpoint, low-photon numbers lead to severe identifiability issues, poor stability, and high uncertainty about the solution. Moreover, low-photon models often exhibit poor regularity properties that make efficient Bayesian computation difficult, e.g., hard nonnegativity constraints, nonsmooth priors, and log-likelihood terms with exploding gradients. More precisely, the lack of suitable regularity properties hinders the use of state-of-the-art Monte Carlo methods based on numerical approximations of the Langevin stochastic differential equation (SDE), as both the SDE and its numerical approximations behave poorly. We address this difficulty by proposing an MCMC methodology based on a reflected and regularized Langevin SDE, which is shown to be well-posed and exponentially ergodic under mild and easily verifiable conditions. This then allows us to derive four reflected proximal Langevin MCMC algorithms to perform Bayesian computation in low-photon imaging problems. The proposed approach is demonstrated with a range of experiments related to image deblurring, denoising, and inpainting under binomial, geometric, and Poisson noise. [ABSTRACT FROM AUTHOR]

The long time behavior and detailed convergence analysis of Langevin equations have received increased attention in recent years. Difficulties arise from a lack of coercivity, usually termed hypocoercivity, of the underlying kinetic Fokker-Planck operator, which is a consequence of the partially deterministic nature of a second order stochastic differential equation. In this manuscript, the effect of controlling the confinement potential without altering the original invariant measure is investigated. This leads to an abstract bilinear control system with an unbounded but infinite-time admissible control operator which, by means of an artificial diffusion approach, is shown to possess a unique solution. The compactness of the underlying semigroup is further used to define an infinite-horizon optimal control problem on an appropriately reduced state space. Under smallness assumptions on the initial data, feasibility of and existence of a solution to the optimal control problem are discussed. Numerical results based on a local approximation based on a shifted Riccati equation illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we derive first-order Pontryagin optimality conditions for risk-averse stochastic optimal control problems subject to final time inequality constraints whose costs are general, possibly nonsmooth finite coherent risk measures. Unlike preexisting contributions covering this situation, our analysis holds for classical stochastic differential equations driven by standard Brownian motions. In addition, it presents the advantages of neither involving second-order adjoint equations nor leading to the so-called weak version of the Pontryagin maximum principle, in which the maximization condition with respect to the control variable is replaced by the stationarity of the Hamiltonian. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study optimal control of conditional McKean-Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffusions, for short). To this end, we first prove a stochastic Fokker-Planck equation for the conditional law of the solution of such equations. Combining this equation with the original state equation, we obtain a Markovian system for the state and its conditional law. Furthermore, we apply this to formulate a Hamilton-Jacobi-Bellman equation for the optimal control of conditional McKean-Vlasov jump diffusions. Then we study the situation when the law is absolutely continuous with respect to Lebesgue measure. In that case the Fokker-Planck equation reduces to a stochastic partial differential equation for the Radon-Nikodym derivative of the conditional law. Finally we apply these results to solve explicitly the linear-quadratic optimal control problem of conditional stochastic McKean-Vlasov jump diffusions, and optimal consumption from a cash flow. [ABSTRACT FROM AUTHOR]

In this work we design an explicit feedback controller u = Φ(ρ) for the exact controllability problem to the nonlinear Fokker-Planck equation ρt - Δβ(ρ) + div(uρ) = 0 in (0, T) x O, ρ(0) = ρ0, ρ(T) = ρ1, with reflecting boundary conditions (∇β(ρ) - uρ) ⋅ n = 0 on ∂O. Here, O ⊂ Rd, 1 ≤ d, is a bounded, open set with smooth boundary ∂O and β is a smooth monotonically increasing function. If ρ0 and ρ1 are probability densities, one obtains in particular the exact controllability for McKean-Vlasov stochastic differential equations on O with reflecting (impenetrable) barrier. [ABSTRACT FROM AUTHOR]

This paper is concerned with a partially observed progressive optimal control problem of forward-backward stochastic differential equations with random jumps, where the control domain is not necessarily convex, and the control variable enters into all the coefficients. In our model, the observation equation is not only driven by a Brownian motion but also a Poisson random measure, which also has correlated noises with the state equation. The partially observed global maximum principle is proved. To show its applications, a partially observed linear quadratic progressive optimal control problem of forward-backward stochastic differential equations with random jumps is investigated by the maximum principle and stochastic filtering. State estimate feedback representation of the optimal control is given in a more explicit form by introducing some ordinary differential equations. [ABSTRACT FROM AUTHOR]

In this paper we present a framework in which one can rigorously study the effect of spatio-temporal noise on traveling waves, stationary patterns, and oscillations that are invariant under the action of a finite-dimensional set of continuous isometries (such as translation or rotation). This formalism can accommodate patterns, waves, and oscillations in reaction-diffusion systems and neural field equations. To do this, we define the phase by precisely projecting the infinite-dimensional system onto the manifold of isometries. We outline a precise stochastic differential equation for the phase. The phase is then used to show that the probability of the system leaving the attracting basin of the manifold after an exponentially long period of time (in ε -2, the magnitude of the noise) is exponentially unlikely. [ABSTRACT FROM AUTHOR]

This paper thoroughly investigates stochastic linear-quadratic optimal control problems with the Markovian regime switching system, where the coefficients of the state equation and the weighting matrices of the cost functional are random. We prove the solvability of the stochastic Riccati equation under the uniform convexity condition and obtain the closed-loop representation of the open-loop optimal control using the unique solvability of the corresponding stochastic Riccati equation. Moreover, by applying Itô's formula with jumps, we get a representation of the cost functional on a Hilbert space, characterized as the adapted solutions of some forward-backward stochastic differential equations. We show that the necessary condition of the open-loop optimal control is the convexity of the cost functional, and the sufficient condition of the open-loop optimal control is the uniform convexity of the cost functional. In addition, we study the properties of the stochastic value flow of the stochastic linear-quadratic optimal control problem. Finally, as an application, we present a continuous-time mean-variance portfolio selection problem and prove its unique solvability. [ABSTRACT FROM AUTHOR]

We study finite-time horizon continuous-time linear-convex reinforcement learning problems in an episodic setting. In this problem, the unknown linear jump-diffusion process is controlled subject to nonsmooth convex costs. We show that the associated linear-convex control problems admit Lipschitz continuous optimal feedback controls and further prove the Lipschitz stability of the feedback controls, i.e., the performance gap between applying feedback controls for an incorrect model and for the true model depends Lipschitz-continuously on the magnitude of perturbations in the model coefficients; the proof relies on a stability analysis of the associated forward-backward stochastic differential equation. We then propose a novel least-squares algorithm which achieves a regret of the order O(√NlnN) on linear-convex learning problems with jumps, where N is the number of learning episodes; the analysis leverages the Lipschitz stability of feedback controls and concentration properties of sub-Weibull random variables. Numerical experiment confirms the convergence and the robustness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we propose a deep learning based numerical scheme for strongly coupled forward backward stochastic differential equations (FBSDEs), stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the mean squared error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDEs fails for a simple linear-quadratic control problem, and we motivate why the new method works. Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems, we provide an error analysis for our method. We show empirically that the method converges for three different problems, one being the one that failed for a direct extension of the deep BSDE method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the density estimation problem associated with the stationary measure of ergodic Itô diffusions from a discrete-time series that approximate the solutions of the stochastic differential equations. To take advantage of the characterization of density function through the stationary solution of a parabolic-type Fokker-Planck PDE, we proceed as follows: First, we employ deep neural networks to approximate the drift and diffusion terms of the SDE by solving appropriate supervised learning tasks. Subsequently, we solve a steady-state Fokker-Planck equation associated with the estimated drift and diffusion coefficients with a neural-network-based least squares method. We establish the convergence of the proposed scheme under appropriate mathematical assumptions, accounting for the generalization errors induced by regressing the drift and diffusion coefficients and the PDE solvers. This theoretical study relies on a recent perturbation theory of Markov chain result that shows a linear dependence of the density estimation to the error in estimating the drift term and generalization error results of nonparametric regression and PDE regression solution obtained with neural-network models. We demonstrate the effectiveness of this method by numerical simulations of a two-dimensional Student t-distribution and a 20-dimensional Langevin dynamics. [ABSTRACT FROM AUTHOR]

This paper focuses on a class of highly nonlinear stochastic differential delay equations (SDDEs) driven by Lévy noise and Markovian chain, where the drift and diffusion coefficients satisfy more general polynomial growth condition (than the classical linear growth condition). Under the local Lipschitz condition, the existence-and-unique theorem of the solution to the highly nonlinear SDDE is established. The key aim is to investigate the stabilization problem by delay feedback controls. The key features include that the time delay in the given system is time-varying and may not be differentiable, while the time lag in the feedback control can also be time-varying as long as it has a sufficiently small upper bound. [ABSTRACT FROM AUTHOR]

The ensemble Kalman inversion (EKI) method is a method for the estimation of unknown parameters in the context of (Bayesian) inverse problems. The method approximates the underlying measure by an ensemble of particles and iteratively applies the ensemble Kalman update to evolve (the approximation of the) prior into the posterior measure. For the convergence analysis of the EKI it is common practice to derive a continuous version, replacing the iteration with a stochastic differential equation. In this paper we validate this approach by showing that the stochastic EKI iteration converges to paths of the continuous time stochastic differential equation by considering both the nonlinear and linear setting, and we prove convergence in probability for the former and convergence in moments for the latter. The methods employed do not rely on the specific structure of the ensemble Kalman method and can also be applied to the analysis of more general numerical schemes for stochastic differential equations. [ABSTRACT FROM AUTHOR]

Localized persistent cortical neural activity is a validated neural substrate of parametric working memory. Such activity "bumps" represent the continuous location of a cue over several seconds. Pyramidal (excitatory (E)) and interneuronal (inhibitory (I)) subpopulations exhibit tuned bumps of activity, linking neural dynamics to behavioral inaccuracies observed in memory recall. However, many bump attractor models collapse these subpopulations into a single joint E/I(lateral inhibitory) population and do not consider the role of interpopulation neural architecture and noise correlations. Both factors have a high potential to impinge upon the stochastic dynamics of these bumps, ultimately shaping behavioral response variance. In our study, we consider a neural field model with separate E/I populations and leverage asymptotic analysis to derive a nonlinear Langevin system describing E/I bump interactions. While the E bump attracts the I bump, the I bump stabilizes but can also repel the E bump, which can result in prolonged relaxation dynamics when both bumps are perturbed. Furthermore, the structure of noise correlations within and between subpopulations strongly shapes the variance in bump position. Surprisingly, higher interpopulation correlations reduce variance. [ABSTRACT FROM AUTHOR]

Data-driven stochastic parameterization methods use observational data to support and improve existing prediction systems. Specifically in atmospheric sciences, uncertainty in observations is a challenge for modeling because several aspects of the physical processes are yet to be understood in some contexts. A key goal in constructing atmospheric models is to express unresolved processes using the resolved variables of a prediction system. Estimating parameterisations for highly nonlinear systems with non stationary data requires application-specific tools. Previous work on methods for analyzing nonstationary data presents a exible model-based nonparametric clustering methodology that can be exploited for parameterization development. This paper adapts the existing framework for stochastic parameterization by combining the continuous-time formulation of stochastic differential equations (SDE) with the clustering method. We use a closed-form likelihood function approach based on a suitable Hermite expansion to approximate the parameter values of the SDE with arbitrary nonlinear drift and nonlinear diffusion. The novel parameterization framework provides a smooth classification function that allows us to recover the underlying temporal parameter modulation of a nonstationary one-dimensional SDE. The numerical examples show that the clustering approach recovers a hidden functional relationship between the parameters of the SDE model and an additional auxiliary process. The study builds upon this functional relationship to develop closed-form, nonstationary, data-driven stochastic models for multiscale dynamical systems in real-world applications. [ABSTRACT FROM AUTHOR]

This paper is devoted to establishing global Carleman estimates for refined stochastic beam equations. First, by establishing a fundamental weighted identity, two Carleman estimates are derived with different weight functions for the refined stochastic beam equation, which is a coupled system consisting of a stochastic ordinary differential equation and a stochastic partial differential equation. As applications of these Carleman estimates, the exact controllability of the refined system is proved by the least controls in some sense. Different from classical stochastic beam equations, the refined one is exactly controllable at any time. Meanwhile, the uniqueness of an inverse source problem for refined stochastic beam equations is obtained without any requirement on the initial and terminal data. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the optimal control problems for a class of systems with fast-slow processes. The problem under consideration is to minimize a functional subject to a system described by a two-time scaled McKean-Vlasov stochastic differential equation whose coefficients depend on state components and their probability distributions. Firstly, we establish the existence and uniqueness of the invariant probability measure for the fast process. Then, by using the relaxed control representation and the martingale method, we prove the weak convergence of the slow process and control process in the original problem, and we obtain an associated limit problem in which the coefficients are determined by the average of those of the original problem with respect to the invariant probability measure. Finally, by establishing the nearly optimal control of the limit problem, we obtain the near optimality of the original problem. [ABSTRACT FROM AUTHOR]

We develop a new iterative method based on the Pontryagin principle to solve stochastic control problems. This method is nothing other than the Newton method extended to the framework of stochastic optimal control, where the state dynamics are given by an ODEs with stochastic coefficients and the cost is random. Each iteration of the method is made of two ingredients: computing the Newton direction, and finding an adapted step length. The Newton direction is obtained by solving an affine-linear Forward-Backward Stochastic Differential Equation (FBSDE) with random coefficients. This is done in the setting of a general filtration. Solving such an FBSDE reduces to solving a Riccati Backward Stochastic Differential Equation (BSDE) and an affine-linear BSDE, as expected in the framework of linear FBSDEs or Linear-Quadratic stochastic control problems. We then establish convergence results for this Newton method. In particular, Lipschitz-continuity of the second-order derivative of the cost functional is established with an appropriate choice of norm and under boundedness assumptions, which is sufficient to prove (local) quadratic convergence of the method in the space of uniformly bounded processes. To choose an appropriate step length while fitting our choice of space of processes, an adapted Backtracking line search method is developed. We then prove global convergence of the Newton method with the proposed line search procedure, which occurs at a quadratic rate after finitely many iterations. An implementation with regression techniques to solve BSDEs arising in the computation of the Newton step is developed. We apply it to the control problem of a large number of batteries providing ancillary services to an electricity network. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of stability of regime-switching stochastic differential equations (SDEs) driven by fractional Brownian motions with Hurst parameter H∈(0,1) perturbed by the transition rate matrix of Markov chain. When H∈(1/2,1), we first obtain the stability result for the multidimensional regime-switching SDE with additive noise under regular conditions in terms of the Wasserstein distance, and then extend the result to the one-dimensional regime-switching SDE with multiplicative noise via a Lamperti transform. In addition, we also derive the stability of the linear regime-switching SDE based on the explicit expression of the solution. While in the case of H∈(0,1/2), we establish the stability of regime-switching SDEs under the locally unbounded condition in terms of the Fortet-Mourier distance. [ABSTRACT FROM AUTHOR]

Based on the stochastic maximum principle for the partially coupled forwardbackward stochastic control system (FBSCS), a modified method of successive approximations (MSA) is established for stochastic recursive optimal control problems. The second-order adjoint processes are introduced in the augmented Hamiltonian minimization step since the control domain is not necessarily convex. Thanks to the theory of bounded mean oscillation martingales, we give a delicate proof of the error estimate and then prove the convergence of the modified MSA algorithm. In a special case, we obtain a logarithmic convergence rate. When the control domain is convex and compact, a sufficient condition which makes the control returned from the MSA algorithm be a near-optimal control is given for a class of linear FBSCSs. [ABSTRACT FROM AUTHOR]

In molecular dynamics, penalized overdamped Langevin dynamics are used to model the motion of a set of particles that follow constraints up to a parameter ε. The most used schemes for simulating these dynamics are the Euler integrator in ℝd and the constrained Euler integrator. Both have weak order one of accuracy, but work properly only in specific regimes depending on the size of the parameter ε. We propose in this paper a new consistent method with an accuracy independent of ε for solving penalized dynamics on a manifold of any dimension. Moreover, this method converges to the constrained Euler scheme when ε goes to zero. The numerical experiments confirm the theoretical findings, in the context of weak convergence and for the invariant measure, on a torus and on the orthogonal group in high dimension and high codimension. [ABSTRACT FROM AUTHOR]

This paper investigates the robust social optimum problem for linear quadratic mean field control systems by the direct approach, where model uncertainty appears in both drift and diffusion terms of each agent. We take a zero-sum game approach by considering local disturbance as the control of an adversarial player. Under centralized information structure, we first obtain the necessary and sufficient condition for the existence of open-loop and closed-loop saddle points, which are characterized by the solvability of forward-backward stochastic differential equations (FBSDEs) and two coupled Riccati equations, respectively. By considering the infinite system, we next design a set of decentralized open-loop strategies based on mean field FBSDEs and obtain closed-loop strategies in terms of two uncoupled Riccati equations. Finally, the performance of the proposed decentralized strategies is analyzed and the efficiency is verified by numerical simulation. [ABSTRACT FROM AUTHOR]

In this article, a class of backward stochastic Volterra integro-differential equations (BSVIDEs) is introduced and studied. It is worthy mentioning that the proposed BSVIDEs cannot be covered by the existing backward stochastic Volterra integral equations (BSVIEs), and they also have the nice flow property such that Itô's formula becomes quite applicable. It is found that BSVIDEs can provide a neat sufficient condition for the solvability of BSVIEs with generator depending on the diagonal value of the solutions. As applications, the optimal control problems in terms of maximum principles and linear quadratic control problems of optimal control for forward stochastic Volterra integro-differential equations are investigated. In contrast with the BSVIEs in current literature, some interesting phenomena and advantages of BSVIDEs are revealed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the tamed stochastic gradient descent method (TSGD) for optimization problems. Inspired by the tamed Euler scheme, which is a commonly used method within the context of stochastic differential equations, TSGD is an explicit scheme that exhibits stability properties similar to those of implicit schemes. As its computational cost is essentially equivalent to that of the well-known stochastic gradient descent method (SGD), it constitutes a very competitive alternative to such methods. We rigorously prove (optimal) sublinear convergence of the scheme for strongly convex objective functions on an abstract Hilbert space. The analysis only requires very mild step size restrictions, which illustrates the good stability properties. The analysis is based on a priori estimates more frequently encountered in a time integration context than in optimization, and this alternative approach provides a different perspective also on the convergence of SGD. Finally, we demonstrate the usability of the scheme on a problem arising in a context of supervised learning. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the global minimization of a possibly non-smooth and nonconvex objective function constrained on the unit hypersphere by means of a multi-agent derivative-free method. The proposed algorithm falls into the class of the recently introduced consensus-based optimization. In fact, agents move on the sphere driven by a drift towards an instantaneous consensus point, which is computed as a convex combination of agent locations, weighted by the cost function according to Laplace's principle, and it represents an approximation to a global minimizer. The dynamics is further perturbed by an anisotropic random vector field to favor exploration. The main results of this paper are about the proof of convergence of the numerical scheme to global minimizers provided conditions of well-preparation of the initial datum. The proof of convergence combines a mean-field limit result with a novel asymptotic analysis and classical convergence results of numerical methods for stochastic differential equations. The main innovation with respect to previous work is the introduction of an anisotropic stochastic term, which allows us to ensure the independence of the parameters of the algorithm from the dimension and to scale the method to work in very high dimension. We present several numerical experiments, which show that the algorithm proposed in the present paper is extremely versatile and outperforms previous formulations with isotropic stochastic noise. [ABSTRACT FROM AUTHOR]