A First-Order Stochastic Primal-Dual Algorithm with Correction Step

In this article, we investigate the convergence properties of a stochastic primal-dual splitting algorithm for solving structured monotone inclusions involving the sum of a cocoercive operator and a composite monotone operator. The proposed method is the stochastic extension to monotone inclusions of a proximal method studied in the literature for saddle point problems. It consists in a forward step determined by the stochastic evaluation of the cocoercive operator, a backward step in the dual variables involving the resolvent of the monotone operator, and an additional forward step using the stochastic evaluation of the cocoercive operator introduced in the first step. We prove weak almost sure convergence of the iterates by showing that the primal-dual sequence generated by the method is stochastic quasi-Fejér-monotone with respect to the set of zeros of the considered primal and dual inclusions. Additional results on ergodic convergence in expectation are considered for the special case of saddle point models.     

A stochastic inertial forward–backward splitting algorithm for multivariate monotone inclusions

We propose an inertial forward–backward splitting algorithm to compute a zero of a sum of two monotone operators allowing for stochastic errors in the computation of the operators. More precisely, we establish almost sure convergence in real Hilbert spaces of the sequence of iterates to an optimal solution. Then, based on this analysis, we introduce two new classes of stochastic inertial primal–dual splitting methods for solving structured systems of composite monotone inclusions and prove their convergence. Our results extend to the stochastic and inertial setting various types of structured monotone inclusion problems and corresponding algorithmic solutions. Application to minimization problems is discussed.     

A Finite Horizon Linear Quadratic Optimal Stochastic Control Problem Driven by Both Brownian Motion and L\'{e}vy Processes

We study the linear quadratic optimal stochastic control problem which is jointly driven by Brownian motion and L\'{e}vy processes. We prove that the new affine stochastic differential adjoint equation exists an inverse process by applying the profound section theorem. Applying for the Bellman's principle of quasilinearization and a monotone iterative convergence method, we prove the existence and uniqueness of the solution of the backward Riccati differential equation. Finally, we prove that the optimal feedback control exists, and the value function is composed of the initial value of the solution of the related backward Riccati differential equation and the related adjoint equation.     

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??We study the linear quadratic optimal stochastic control problem which is jointly driven by Brownian motion and L\'{e}vy processes. We prove that the new affine stochastic differential adjoint equation exists an inverse process by applying the profound section theorem. Applying for the Bellman's principle of quasilinearization and a monotone iterative convergence method, we prove the existence and uniqueness of the solution of the backward Riccati differential equation. Finally, we prove that the optimal feedback control exists, and the value function is composed of the initial value of the solution of the related backward Riccati differential equation and the related adjoint equation.     

由Lévy过程驱动的一类特殊的高维的BSRDE解的存在唯一性

讨论由Brownian运动和Lévy过程共同驱动的线性随机系统的随机LQ问题,其中代价泛函是关于Lévy过程生成的σ-代数取条件期望.得到由Lévy过程驱动的新的多维的倒向随机Riccati方程,利用Bellman拟线性原理和单调收敛方法证明了此随机Riccati方程的解的存在性.     

Monotone quantum stochastic processes and covariant dynamical hemigroups

Based on the Hilbert C^?-module structure we study non-stationary monotone quantum stochastic processes and general Markov processes constructed from quantum dynamical hemigroups indexed by a totally ordered set. We prove that the quantum stochastic monotone process implementing the weakly covariant process described by a covariant quantum dynamical hemigroup with respect to a symmetry semigroup is again covariant in the strong sense.     

Yosida approximations for multivalued stochastic partial differential equations driven by Lévy noise on a Gelfand triple

We prove the existence and uniqueness of solutions for a class of multivalued stochastic partial differential equations with maximal monotone drift on Banach space driven by multiplicative Lévy noise. We also establish the strong convergence result for solutions of the approximating equations where the maximal monotone drift operator is replaced by its Yosida approximation. As an application, the existence and uniqueness of solutions for multivalued stochastic porous medium equations is obtained.     

SINGLE PROJECTION ALGORITHM FOR VARIATIONAL INEQUALITIES IN BANACH SPACES WITH APPLICATION TO CONTACT PROBLEM

We study the single projection algorithm of Tseng for solving a variational inequality problem in a 2-uniformly convex Banach space. The underline cost function of the variational inequality is assumed to be monotone and Lipschitz continuous. A weak convergence result is obtained under reasonable assumptions on the variable step-sizes. We also give the strong convergence result for when the underline cost function is strongly monotone and Lipchitz continuous. For this strong convergence case, the proposed method does not require prior knowledge of the modulus of strong monotonicity and the Lipschitz constant of the cost function as input parameters, rather, the variable step-sizes are diminishing and non-summable. The asymptotic estimate of the convergence rate for the strong convergence case is also given. For completeness, we give another strong convergence result using the idea of Halpern's iteration when the cost function is monotone and Lipschitz continuous and the variable step-sizes are bounded by the inverse of the Lipschitz constant of the cost function.Finally, we give an example of a contact problem where our proposed method can be applied.     

Emergent dynamics of the first‐order stochastic Cucker‐Smale model and application to finance

In this paper, we study stochastic aggregation properties of the financial model for the N‐asset price process whose dynamics is modeled by the weakly geometric Brownian motions with stochastic drifts. For the temporal evolution of stochastic components of drift coefficients, we employ a stochastic first‐order Cucker‐Smale model with additive noises. The asset price processes are weakly interacting via the stochastic components of drift coefficients. For the aggregation estimates, we use the macro‐micro decomposition of the fluctuations around the average process and show that the fluctuations around the average value satisfies a practical aggregation estimate over a time‐independent symmetric network topology so that we can control the differences of drift coefficients by tuning the coupling strength. We provide numerical examples and compare them with our analytical results. We also discuss some financial implications of our proposed model.     

A Proximal Point Method in Nonreflexive Banach Spaces

We propose an inexact version of the proximal point method and study its properties in nonreflexive Banach spaces which are duals of separable Banach spaces, both for the problem of minimizing convex functions and of finding zeroes of maximal monotone operators. By using surjectivity results for enlargements of maximal monotone operators, we prove existence of the iterates in both cases. Then we recover most of the convergence properties known to hold in reflexive and smooth Banach spaces for the convex optimization problem. When dealing with zeroes of monotone operators, our convergence result requests that the regularization parameters go to zero, as is the case for standard (non-proximal) regularization schemes.     

Existence,uniqueness, and numerical approximations for stochastic Burgers equations

Abstract

In this article, we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities. The proof of this result exploits the properties of an existing fully explicit space-time discrete approximation scheme, in particular the fact that it satisfies suitable a priori estimates. We also obtain almost sure and strong convergence of the approximation scheme to the mild solutions of the considered SPDEs. We conclude by applying the main result of the article to the stochastic Burgers equations with additive space-time white noise.     

Strong convergence for explicit space–time discrete numerical approximation methods for stochastic Burgers equations

In this paper we propose and analyze explicit space–time discrete numerical approximations for additive space–time white noise driven stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities such as the stochastic Burgers equation with space–time white noise. The main result of this paper proves that the proposed explicit space–time discrete approximation method converges strongly to the solution process of the stochastic Burgers equation with space–time white noise. To the best of our knowledge, the main result of this work is the first result in the literature which establishes strong convergence for a space–time discrete approximation method in the case of the stochastic Burgers equations with space–time white noise.     

Approximating Stationary Points of Stochastic Mathematical Programs with Equilibrium Constraints via Sample Averaging

We investigate sample average approximation of a general class of one-stage stochastic mathematical programs with equilibrium constraints. By using graphical convergence of unbounded set-valued mappings, we demonstrate almost sure convergence of a sequence of stationary points of sample average approximation problems to their true counterparts as the sample size increases. In particular we show the convergence of M(Mordukhovich)-stationary point and C(Clarke)-stationary point of the sample average approximation problem to those of the true problem. The research complements the existing work in the literature by considering a general constraint to be represented by a stochastic generalized equation and exploiting graphical convergence of coderivative mappings.     

Almost sure convergence of the forward–backward–forward splitting algorithm

In this paper, we propose a stochastic forward–backward–forward splitting algorithm and prove its almost sure weak convergence in real separable Hilbert spaces. Applications to composite monotone inclusion and minimization problems are demonstrated.     

Rearrangements of Gaussian fields

The monotone rearrangement of a function is the non-decreasing function with the same distribution. The convex rearrangement of a smooth function is obtained by integrating the monotone rearrangement of its derivative. This operator can be applied to regularizations of a stochastic process to measure quantities of interest in econometrics.A multivariate generalization of these operators is proposed, and the almost sure convergence of rearrangements of regularized Gaussian fields is given. For the fractional Brownian field or the Brownian sheet approximated on a simplicial grid, it appears that the limit object depends on the orientation of the simplices.     

Stable limits for sums of dependent infinite variance random variables

The aim of this paper is to provide conditions which ensure that the affinely transformed partial sums of a strictly stationary process converge in distribution to an infinite variance stable distribution. Conditions for this convergence to hold are known in the literature. However, most of these results are qualitative in the sense that the parameters of the limit distribution are expressed in terms of some limiting point process. In this paper we will be able to determine the parameters of the limiting stable distribution in terms of some tail characteristics of the underlying stationary sequence. We will apply our results to some standard time series models, including the GARCH(1, 1) process and its squares, the stochastic volatility models and solutions to stochastic recurrence equations.     

Strong convergence studied by a hybrid type method for monotone operators in a Banach space

In this paper, we study a strong convergence for monotone operators. We first introduce the hybrid type algorithm for monotone operators. Next, we obtain a strong convergence theorem (Theorem 3.3) for finding a zero point of an inverse-strongly monotone operator in a Banach space. Finally, we apply our convergence theorem to the problem of finding a minimizer of a convex function.     

Uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming

Sample average approximation (SAA) is one of the most popular methods for solving stochastic optimization and equilibrium problems. Research on SAA has been mostly focused on the case when sampling is independent and identically distributed (iid) with exceptions (Dai et al. (2000) [9], Homem-de-Mello (2008) [16]). In this paper we study SAA with general sampling (including iid sampling and non-iid sampling) for solving nonsmooth stochastic optimization problems, stochastic Nash equilibrium problems and stochastic generalized equations. To this end, we first derive the uniform exponential convergence of the sample average of a class of lower semicontinuous random functions and then apply it to a nonsmooth stochastic minimization problem. Exponential convergence of estimators of both optimal solutions and M-stationary points (characterized by Mordukhovich limiting subgradients (Mordukhovich (2006) [23], Rockafellar and Wets (1998) [32])) are established under mild conditions. We also use the unform convergence result to establish the exponential rate of convergence of statistical estimators of a stochastic Nash equilibrium problem and estimators of the solutions to a stochastic generalized equation problem.     

Interior Proximal Algorithm for Quasiconvex Programming Problems and Variational Inequalities with Linear Constraints

In this paper, we propose two interior proximal algorithms inspired by the logarithmic-quadratic proximal method. The first method we propose is for general linearly constrained quasiconvex minimization problems. For this method, we prove global convergence when the regularization parameters go to zero. The latter assumption can be dropped when the function is assumed to be pseudoconvex. We also obtain convergence results for quasimonotone variational inequalities, which are more general than monotone ones.     

Regime-switching stochastic volatility model: estimation and calibration to VIX options

We develop and implement a method for maximum likelihood estimation of a regime-switching stochastic volatility model. Our model uses a continuous time stochastic process for the stock dynamics with the instantaneous variance driven by a Cox–Ingersoll–Ross process and each parameter modulated by a hidden Markov chain. We propose an extension of the EM algorithm through the Baum–Welch implementation to estimate our model and filter the hidden state of the Markov chain while using the VIX index to invert the latent volatility state. Using Monte Carlo simulations, we test the convergence of our algorithm and compare it with an approximate likelihood procedure where the volatility state is replaced by the VIX index. We found that our method is more accurate than the approximate procedure. Then, we apply Fourier methods to derive a semi-analytical expression of S&P500 and VIX option prices, which we calibrate to market data. We show that the model is sufficiently rich to encapsulate important features of the joint dynamics of the stock and the volatility and to consistently fit option market prices.