Sublinear expectation relaxes the linear property of classical expectation to subadditivity and positive homogeneity, which can be expressed as E(⋅)=sup_θ∈ΘE_θ(⋅) for a certain set of linear expectations {E_θ:θ∈Θ}. Such a framework can capture the uncertainty and facilitate a robust method of measuring risk loss reasonably. This study established a law of large numbers for m-dependent random vectors within the framework of sublinear expectation. Consequently, the corresponding explicit rate of convergence were derived. The results of this study can be considered as an extension of the Peng’s law of large numbers [22].

In this paper, we explore non-homogeneous stochastic linear-quadratic (LQ) optimal control problems with multidimensional states and regime switching. We focus on the corresponding stochastic Riccati equation (SRE), which mirrors that of the homogeneous stochastic LQ optimal control problem, and the adjoint backward stochastic differential equation (BSDE), which arises from the non-homogeneous terms in the state equation and cost functional. We solve both the SRE and adjoint BSDE using the contraction mapping method, which helps represent the closed-loop optimal control and the optimal value of our problems. In particular, we extend some results of Hu et al. [7] to the multidimensional case.

This study addresses the existence, uniqueness, and comparison theorem for unbounded solutions of one-dimensional backward stochastic differential equations (BSDEs) with sub-quadratic generators, considering both finite and infinite terminal times. Initially, we establish the existence of unbounded solutions for BSDEs where the generator g satisfies a time-varying one-sided linear growth condition in the first unknown variable y and a time-varying sub-quadratic growth condition in the second unknown variable z. Next, the uniqueness and comparison theorems for unbounded solutions are proven under a time-varying extended convexity assumption. These findings extend the results in [12] to the general time-interval BSDEs. Finally, we propose and verify several sufficient conditions for ensuring uniqueness, utilizing innovative approaches applied for the first time, even in the context of finite time-interval BSDEs.

The concept of upper variance under multiple probabilities is defined through a corresponding minimax optimization problem. This study proposes a simple algorithm to solve this optimization problem exactly. Additionally, we provide a probabilistic representation for a class of quadratic programming problems, demonstrating the practical application of our approach.

This paper investigates the optimal control problem for a class of fully coupled forward-backward stochastic partial differential equations (FBSPDEs). Based on the existence of a unique solution to such equations, we formulated the associated optimal control problem within a convex control domain. By employing the convex variational method, we derive the associated stochastic maximum principle (SMP) for the optimal control problem intrinsic to this system. Finally, to demonstrate the applicability of our theoretical results, we apply SMP to a class of linear quadratic problems and obtain explicit expressions for the unique optimal control.

We introduce and analyze a family of linear least-squares Monte Carlo schemes for backward SDEs, which interpolate between the one-step dynamic programming scheme of Lemor, Warin, and Gobet (Bernoulli, 2006) and the multi-step dynamic programming scheme of Gobet and Turkedjiev (Mathematics of Computation, 2016). Our algorithm approximates conditional expectations over segments of the time grid. We discuss the optimal choice of the segment length depending on the ‘smoothness’ of the problem and show that, in typical situations, the complexity can be reduced compared to the state-of-the-art multi-step dynamic programming scheme.

Hanson-Wright inequality provides a powerful tool for bounding the norm ‖ξ‖ of a centered stochastic vector ξ with independent entries and sub-gaussian behavior. This paper extends the bounds to the case when ξ only has bounded exponential moments of the form log E exp⟨V⁻¹ξ,u⟩≤‖u‖²/2, where V²≥Var(ξ) and ‖u‖≤g for some fixed g. For a linear mapping Q, we present an upper quantile function z_c(B,x) ensuring P(‖Qξ‖>z_c(B,x))≤3e^−x with B=QV²Q^T. The obtained results exhibit a phase transition effect: with a value xc depending on g and B, for x≤x_c, the function z_c(B,x) replicates the case of a Gaussian vector ξ, that is, $z_{c}^{2}(B, x)=\operatorname{tr}(B)+ 2 \sqrt{x \operatorname{tr}\left(B^{2}\right)}+2 x\|B\|. $ For x>x_c, the function z_c(B,x) grows linearly in x. The results are specified to the case of Bernoulli vector sums and to covariance estimation in Frobenius norm.