Bounding stationary averages of polynomial diffusions via semidefinite programming

We introduce an algorithm based on semidefinite programming that yields increasing (resp. decreasing) sequences of lower (resp. upper) bounds on polynomial stationary averages of diffusions with polynomial drift vector and diffusion coefficients. The bounds are obtained by optimising an objective, determined by the stationary average of interest, over the set of real vectors defined by certain linear equalities and semidefinite inequalities which are satisfied by the moments of any stationary measure of the diffusion. We exemplify the use of the approach through several applications: a Bayesian inference problem; the computation of Lyapunov exponents of linear ordinary differential equations perturbed by multiplicative white noise; and a reliability problem from structural mechanics. Additionally, we prove that the bounds converge to the infimum and supremum of the set of stationary averages for certain SDEs associated with the computation of the Lyapunov exponents, and we provide numerical evidence of convergence in more general settings

The Euler-Maruyama approximation for the absorption time of the CEV diffusion

A standard convergence analysis of the simulation schemes for the hitting times of diffusions typically requires non-degeneracy of their coefficients on the boundary, which excludes the possibility of absorption. In this paper we consider the CEV diffusion from the mathematical finance and show how a weakly consistent approximation for the absorption time can be constructed, using the Euler-Maruyama scheme

From Infinite to Finite Programs: Explicit Error Bounds with Applications to Approximate Dynamic Programming

We consider linear programming (LP) problems in infinite dimensional spaces that are in general computationally intractable. Under suitable assumptions, we develop an approximation bridge from the infinite-dimensional LP to tractable finite convex programs in which the performance of the approximation is quantified explicitly. To this end, we adopt the recent developments in two areas of randomized optimization and first order methods, leading to a priori as well as a posterior performance guarantees. We illustrate the generality and implications of our theoretical results in the special case of the long-run average cost and discounted cost optimal control problems for Markov decision processes on Borel spaces. The applicability of the theoretical results is demonstrated through a constrained linear quadratic optimal control problem and a fisheries management problem.Comment: 30 pages, 5 figure

Optimization of mesh hierarchies in Multilevel Monte Carlo samplers

We perform a general optimization of the parameters in the Multilevel Monte Carlo (MLMC) discretization hierarchy based on uniform discretization methods with general approximation orders and computational costs. We optimize hierarchies with geometric and non-geometric sequences of mesh sizes and show that geometric hierarchies, when optimized, are nearly optimal and have the same asymptotic computational complexity as non-geometric optimal hierarchies. We discuss how enforcing constraints on parameters of MLMC hierarchies affects the optimality of these hierarchies. These constraints include an upper and a lower bound on the mesh size or enforcing that the number of samples and the number of discretization elements are integers. We also discuss the optimal tolerance splitting between the bias and the statistical error contributions and its asymptotic behavior. To provide numerical grounds for our theoretical results, we apply these optimized hierarchies together with the Continuation MLMC Algorithm. The first example considers a three-dimensional elliptic partial differential equation with random inputs. Its space discretization is based on continuous piecewise trilinear finite elements and the corresponding linear system is solved by either a direct or an iterative solver. The second example considers a one-dimensional It\^o stochastic differential equation discretized by a Milstein scheme

Risk-Sensitive Reinforcement Learning: A Constrained Optimization Viewpoint

The classic objective in a reinforcement learning (RL) problem is to find a policy that minimizes, in expectation, a long-run objective such as the infinite-horizon discounted or long-run average cost. In many practical applications, optimizing the expected value alone is not sufficient, and it may be necessary to include a risk measure in the optimization process, either as the objective or as a constraint. Various risk measures have been proposed in the literature, e.g., mean-variance tradeoff, exponential utility, the percentile performance, value at risk, conditional value at risk, prospect theory and its later enhancement, cumulative prospect theory. In this article, we focus on the combination of risk criteria and reinforcement learning in a constrained optimization framework, i.e., a setting where the goal to find a policy that optimizes the usual objective of infinite-horizon discounted/average cost, while ensuring that an explicit risk constraint is satisfied. We introduce the risk-constrained RL framework, cover popular risk measures based on variance, conditional value-at-risk and cumulative prospect theory, and present a template for a risk-sensitive RL algorithm. We survey some of our recent work on this topic, covering problems encompassing discounted cost, average cost, and stochastic shortest path settings, together with the aforementioned risk measures in a constrained framework. This non-exhaustive survey is aimed at giving a flavor of the challenges involved in solving a risk-sensitive RL problem, and outlining some potential future research directions

An Optimization Approach to Weak Approximation of Lévy-Driven Stochastic Differential Equations

We propose an optimization approach to weak approximation of Lévy-driven stochastic differential equations. We employ a mathematical programming framework to obtain numerically upper and lower bound estimates of the target expectation, where the optimization procedure ends up with a polynomial programming problem. An advantage of our approach is that all we need is a closed form of the Lévy measure, not the exact simulation knowledge of the increments or of a shot noise representation for the time discretization approximation. We also investigate methods for approximation at some different intermediate time points simultaneously

Peak Value-at-Risk Estimation for Stochastic Differential Equations using Occupation Measures

This paper proposes an algorithm to upper-bound maximal quantile statistics of a state function over the course of a Stochastic Differential Equation (SDE) system execution. This chance-peak problem is posed as a nonconvex program aiming to maximize the Value-at-Risk (VaR) of a state function along SDE state distributions. The VaR problem is upper-bounded by an infinite-dimensional Second-Order Cone Program in occupation measures through the use of one-sided Cantelli or Vysochanskii-Petunin inequalities. These upper bounds on the true quantile statistics may be approximated from above by a sequence of Semidefinite Programs in increasing size using the moment-Sum-of-Squares hierarchy when all data is polynomial. Effectiveness of this approach is demonstrated on example stochastic polynomial dynamical systems.Comment: 21 pages, 4 figures, 10 table