We take a new look at the relation between the optimal transport problem and the Schrödinger bridge problem from a stochastic control perspective. Our aim is to highlight new connections between the ...two that are richer and deeper than those previously described in the literature. We begin with an elementary derivation of the Benamou–Brenier fluid dynamic version of the optimal transport problem and provide, in parallel, a new fluid dynamic version of the Schrödinger bridge problem. We observe that the latter establishes an important connection with optimal transport without zero-noise limits and solves a question posed by Eric Carlen in 2006. Indeed, the two variational problems differ by a
Fisher information functional
. We motivate and consider a generalization of optimal mass transport in the form of a (fluid dynamic) problem of
optimal transport with prior
. This can be seen as the zero-noise limit of Schrödinger bridges when the prior is any Markovian evolution. We finally specialize to the Gaussian case and derive an explicit computational theory based on matrix Riccati differential equations. A numerical example involving Brownian particles is also provided.
The purpose of this paper is to provide a detailed probabilistic analysis of the optimal control of nonlinear stochastic dynamical systems of McKean–Vlasov type. Motivated by the recent interest in ...mean-field games, we highlight the connection and the differences between the two sets of problems. We prove a new version of the stochastic maximum principle and give sufficient conditions for existence of an optimal control. We also provide examples for which our sufficient conditions for existence of an optimal solution are satisfied. Finally we show that our solution to the control problem provides approximate equilibria for large stochastic controlled systems with mean-field interactions when subject to a common policy.
We discuss a continuous-time portfolio optimization problem to beat a stochastic benchmark. The model considers non-linear stochastic differential equations (SDEs) to model the dynamics of assets and ...economic factors. Unlike existing literature on risk-sensitive criteria, the proposed framework allows the model to capture the non-linearity in assets and factors dynamics. This article contributes to the two essential aspects of the problem; first, the existence and uniqueness of optimal investment strategy, which we prove using stochastic control theory and shows optimal strategies remain unchanged for the finite and infinite time horizon problems. Second, we use forecasting to compare investment performance under different economic factors. We analyze returns for the proposed model for three years in two important financial markets; S&P 100 and Dow 30. Risk versus return analysis indicates the importance of choosing relevant economic factors and their model for better portfolio performance. The results suggest an excellent bet to select the non-linear model over the linear one.
•This article examines a benchmarked asset management problem using a non-linear framework, assuming that the residual of economic factors and asset returns are uncorrelated.•The study demonstrates that the optimal strategies remain invariant, regardless of whether they are applied to risk-sensitive problems over finite or infinite time horizons.•The work also evaluates the performance of optimal investment strategies under various economic factors through forecasting.•The empirical analysis covers three years and includes two major financial markets: the S&P 100 and the Dow 30.•The risk-versus-return analysis emphasizes the critical importance of selecting relevant economic factors and their corresponding models.
Linear-Quadratic Mean Field Games Bensoussan, A.; Sung, K. C. J.; Yam, S. C. P. ...
Journal of optimization theory and applications,
05/2016, Letnik:
169, Številka:
2
Journal Article
Recenzirano
We provide a comprehensive study of a general class of linear-quadratic mean field games. We adopt the adjoint equation approach to investigate the unique existence of their equilibrium strategies. ...Due to the linearity of the adjoint equations, the optimal mean field term satisfies a forward–backward ordinary differential equation. For the one-dimensional case, we establish the unique existence of the equilibrium strategy. For a dimension greater than one, by applying the Banach fixed point theorem under a suitable norm, a sufficient condition for the unique existence of the equilibrium strategy is provided, which is independent of the coefficients of controls in the underlying dynamics and is always satisfied whenever the coefficients of the mean field term are vanished, and hence, our theories include the classical linear-quadratic stochastic control problems as special cases. As a by-product, we also establish a neat and instructive sufficient condition, which is apparently absent in the literature and only depends on coefficients, for the unique existence of the solution for a class of nonsymmetric Riccati equations. Numerical examples of nonexistence of the equilibrium strategy will also be illustrated. Finally, a similar approach has been adopted to study the linear-quadratic mean field type stochastic control problems and their comparisons with mean field games.
In this paper, we study de Finetti’s optimal dividend problem with capital injection under the assumption that the dividend strategies are absolutely continuous. In many previous studies, the process ...before being controlled was assumed to be a spectrally one-sided Lévy process, however in this paper we use a Lévy process that may have both positive and negative jumps. In the main theorem, we show that a refraction–reflection strategy is an optimal strategy. We also mention the existence and uniqueness of solutions of the stochastic differential equations that define refracted Lévy processes.
In this paper, a general event-triggered framework is set up to deal with the variance-constrained H∞ control problem for a class of discrete time-varying systems with randomly occurring saturations, ...stochastic nonlinearities and state-multiplicative noises. Based on the relative error with respect to the measurement signal, an event indicator variable is introduced and the corresponding event-triggered scheme is proposed in order to determine whether the measurement output is transmitted to the controller or not. The stochastic nonlinearities under consideration are characterized by statistical means which can cover several classes of well-studied nonlinearities. A set of unrelated random variables is exploited to govern the phenomena of randomly occurring saturations, stochastic nonlinearities and state-dependent noises. The purpose of the addressed multiobjective control problem is to design a set of time-varying output feedback controller such that, over a finite horizon, the closed-loop system achieves both the prescribed H∞ noise attenuation level and the state covariance constraints. A recursive matrix inequality approach is developed to derive the sufficient conditions for the existence of the desired finite-horizon controllers, and the analytical characterization of such controllers is also given. Simulation studies are conducted to demonstrate the effectiveness of the developed controller design scheme.
Control Barrier Functions (CBFs) aim to ensure safety by constraining the control input at each time step so that the system state remains within a desired safe region. This paper presents a ...framework for CBFs in stochastic systems in the presence of Gaussian process and measurement noise. We first consider the case where the system state is known at each time step, and present reciprocal and zero CBF constructions that guarantee safety with probability 1. We extend our results to high relative degree systems and present both general constructions and the special case of linear dynamics and affine safety constraints. We then develop CBFs for incomplete state information environments, in which the state must be estimated using sensors that are corrupted by Gaussian noise. We prove that our proposed CBF ensures safety with probability 1 when the state estimate is within a given bound of the true state, which can be achieved using an Extended Kalman Filter when the system is linear or the process and measurement noise are sufficiently small. We propose control policies that combine these CBFs with Control Lyapunov Functions in order to jointly ensure safety and stochastic stability. Our results are validated via numerical study on a multi-agent collision avoidance scenario.
In this work, we investigate the problem of event-triggered stabilization for a class of stochastic nonlinear systems. An event-triggered control (ETC) approach is proposed by introducing an ...additional internal dynamic variable. The presented event-triggered mechanism (ETM) can guarantee the existence of a positive lower bound on inter-event times (or called inter-execution times). In addition, the presented technique can ensure the second moment asymptotic stability of the closed-loop stochastic nonlinear system.
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