6,024 research outputs found
Large deviations for some fast stochastic volatility models by viscosity methods
We consider the short time behaviour of stochastic systems affected by a
stochastic volatility evolving at a faster time scale. We study the asymptotics
of a logarithmic functional of the process by methods of the theory of
homogenisation and singular perturbations for fully nonlinear PDEs. We point
out three regimes depending on how fast the volatility oscillates relative to
the horizon length. We prove a large deviation principle for each regime and
apply it to the asymptotics of option prices near maturity
ES Is More Than Just a Traditional Finite-Difference Approximator
An evolution strategy (ES) variant based on a simplification of a natural
evolution strategy recently attracted attention because it performs
surprisingly well in challenging deep reinforcement learning domains. It
searches for neural network parameters by generating perturbations to the
current set of parameters, checking their performance, and moving in the
aggregate direction of higher reward. Because it resembles a traditional
finite-difference approximation of the reward gradient, it can naturally be
confused with one. However, this ES optimizes for a different gradient than
just reward: It optimizes for the average reward of the entire population,
thereby seeking parameters that are robust to perturbation. This difference can
channel ES into distinct areas of the search space relative to gradient
descent, and also consequently to networks with distinct properties. This
unique robustness-seeking property, and its consequences for optimization, are
demonstrated in several domains. They include humanoid locomotion, where
networks from policy gradient-based reinforcement learning are significantly
less robust to parameter perturbation than ES-based policies solving the same
task. While the implications of such robustness and robustness-seeking remain
open to further study, this work's main contribution is to highlight such
differences and their potential importance
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Robust H2/H∞-state estimation for systems with error variance constraints: the continuous-time case
Copyright [1999] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.The paper is concerned with the state estimator design problem for perturbed linear continuous-time systems with H∞ norm and variance constraints. The perturbation is assumed to be time-invariant and norm-bounded and enters into both the state and measurement matrices. The problem we address is to design a linear state estimator such that, for all admissible measurable perturbations, the variance of the estimation error of each state is not more than the individual prespecified value, and the transfer function from disturbances to error state outputs satisfies the prespecified H∞ norm upper bound constraint, simultaneously. Existence conditions of the desired estimators are derived in terms of Riccati-type matrix inequalities, and the analytical expression of these estimators is also presented. A numerical example is provided to show the directness and effectiveness of the proposed design approac
Generalized Stochastic Gradient Learning
We study the properties of generalized stochastic gradient (GSG) learning in forwardlooking models. We examine how the conditions for stability of standard stochastic gradient (SG) learning both di1er from and are related to E-stability, which governs stability under least squares learning. SG algorithms are sensitive to units of measurement and we show that there is a transformation of variables for which E-stability governs SG stability. GSG algorithms with constant gain have a deeper justification in terms of parameter drift, robustness and risk sensitivity
Robust Mean Square Stability of Open Quantum Stochastic Systems with Hamiltonian Perturbations in a Weyl Quantization Form
This paper is concerned with open quantum systems whose dynamic variables
satisfy canonical commutation relations and are governed by quantum stochastic
differential equations. The latter are driven by quantum Wiener processes which
represent external boson fields. The system-field coupling operators are linear
functions of the system variables. The Hamiltonian consists of a nominal
quadratic function of the system variables and an uncertain perturbation which
is represented in a Weyl quantization form. Assuming that the nominal linear
quantum system is stable, we develop sufficient conditions on the perturbation
of the Hamiltonian which guarantee robust mean square stability of the
perturbed system. Examples are given to illustrate these results for a class of
Hamiltonian perturbations in the form of trigonometric polynomials of the
system variables.Comment: 11 pages, Proceedings of the Australian Control Conference, Canberra,
17-18 November, 2014, pp. 83-8
Relative Value Iteration for Stochastic Differential Games
We study zero-sum stochastic differential games with player dynamics governed
by a nondegenerate controlled diffusion process. Under the assumption of
uniform stability, we establish the existence of a solution to the Isaac's
equation for the ergodic game and characterize the optimal stationary
strategies. The data is not assumed to be bounded, nor do we assume geometric
ergodicity. Thus our results extend previous work in the literature. We also
study a relative value iteration scheme that takes the form of a parabolic
Isaac's equation. Under the hypothesis of geometric ergodicity we show that the
relative value iteration converges to the elliptic Isaac's equation as time
goes to infinity. We use these results to establish convergence of the relative
value iteration for risk-sensitive control problems under an asymptotic
flatness assumption
Optimal control of multiscale systems using reduced-order models
We study optimal control of diffusions with slow and fast variables and
address a question raised by practitioners: is it possible to first eliminate
the fast variables before solving the optimal control problem and then use the
optimal control computed from the reduced-order model to control the original,
high-dimensional system? The strategy "first reduce, then optimize"--rather
than "first optimize, then reduce"--is motivated by the fact that solving
optimal control problems for high-dimensional multiscale systems is numerically
challenging and often computationally prohibitive. We state sufficient and
necessary conditions, under which the "first reduce, then control" strategy can
be employed and discuss when it should be avoided. We further give numerical
examples that illustrate the "first reduce, then optmize" approach and discuss
possible pitfalls
Robustifying Learnability
In recent years, the learnability of rational expectations equilibria (REE) and determinacy of economic structures have rightfully joined the usual performance criteria among the sought after goals of policy design. And while some contributions to the literature (for example Bullard and Mitra (2001) and Evans and Honkapohja (2002)) have made significant headway in establishing certain features of monetary policy rules that facilitate learning, a comprehensive treatment of policy design for learnability has yet to surface, especially for cases in which agents have potentially misspecified their learning models. This paper provides such a treatment. We argue that since even among professional economists a generally acceptable workhorse model of the economy has not been agreed upon, it is unreasonable to expect private agents to have collective rational expectations. We assume instead that agents have an approximate understanding of the workings of the economy and that their task of learning true reduced forms of the economy is subject to potentially destabilizing errors. We then ask: can a central bank set policy that accounts for learning errors but also succeeds in bounding them in a way that allows eventual learnability of the model, given policy. For different parameterizations of a given policy rule applied to a New Keynesian model, we use structured singular value analysis (from robust control) to find the largest ranges of misspecifications that can be tolerated in a learning model without compromising convergence to an REE. A parallel set of experiments seeks to determine the optimal stance (strong inflation as opposed to strong output stabilization) that allows for the greatest scope of errors in learning without leading to expectational instabilty in cases when the central bank designs both optimal and robust policy rules with commitment. We compare the features of all the rules contemplated in the paper with those that maximize economic performance in the true model, and we measure the performance cost of maximizing learnability under the various conditions mentioned here.monetary policy, learning, E-stability, model uncertainty, robustness
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