Small Noise Diffusions Research Articles

Functional Large Deviations for Kac–Stroock Approximation to a Class of Gaussian Processes with Application to Small Noise Diffusions

Functional Large Deviations for Kac–Stroock Approximation to a Class of Gaussian Processes with Application to Small Noise Diffusions

Large deviations for small noise diffusions over long time

We study two problems. First, we consider the large deviation behavior of empirical measures of certain diffusion processes as, simultaneously, the time horizon becomes large and noise becomes vanishingly small. The law of large numbers (LLN) of the empirical measure in this asymptotic regime is given by the unique equilibrium of the noiseless dynamics. Due to degeneracy of the noise in the limit, the methods of Donsker and Varadhan [Comm. Pure Appl. Math. 29 (1976), pp. 389–461] are not directly applicable and new ideas are needed. Second, we study a system of slow-fast diffusions where both the slow and the fast components have vanishing noise on their natural time scales. This time the LLN is governed by a degenerate averaging principle in which local equilibria of the noiseless system obtained from the fast dynamics describe the asymptotic evolution of the slow component. We establish a large deviation principle that describes probabilities of divergence from this behavior. On the one hand our methods require stronger assumptions than the nondegenerate settings, while on the other hand the rate functions take simple and explicit forms that have striking differences from their nondegenerate counterparts.

Analysis and Optimization of Certain Parallel Monte Carlo Methods in the Low Temperature Limit

Metastability is a formidable challenge to Markov chain Monte Carlo methods. In this paper we present methods for algorithm design to meet this challenge. The design problem we consider is temperature selection for the infinite swapping scheme, which is the limit of the widely used parallel tempering scheme obtained when the swap rate tends to infinity. We use a recently developed tool for the analysis of the empirical measure of a small noise diffusion to transform the variance reduction problem into an explicit optimization problem. Our first analysis of the optimization problem is in the setting of a double-well model, and it shows that the optimal selection of temperature ratios is a geometric sequence except possibly the highest temperature. In the same setting we identify two different sources of variance reduction and show how their competition determines the optimal highest temperature. In the general multiwell setting we prove that the same geometric sequence of temperature ratios as in the two-well case is always nearly optimal, with a performance gap that decays geometrically in the number of temperatures.

Large Deviation Properties of the Empirical Measure of a Metastable Small Noise Diffusion

The aim of this paper is to develop tractable large deviation approximations for the empirical measure of a small noise diffusion. The starting point is the Freidlin–Wentzell theory, which shows how to approximate via a large deviation principle the invariant distribution of such a diffusion. The rate function of the invariant measure is formulated in terms of quasipotentials, quantities that measure the difficulty of a transition from the neighborhood of one metastable set to another. The theory provides an intuitive and useful approximation for the invariant measure, and along the way many useful related results (e.g., transition rates between metastable states) are also developed. With the specific goal of design of Monte Carlo schemes in mind, we prove large deviation limits for integrals with respect to the empirical measure, where the process is considered over a time interval whose length grows as the noise decreases to zero. In particular, we show how the first and second moments of these integrals can be expressed in terms of quasipotentials. When the dynamics of the process depend on parameters, these approximations can be used for algorithm design, and applications of this sort will appear elsewhere. The use of a small noise limit is well motivated, since in this limit good sampling of the state space becomes most challenging. The proof exploits a regenerative structure, and a number of new techniques are needed to turn large deviation estimates over a regenerative cycle into estimates for the empirical measure and its moments.

Asymptotic behaviour of randomised fractional volatility models

AbstractWe study the asymptotic behaviour of a class of small-noise diffusions driven by fractional Brownian motion, with random starting points. Different scalings allow for different asymptotic properties of the process (small-time and tail behaviours in particular). In order to do so, we extend some results on sample path large deviations for such diffusions. As an application, we show how these results characterise the small-time and tail estimates of the implied volatility for rough volatility models, recently proposed in mathematical finance.

Large deviations for small noise diffusions in a fast markovian environment

A large deviation principle is established for a two-scale stochastic system in which the slow component is a continuous process given by a small noise finite dimensional Ito stochastic differential equation, and the fast component is a finite state pure jump process. Previous works have considered settings where the coupling between the components is weak in a certain sense. In the current work we study a fully coupled system in which the drift and diffusion coefficient of the slow component and the jump intensity function and jump distribution of the fast process depend on the states of both components. In addition, the diffusion can be degenerate. Our proofs use certain stochastic control representations for expectations of exponential functionals of finite dimensional Brownian motions and Poisson random measures together with weak convergence arguments. A key challenge is in the proof of the large deviation lower bound where, due to the interplay between the degeneracy of the diffusion and the full dependence of the coefficients on the two components, the associated local rate function has poor regularity properties.

Asymptotic Behaviour of Randomised Fractional Volatility Models

We study the asymptotic behaviour of a class of small-noise diffusions driven by fractional Brownian motion, with random starting points. Different scalings allow for different asymptotic properties of the process (small-time and tail behaviours in particular). In order to do so, we extend some results on sample path large deviations for such diffusions. As an application, we show how these results characterise the small-time and tail estimates of the implied volatility for rough volatility models, recently proposed in mathematical finance.

Gaussian approximations of small noise diffusions in Kullback–Leibler divergence

We study Gaussian approximations to the distribution of a diffusion. The approximations are easy to compute: they are defined by two simple ordinary differential equations for the mean and the covariance. Time correlations can also be computed via solution of a linear stochastic differential equation. We show, using the Kullback-Leibler divergence, that the approximations are accurate in the small noise regime. An analogous discrete time setting is also studied. The results provide both theoretical support for the use of Gaussian processes in the approximation of diffusions, and methodological guidance in the construction of Gaussian approximations in applications.

On score-functions and goodness-of-fit tests for stochastic processes

The problems of the construction of asymptotically distribution free goodness-of-fit tests for two diffusion processes are considered. The null hypothesis is composite parametric. All tests are based on the score-function processes, where the unknown parameter is replaced by the maximum likelihood estimators. We show that a special change of time transforms the limit score-function processes into the Brownian bridge. This property allows us to construct asymptotically distribution-free tests for dynamical systems with small noise and ergodic diffusion processes. The proposed tests are in some sense universal. We discuss the possibilities of the construction of asymptotically distribution free tests for inhomogeneous Poisson processes and nonlinear AR time series.

Escaping from an attractor: Importance sampling and rest points I

We discuss importance sampling schemes for the estimation of finite time exit probabilities of small noise diffusions that involve escape from an equilibrium. A factor that complicates the analysis is that rest points are included in the domain of interest. We build importance sampling schemes with provably good performance both pre-asymptotically, that is, for fixed size of the noise, and asymptotically, that is, as the size of the noise goes to zero, and that do not degrade as the time horizon gets large. Simulation studies demonstrate the theoretical results.

Nonasymptotic performance analysis of importance sampling schemes for small noise diffusions

In this paper we develop a prelimit analysis of performance measures for importance sampling schemes related to small noise diffusion processes. In importance sampling the performance of any change of measure is characterized by its second moment. For a given change of measure, we characterize the second moment of the corresponding estimator as the solution to a partial differential equation, which we analyze via a full asymptotic expansion with respect to the size of the noise and obtain a precise statement on its accuracy. The main correction term to the decay rate of the second moment solves a transport equation that can be solved explicitly. The asymptotic expansion that we obtain identifies the source of possible poor performance of nevertheless asymptotically optimal importance sampling schemes and allows for a more accurate comparison among competing importance sampling schemes.

Nonasymptotic performance analysis of importance sampling schemes for small noise diffusions

In this paper we develop a prelimit analysis of performance measures for importance sampling schemes related to small noise diffusion processes. In importance sampling the performance of any change of measure is characterized by its second moment. For a given change of measure, we characterize the second moment of the corresponding estimator as the solution to a partial differential equation, which we analyze via a full asymptotic expansion with respect to the size of the noise and obtain a precise statement on its accuracy. The main correction term to the decay rate of the second moment solves a transport equation that can be solved explicitly. The asymptotic expansion that we obtain identifies the source of possible poor performance of nevertheless asymptotically optimal importance sampling schemes and allows for a more accurate comparison among competing importance sampling schemes.

Fluctuation analysis and short time asymptotics for multiple scales diffusion processes

We consider the limiting behavior of fluctuations of small noise diffusions with multiple scales around their homogenized deterministic limit. We allow full dependence of the coefficients on the slow and fast motion. These processes arise naturally when one is interested in short time asymptotics of multiple scale diffusions. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. Depending on the order of interaction between the fast scale and the size of the noise, we get different behavior. In certain cases additional drift terms arise in the limiting process, which are explicitly characterized. These results provide a better approximation to the limiting behavior of such processes when compared to the law of large numbers homogenization limit.

Large Deviations and Importance Sampling for Systems of Slow-Fast Motion

In this paper we develop the large deviations principle and a rigorous mathematical framework for asymptotically efficient importance sampling schemes for general, fully dependent systems of stochastic differential equations of slow and fast motion with small noise in the slow component. We assume periodicity with respect to the fast component. Depending on the interaction of the fast scale with the smallness of the noise, we get different behavior. We examine how one range of interaction differs from the other one both for the large deviations and for the importance sampling. We use the large deviations results to identify asymptotically optimal importance sampling schemes in each case. Standard Monte Carlo schemes perform poorly in the small noise limit. In the presence of multiscale aspects one faces additional difficulties and straightforward adaptation of importance sampling schemes for standard small noise diffusions will not produce efficient schemes. It turns out that one has to consider the so called cell problem from the homogenization theory for Hamilton-Jacobi-Bellman equations in order to guarantee asymptotic optimality. We use stochastic control arguments.

Rare Event Simulation of Small Noise Diffusions

AbstractAn important sampling method for certain rare event problems involving small noise diffusions is proposed. Standard Monte Carlo schemes for these problems behave exponentially poorly in the small noise limit. Previous work in rare event simulation has focused on developing estimators with optimal exponential variance decay rates. This criterion still allows for exponential growth of the statistical relative error. We show that an estimator related to a deterministic control problem not only has an optimal variance decay rate but can have vanishingly small relative statistical error in the small noise limit. The sampling method based on this estimator can be seen as the limit of the zero variance importance sampling scheme, which uses the solution of the second‐order partial differential equation (PDE) associated with the diffusion. In the scheme proposed here this PDE is replaced by a Hamilton‐Jacobi equation whose solution is computed pointwise on the fly from its variational formulation, an operation that remains practical even in high‐dimensional problems. We test the scheme on several simple illustrative examples as well as a stochastic PDE, the noisy Allen‐Cahn equation. © 2012 Wiley Periodicals, Inc.

Large deviations for multiscale diffusion via weak convergence methods

We study the large deviations principle for locally periodic SDEs with small noise and fast oscillating coefficients. There are three regimes depending on how fast the intensity of the noise goes to zero relative to homogenization parameter. We use weak convergence methods which provide convenient representations for the action functional for all regimes. Along the way, we study weak limits of controlled SDEs with fast oscillating coefficients. We derive, in some cases, a control that nearly achieves the large deviations lower bound at prelimit level. This control is useful for designing efficient importance sampling schemes for multiscale small noise diffusion.

Importance Sampling for Multiscale Diffusions

We construct importance sampling schemes for stochastic differential equations with small noise and fast oscillating coefficients. Standard Monte Carlo methods perform poorly for these problems in the small noise limit. With multiscale processes there are additional complications, and indeed the straightforward adaptation of importance sampling methods for standard small noise diffusions will not produce efficient schemes. Using the subsolution approach we construct schemes and identify conditions under which the schemes will be asymptotically optimal. Examples and simulation results are provided.

Learning, Large Deviations, And Recurrent Currency Crises*

This article studies a version of Obstfeld's (Journal of International Economics 43 (1997), 61–77) “escape clause” model. The model is calibrated to produce three rational expectations equilibria. Two of these equilibria are E‐stable and one is unstable. Dynamics are introduced by assuming that agents must learn about the government's decision rule. It is assumed they do this using a stochastic approximation algorithm. It turns out that as a certain parameter describing the sensitivity of beliefs to new information gets small, the algorithm converges to a small noise diffusion process. The dynamics of exchange rate changes are then characterized using large deviation techniques from Freidlin and Wentzell (Random Perturbations of Dynamical Systems, Second Edition, Berlin: Springer‐Verlag, 1998). These methods describe the sense in which the limiting distribution of exchange rate changes is approximated by a two‐state Markov‐Switching process, where the two states correspond to the two E‐stable equilibria. The model is calibrated to the exchange rate histories of Argentina, Brazil, and Mexico. Currency crises in these countries resemble the predicted “escape routes” of the model. A key feature of these escape routes is that expectations of a devaluation erupt suddenly, without large contemporaneous shocks. This is consistent with evidence showing that crises are often poorly anticipated by financial markets.

Risk-Sensitive and Robust Escape Control for Degenerate Diffusion Processes

This paper considers the problem of controlling a possibly degenerate small noise diffusion so as to prevent it from leaving a prescribed set. The criterion of interest is a risk-sensitive version of the mean escape time criterion. Using a general representation formula, this criterion is expressed as the upper value of a stochastic differential game. It is shown that in the small noise limit this upper value converges to the value of an associated deterministic differential game. Our approach differs from standard PDE approaches in a number of ways. For example, the upper game representation allows one to relate directly the prelimit and the limit controls and, in fact, strategies that are nearly maximizing for the robust problem can be used to define nearly minimizing controls for the risk-sensitive control problem for sufficiently small e>0. The result provides a canonical example of the use of variational representations in connecting risk-sensitive and robust control.

Large deviations for small noise diffusions with discontinuous statistics

This paper proves the large deviation principle for a class of non-degenerate small noise diffusions with discontinuous drift and with state-dependent diffusion matrix. The proof is based on a variational representation for functionals of strong solutions of stochastic differential equations and on weak convergence methods.