Moderate deviations for systems of slow-fast diffusions

Abstract

In this paper, we prove the moderate deviations principle (MDP) for a general system of slow-fast dynamics. We provide a unified approach, based on weak convergence ideas and stochastic control arguments, that cover both the averaging and the homogenization regimes. We allow the coefficients to be in the whole space and not just the torus and allow the noises driving the slow and fast processes to be correlated arbitrarily. Similar to the large deviation case, the methodology that we follow allows construction of provably efficient Monte Carlo methods for rare events that fall into the moderate deviations regime.For convenience, we refer to the state space of Y ε as Y. The parameter ε ≪ 1 represents the strength of the noise while δ ≪ 1 is the time-scale separation parameter. W t and B t are independent m-dimensional Brownian motions.In (1), X ε is the slow motion and Y ε is the fast motion. Depending on the order in which ε, δ go to zero, we get different behavior, and in particular

Simultaneous Small Noise Limit for Singularly Perturbed Slow-Fast Coupled Diffusions

“…where ν 0,Xt t (dy) is a probability measure supported on finitely many global minima of U (X t , ·) (see Theorem 1.3). If an additional assumption on the behaviour of U (x, ·) at its global minima is made then we show that ν 0,Xt t (dy) is time independent and given by a determinantal formula arising from In related works, Spiliopoulos in [Spi13,Spi14], Morse and Spiliopolous in [MS17], and Gailus and Spiliopoulous in [GS17] considered a class of coupled diffusions with multiple time scales in the full dependence setting. Contained therein, after suitable relabelling of the parameters and appropriate choice of coefficients, are results that will apply to (1)-(2) for specific b, ∇ y U and with s(ε) = ε α− 1 2 .…”

“…In [Spi13], an LDP is shown for the slow process under periodicity assumptions for all the three regimes. Without the periodicity assumption on the coefficients, in [Spi14] fluctuation results for the slow process are shown in the homogenization and s(ε) = 1 regimes, while in [MS17] moderate deviations for the slow process are shown for these two regimes. In [GS17] parameter estimation results are obtained when s(ε) = 1.…”

Pathwise moderate deviations for option pricing

“…r = 1 in the notation of [28]. In addition, we would like to note that Theorem 2.1 of [28] is proven under slightly stronger conditions on the growth of the coefficients than the ones made in the current Assumption 3.1. There are two reasons for this.…”

“…Theorem 3.3 corresponds to the setting of Regime 2 in [28] and in the special case where the drift of the fast motion, Y ε grows at most linearly in y, i.e. r = 1 in the notation of [28]. In addition, we would like to note that Theorem 2.1 of [28] is proven under slightly stronger conditions on the growth of the coefficients than the ones made in the current Assumption 3.1.…”

“…Even though this is to be intuitively expected, it shows that one can use moderate deviations to get an approximation to large deviations asymptotics, and is useful in practice as the moderate deviations rate function is usually available in closed form as a quadratic, whereas the large deviations rate function is often not explicit. Lastly, we mention that even though the results in this paper are mostly presented for X ∈ R 2 , the actual theorems hold in any finite dimension, as in [1,5,14,28]. We restrict ourselves here to this setting, solely because of the financial motivation.…”