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Unformatted text preview: Chapter 35 Large Deviations for Stochastic Differential Equations This last chapter revisits large deviations for stochastic differen- tial equations in the small-noise limit, first raised in Chapter 22. Section 35.1 establishes the LDP for the Wiener process (Schilder’s Theorem). Section 35.2 proves the LDP for stochastic differential equations where the driving noise is independent of the state of the process. Section 35.3 states the corresponding result for SDEs when the noise is state-dependent, and gestures in the direction of the proof. In Chapter 22, we looked at how the diffusions X which solve the SDE dX = a ( X ) dt + dW , X (0) = x (35.1) converge on the trajectory x ( t ) solving the ODE dx dt = a ( x ( t )) , x (0) = x (35.2) in the “small noise” limit, → 0. Specifically, Theorem 256 gave a (fairly crude) upper bound on the probability of deviations: lim → 2 log P sup ≤ t ≤ T Δ ( t ) > δ ≤ - δ 2 e- 2 K a T (35.3) where K a depends on the Lipschitz coefficient of the drift function a . The the- ory of large deviations for stochastic differential equations, known as Freidlin- Wentzell theory for its original developers, shows that, using the metric implicit 238 CHAPTER 35. FREIDLIN-WENTZELL THEORY 239 in the left-hand side of Eq. 35.3, the family of processes X obey a large devia- tions principle with rate- 2 , and a good rate function. (The full Freidlin-Wentzell theory actually goes somewhat further than just SDEs, to consider small-noise perturbations of dynamical systems of many sorts, perturbations by Markov processes (rather than just white noise), etc. Time does not allow us to consider the full theory (Freidlin and Wentzell, 1998), or its many applications to nonparametric estimation (Ibragimov and Has’minskii, 1979/1981), systems analysis and signal processing (Kushner, 1984), statistical mechanics (Olivieri and Vares, 2005), etc.) As in Chapter 31, the strategy is to first prove a large deviations principle for a comparatively simple case, and then transfer it to more subtle processes which can be represented as appropriate functionals of the basic case. Here, the basic case is the Wiener process W ( t ), with t restricted to the unit interval [0 , 1]. 35.1 Large Deviations of the Wiener Process We start with a standard d-dimensional Wiener process W , and consider its di- lation by a factor , X ( t ) = W ( t ). There are a number of ways of establishing that X obeys a large deviation principle as → 0. One approach (see Dembo and Zeitouni (1998, ch. 5) starts with establishing an LDP for continuous-time random walks, ultimately based on the G¨ artner-Ellis Theorem, and then show- ing that the convergence of such processes to the Wiener process (the Functional Central Limit Theorem, Theorem 174 of Chapter 16) is sufficiently fast that the LDP carries over. However, this approach involves a number of surprisingly tricky topological issues, so I will avoid it, in favor of a more probabilistic path, marked out by Freidlin and Wentzell (Freidlin and Wentzell, 1998, sec. 3.2).marked out by Freidlin and Wentzell (Freidlin and Wentzell, 1998, sec....
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- Spring '06
- LDP, large deviations, Rate function, Large deviations theory, Contraction Principle, freidlin-wentzell theory