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Unformatted text preview: Chapter 22 Large Deviations for Small-Noise Stochastic Differential Equations This lecture is at once the end of our main consideration of dif- fusions and stochastic calculus, and a first taste of large deviations theory. Here we study the divergence between the trajectories pro- duced by an ordinary differential equation, and the trajectories of the same system perturbed by a small amount of white noise. Section 22.1 establishes that, in the small noise limit, the SDE’s trajectories converge in probability on the ODE’s trajectory. This uses Feller-process convergence. Section 22.2 upper bounds the rate at which the probability of large deviations goes to zero as the noise vanishes. The methods are elementary, but illustrate deeper themes to which we will recur once we have the tools of ergodic and information theory. In this chapter, we will use the results we have already obtained about SDEs to give a rough estimate of a basic problem, frequently arising in practice 1 namely taking a system governed by an ordinary differential equation and seeing how much effect injecting a small amount of white noise has. More exactly, we will put an upper bound on the probability that the perturbed trajectory goes very far from the unperturbed trajectory, and see the rate at which this probability goes to zero as the amplitude of the noise shrinks; this will be 1 For applications in statistical physics and chemistry, see Keizer (1987). For applications in signal processing and systems theory, see Kushner (1984). For applications in nonparamet- ric regression and estimation, and also radio engineering (!) see Ibragimov and Has’minskii (1979/1981). The last book is especially recommended for those who care about the connec- tions between stochastic process theory and statistical inference, but unfortunately expound- ing the results, or even just the problems, would require a too-long detour through asymptotic statistical theory. 144 CHAPTER 22. SMALL-NOISE SDES 145 O ( e- C 2 ). This will be our first illustration of a large deviations calculation. It will be crude, but it will also introduce some themes to which we will return (inshallah!) at greater length towards the end of the course. Then we will see that the major improvement of the more refined tools is to give a lower bound to match the upper bound we will calculate now, and see that we at least got the logarithmic rate right. I should say before going any further that this example is shamelessly ripped off from Freidlin and Wentzell (1998, ch. 3, sec. 1, pp. 70–71), which is the book on the subject of large deviations for continuous-time processes....
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This note was uploaded on 12/20/2011 for the course STAT 36-754 taught by Professor Schalizi during the Spring '06 term at University of Michigan.
- Spring '06