Scaling laws for density correlations and fluctuations in multiparticle dynamics

On large deviations of empirical measures for stationary Gaussian processes

In this paper we obtain three results concerning laws of the iterated logarithms (LILs) for certain functionals of some Markov processes. The first is for symmetric diffusions whose transition densities satisfy upper and lower bounds similar to those of Aronson for uniformly elliptic divergence form operators in R^. We suppose the transition densities pt(x, y) are symmetric in x and y and satisfy an estimate of the form

https://ir.library.osaka-u.ac.jp/repo/ouka/all/9960/ojm37_03_06.pdf

Laws of the iterated logarithm for some symmetric diffusion processes

The one-dimensional SDE with non Lipschitz diffusion coefficient 

$$\begin{aligned} \textit{dX}_{t} = b(X_{t})\textit{dt} + \sigma X_{t}^{\gamma } \textit{dB}_{t}, \quad X_{0}=x, \quad \gamma <1 \end{aligned}$$

(1)

is widely studied in mathematical finance Several works have proposed asymptotic analysis of densities and implied volatilities in models involving instances of (1), based on a careful implementation of saddle-point methods and (essentially) the explicit knowledge of Fourier transforms Recent research on tail asymptotics for heat kernels (Deuschel et al Comm in Pure and Applied Math, 67(1):40–82, 2014, [11]) suggests to work with the rescaled variable $ X^{\varepsilon }:=\varepsilon ^{1/(1-\gamma )} X$: while allowing to turn a space asymptotic problem into a small-$\varepsilon $ problem, the process $X^{\varepsilon }$ satisfies a SDE in Wentzell–Freidlin form (ie with driving noise $\varepsilon \textit{dB}$) We prove a pathwise large deviation principle for the process $X^{\varepsilon }$ as $\varepsilon \rightarrow 0$ As it will be seen, the limiting ODE governing the large deviations admits infinitely many solutions, a non-standard situation in the Wentzell–Freidlin theory As for applications, the $\varepsilon $-scaling allows to derive leading order asymptotics for path functionals: while on the one hand the resulting formulae are confirmed by the CIR-CEV benchmarks, on the other hand the large deviation approach (i) applies to equations with a more general drift term and (ii) potentially opens the way to heat kernel analysis for higher-dimensional diffusions involving (1) as a component

On Small-Noise Equations with Degenerate Limiting System Arising from Volatility Models

https://www.cims.nyu.edu/~benarous/Publications/benarous_26.pdf

A Probabilistic Approach to Semi-classical Approximations

An introduction to the theory of large deviations

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An introduction to the theory of large deviations