Large deviations for two scaled diffusions

We formulate large deviations principle (LDP) for diffusion pair $(X^\epsilon,\xi^\epsilon)=(X_t^\epsilon,\xi_t^\epsilon)$, where first component has a small diffusion parameter while the second is ergodic Markovian process with fast time. More exactly, the LDP is established for $(X^\epsilon,\nu^\epsilon)$ with $\nu^\epsilon(dt,dz)$ being an occupation type measure corresponding to $\xi_t^\epsilon$. In some sense we obtain a combination of Freidlin-Wentzell's and Donsker-Varadhan's results. Our approach relies the concept of the exponential tightness and Puhalskii's theorem

On a role of predictor in the filtering stability

When is a nonlinear filter stable with respect to its initial condition? In spite of the recent progress, this question still lacks a complete answer in general. Currently available results indicate that stability of the filter depends on the signal ergodic properties and the observation process regularity and may fail if either of the ingredients is ignored. In this note we address the question of stability in a particular weak sense and show that the estimates of certain functions are always stable. This is verified without dealing directly with the filtering equation and turns to be inherited from certain one-step predictor estimates.Comment: the final versio

On exponential stability of Wonham filter

We give elementary proof of a stability result concerning an exponential asymptotic ($t\to\infty$) for filtering estimates generated by wrongly initialized Wonham filter. This proof is based on new exponential bound having independent interest.Comment: 6 page

On tail distributions of supremum and quadratic variation of local martingales

We extend some known results relating the distribution tails of a continuous local martingale supremum and its quadratic variation to the case of locally square integrable martingales with bounded jumps. The predictable and optional quadratic variations are involved in the main result

Cramer's theorem for nonnegative multivariate point processes with independent increments

We consider a continuous time version of Cramer's theorem with nonnegative summands $S_t=\frac{1}{t}\sum_{i:\tau_i\le t}\xi_i, t \to\infty,$ where $(\tau_i,\xi_i)_{i\ge 1}$ is a sequence of random variables such that $tS_t$ is a random process with independent increments.Comment: 8 ppages, 2 figure

Large Deviations for Past-Dependent Recursions

The Large Deviation Principle is established for stochastic models defined by past-dependent non linear recursions with small noise. In the Markov case we use the result to obtain an explicit expression for the asymptotics of exit time.Comment: Revised versio

The Freidlin-Wentzell LDP with rapidly growing coefficients

The Large Deviations Principle (LDP) is verified for a homogeneous diffusion process with respect to a Brownian motion $B_t$, X^\eps_t=x_0+\int_0^tb(X^\eps_s)ds+ \eps\int_0^t\sigma(X^\eps_s)dB_s, where $b(x)$ and $\sigma(x)$ are are locally Lipschitz functions with super linear growth. We assume that the drift is directed towards the origin and the growth rates of the drift and diffusion terms are properly balanced. Nonsingularity of $a=\sigma\sigma^*(x)$ is not required.Comment: 20 page

On-line tracking of a smooth regression function

We construct an on-line estimator with equidistant design for tracking a smooth function from Stone-Ibragimov-Khasminskii class. This estimator has the optimal convergence rate of risk to zero in sample size. The procedure for setting coefficients of the estimator is controlled by a single parameter and has a simple numerical solution. The off-line version of this estimator allows to eliminate a boundary layer. Simulation results are given.Comment: 13 pages, 2 figure

Asymptotic analysis of ruin in CEV model

We give asymptotic analysis for probability of absorbtion $\mathsf{P}(\tau_0\le T)$ on the interval $[0,T]$, where $\tau_0=\inf\{t:X_t=0\}$ and $X_t$ is a nonnegative diffusion process relative to Brownian motion $B_t$, dX_t&=\mu X_tdt+\sigma X^\gamma_tdB_t. X_0&=K>0 Diffusion parameter $\sigma x^\gamma$, $\gamma\in [{1/2},1)$ is not Lipschitz continuous and assures $\mathsf{P}(\tau_0>T)>0$. Our main result: \lim\limits_{K\to\infty} \frac{1}{K^{2(1-\gamma)}}\log\mathsf{P}(\tau_{0}\le T) =-\frac{1}{2\E M^2_T}, where $M_T=\int_0^T\sigma(1-\gamma)e^{-(1-\gamma)\mu s}dB_s$. Moreover we describe the most likely path to absorbtion of the normed process $\frac{X_t}{K}$ for $K\to\infty$.Comment: 10 page

Tracking of Historical Volatility

We propose an adaptive algorithm for tracking of historical volatility. The algorithm is built under the assumption that the historical volatility function belongs to the Stone-Ibragimov-Khasminskii class of $k$ times differentiable functions with bounded highest derivative and its subclass of functions satisfying a differential inequalities. We construct an estimator of the Kalman filter type and show optimality of the estimator's convergence rate to zero as sample size $n\to\infty$. This estimator is in the framework of GARCH design, but a tuning procedure of its parameters is faster than with traditional GARCH techniques.Comment: 20 pages, 4 figure