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The Ornstein-Uhlenbeck process may be used to generate a noise signal with a finite correlation time. If a one-dimensional stochastic process is driven by such a noise source, it may be analysed by solving a Fokker-Planck equation in two…
This paper is concerned with nonlinear filtering of the coefficients in asset price models with stochastic volatility. More specifically, we assume that the asset price process $ S=(S_{t})_{t\geq0} $ is given by \[…
This paper investigates the asymptotic behavior of stochastic recursive inclusions in the presence of non-zero, non-diminishing bias, a setting that frequently arises in zeroth-order optimization, stochastic approximation with…
Consider a noisy linear observation model with an unknown permutation, based on observing $y = \Pi^* A x^* + w$, where $x^* \in \mathbb{R}^d$ is an unknown vector, $\Pi^*$ is an unknown $n \times n$ permutation matrix, and $w \in…
Given a sequence $(M_{k}, Q_{k})_{k\ge 1}$ of independent, identically distributed ran\-dom vectors with nonnegative components, we consider the recursive Markov chain $(X_{n})_{n\ge 0}$, defined by the random difference equation…
This paper presents a general approach to linear stochastic processes driven by various random noises. Mathematically, such processes are described by linear stochastic differential equations of arbitrary order (the simplest non-trivial…
We develop a convergent variational perturbation theory for conditional probability densities of Markov processes. The power of the theory is illustrated by applying it to the diffusion of a particle in an anharmonic potential.
We consider a diffusion $(\xi_t)_{t\ge 0}$ whose drift involves a $T$-periodic signal. $T$ is fixed and known, whereas the signal depends on an unknown $d$-dimensional parameter $\vartheta\in\Theta$. Assuming positive Harris recurrence of…
This paper is concerned with nonlinear filtering of the coefficients in asset price models with stochastic volatility. More specifically, we assume that the asset price process $S=(S_{t})_{t\geq0}$ is given by \[ dS_{t}=m(\theta_{t})S_{t}…
We study the $\epsilon \to 0$ behavior of recurrence relations of the type $\sum_{j=0}^l a_j(k\epsilon,\epsilon)y_{k+j}=0,$ $k\in \zdd$ ($l$ fixed). The $a_j$ are $C^{\infty}$ functions in each variable on $I\times [0,\e_0]$ for a bounded…
By well known results of probability theory, any sequence of random variables with bounded second moments has a subsequence satisfying the central limit theorem and the law of the iterated logarithm in a randomized form. In this paper we…
We consider stability and uniqueness in real phase retrieval problems over general input sets. Specifically, we assume the data consists of noisy quadratic measurements of an unknown input x in R^n that lies in a general set T and study…
A three term recurrence relation is derived for a basis consisting of polynomials multiplied by sines and cosines with large, but fixed frequencies. A numerical method for computing the coefficients of the three term recurrence relation is…
We consider a problem of estimation for the telegrapher's process on the line, say X(t), driven by a Poisson process with non constant rate. It turns out that the finite-dimensional law of the process X(t) is a solution to the telegraph…
We present a finite-order system of recurrence relations for a permanent of circulant matrices containing a band of k any-value diagonals on top of a uniform matrix (for k = 1, 2, and 3) as well as the method for deriving such recurrence…
Let $(Y_n)$ be a sequence of i.i.d. real valued random variables. Reflected random walk $(X_n)$ is defined recursively by $X_0=x \ge 0$, $X_{n+1} = |X_n - Y_{n+1}|$. In this note, we study recurrence of this process, extending a previous…
We consider k-step recurrences of the form $z_{n+k} = A(z)/B(z)$, where A and B are linear functions of $z_n, z_{n+1}, ..., z_{n+k-1}$, which we call k-step linear fractional recurrences. The first Theorem in this paper shows that for each…
By using variational methods, the existence of infinitely many solutions for a nonlinear algebraic system with a parameter is established in presence of a perturbed Lipschitz term. Our goal was achieved requiring an appropriate behavior of…
We study the problem of nonparametric estimation of the linear multiplier function $\theta(t)$ for processes satisfying stochastic differential equations of the type $$dX_t= \theta(t)X_t dt+ \epsilon\; \sigma_1(t,X_t)\sigma_2(t,Y_t)dW_t,…
Filtered Poisson processes are often used as reference models for intermittent fluc- tuations in physical systems. Such a process is here extended by adding a noise term, either as a purely additive term to the process or as a dynamical…