A symmetry-based constructive approach to probability densities for one-dimensional diffusion processes

In the present paper, the maximal Lyapunov exponent is investigated for a co-dimension two bifurcation system that is on a three-dimensional central manifold and subjected to parametric excitation by a bounded noise. By using a perturbation method, the expressions of the invariant measure of a one-dimensional phase diffusion process are obtained for three cases, in which different forms of the matrix B, that is included in the noise excitation term, are assumed and then, as a result, all the three kinds of singular boundaries for one-dimensional phase diffusion process are analyzed. Via Monte-Carlo simulation, we find that the analytical expressions of the invariant measures meet well the numerical ones. And furthermore, the P-bifurcation behaviors are investigated for the one-dimensional phase diffusion process. Finally, for the three cases of singular boundaries for one-dimensional phase diffusion process, analytical expressions of the maximal Lyapunov exponent are presented for the stochastic bifurcation system.

Occupation time theorems for one-dimensional random walks and diffusion processes in random environments

A stochastic model of tumor growth incorporating several key elements of the growth processes is presented. Generalizing a previous work by the authors, two one-dimensional diffusion processes representing populations of proliferating and quiescent cells are obtained. Their forms turn out by their relation with total tumor population analysed in (1). The proposed model is able to incorporate the effects of the mutual interactions between the two subpopulations. It is also used to simulate the effects of two kinds of time-dependent therapies: non-specific cycle and specific cycle drugs. Moreover, the first-exit-time problem is analyzed to study cancer evolution in the presence of a time-dependent therapy. 1 Introduction and background Tumor is one of the main causes of death in our so- ciety so, in the last twenty years a lot of attempts have been made to describe the tumor kinetic. Cancer cells population consists of a combination of proliferating, quiescent and dead cells that determine tumor growth based on surrounding environmental conditions (cf. (5)). Furthermore, since experimental data show the existence of more or less intense random fluctuations in tumor growth, in a previous work (see (1)) the authors provided a stochastic generalization of the Gompertz law in order to model monoclonal tumor growth. So tumor size is described by means of a one-dimensional diffusion process X(t) and the first exit time problem (FET) for X(t) from a region D has been analysed. In particular, D is restricted by two absorbing boundaries representing healing threshold and the carrying capacity. A first natural generalization consists of including all the essential biological phenomena of a cellular population. To this aim, following Kozusko and Bajzer (cf. (7)), we split tumor population in two subpopulations: proliferating and quiescent cells. In this direction, in (7), the authors proposed a deterministic model to describe tumor dynamics, assuming that the transition rates between proliferating and quiescent populations depend on the total population size. By imposing that the total population is governed by the Gompertz law, Kozusko and Bajzer resolved analytically the model and obtained the dynamics of prolif- erating and quiescent populations as functions of the whole population size. Following this approach, we describe proliferating and quiescent populations via two new diffusion pro- cesses, generally time-non-homogeneous, connected to the process X(t). The FET problem for such processes through suitable boundaries is considered. The followed approach per- mits to analyse the effect of different therapies on tumor cells. In Section 2 we will review the main characteristics of the model proposed in (1) in which the effect of a therapy is seen as a moderation term of the growth rate of the tumor cells. In Section 3 we generalize the deterministic model by Kozusko and Bajzer by obtaining two diffusion processes P(t) and Q(t) representing the populations of proliferating and quies- cent cells, respectively. Finally, in Section 4 the effect of a therapy on X(t) is investigated

Stochastic control of one-dimensional singular diffusion processes on an open interval $(\alpha ,\beta )$ is studied. Coefficients $m_i (x)$, $s_i (x)$ of those generators $({1 / {m_i (x)}})({d / {dx}})({1 /{s_i (x)}})({d / {dx}})$ are so singular that controlled processes cannot be defined by solutions of stochastic differential equations. We take up Markovian switching controlled processes and characterize the pay-off function, which is defined by additive functionals of the diffusion processes corresponding to given measures $\mu ^i $, by the unique solution of \[\begin{gathered} \mathcal{E}^i (u,v - u) \geqq \left\langle {\mu ^i ,v - u} \right\rangle \quad \forall v \in \mathcal{S},\quad i = 1,2,3, \cdots , \hfill u \in \mathcal{S}. \hfill \end{gathered} \] Here \[ \begin{gathered} \mathcal{E}^i (u,v) = \int_\alpha ^\beta {\frac{{du}} {{dx}}\frac{{dv}} {{dx}}\frac{1} {{s_i (x)}}dx,} \hfill \mathcal{S} = \left\{ {v \in H_0^1 (\alpha ,\beta );\mathcal{E}^i (u,\phi ) \leqq \left\langle {\mu ^i ,...

We prove in this note that, for general one dimensional control problems of diffusion processes, the associated Hamilton-Jacobi-Bellman equations hold in the sense of measures for the value function.

For a, b > 0, we consider a temporally homogeneous, one-dimensional diffusion process X(t) defined over I = (-b, a), with infinitesimal parameters depending on the sign of X(t). We suppose that, when X(t) reaches the position 0, it is reflected rightward to δ with probability p > 0 and leftward to -δ with probability 1 - p, where δ > 0. Closed analytical expressions are found for the mean exit time from the interval (-b, a), and for the probability of exit through the right end a, in the limit δ → 0+, generalizing the results of Lefebvre, holding for asymmetric Wiener process. Moreover, in alternative to the heavy analytical calculations, a numerical method is presented to estimate approximately the quantities above. Furthermore, on the analogy of skew Brownian motion, the notion of skew diffusion process is introduced. Some examples and numerical results are also reported.

A one-dimensional diffusion process X = {X t , 0 ≤ t ≤ T}, with drift b(x) and diffusion coefficient known up to θ > 0, is supposed to switch volatility regime at some point t* ∈ (0,T). On the basis of discrete time observations from X, the problem is the one of estimating the instant of change in the volatility structure t* as well as the two values of θ, say θ1 and θ2, before and after the change point. It is assumed that the sampling occurs at regularly spaced times intervals of length Δ n with nΔ n = T. To work out our statistical problem we use a least squares approach. Consistency, rates of convergence and distributional results of the estimators are presented under an high frequency scheme. We also study the case of a diffusion process with unknown drift and unknown volatility but constant.

This paper proposes a visual servoing algorithm for mobile robot navigation based on the epipolar geometry retrieved by object profiles. The main motivation for this approach is that for unstructured scenes the task of solving correspondences is definitely a harder task than contour detection. An unstructured three-dimensional (3-D) scene consists mainly of objects whose most important 2-D features are their apparent contours in the image plane. Apparent contours are used to estimate the positions of the epipoles and some special symmetry conditions whereby the visual servoing is able to steer the mobile robot to the desired position.

We investigate the critical behavior of a one-dimensional diffusive epidemic propagation process by means of a Monte Carlo procedure. In the model, healthy (A) and sick (B) individuals diffuse on a lattice with diffusion constants D(A) and D(B), respectively. According to a Wilson renormalization calculation, the system presents a second-order phase transition between a steady reactive state and a vacuum state, with distinct universality classes for the cases D(A)=D(B) and D(A)<D(B). A first-order transition has been conjectured for D(A)>D(B). In this work we perform a finite size scaling analysis of order parameter data at the vicinity of the critical point in dimension d=1. Our results show no signature of a first-order transition in the case of D(A)>D(B). A finite size scaling typical of second-order phase transitions fits well the data from all three regimes. We found that the correlation exponent nu=2 as predicted by field-theoretical arguments. Estimates for beta/nu are given for all relevant regimes.

We investigate the exit times from an interval for a general one-dimensional time-homogeneous diffusion process and their applications to the dividend problem in risk theory. Specifically, we first use Dynkin’s formula to derive the ordinary differential equations satisfied by the Laplace transform of the exit times. Then, as some examples, we solve the closed-form expression of the Laplace transform of the exit times for several popular diffusions, which are commonly used in modelling of finance and insurance market. Most interestingly, as the applications of the exit times, we create the connect between the dividend value function and the Laplace transform of the exit times. Both the barrier and threshold dividend value function are clearly expressed in terms of the Laplace transform of the exit times.

Nonlinear stochastic optimal control of Preisach hysteretic systems

The maximal Lyapunov exponent for a three-dimensional system driven by white noise

On definit et on construit le processus de Markov a une dimension correspondant a une fonction d'echelle non decroissante s et a une mesure de vitesse non negative dm, que l'on appellera processus de diffusion bigeneralise

The motion of ions, molecules or proteins in dendrites is restricted by cytoplasmic obstacles such as organelles, microtubules and actin network. To account for molecular crowding, we study the effect of diffusion barriers on local calcium spread in a dendrite. We first present a model based on a dimension reduction approach to approximate a three dimensional diffusion in a cylindrical dendrite by a one-dimensional effective diffusion process. By comparing uncaging experiments of an inert dye in a spiny dendrite and in a thin glass tube, we quantify the change in diffusion constants due to molecular crowding as Dcyto/Dwater = 1/20. We validate our approach by reconstructing the uncaging experiments using Brownian simulations in a realistic 3D model dendrite. Finally, we construct a reduced reaction-diffusion equation to model calcium spread in a dendrite under the presence of additional buffers, pumps and synaptic input. We find that for moderate crowding, calcium dynamics is mainly regulated by the buffer concentration, but not by the cytoplasmic crowding, dendritic spines or synaptic inputs. Following high frequency stimulations, we predict that calcium spread in dendrites is limited to small microdomains of the order of a few microns (<5 μm).

Let $R_t$ be the position of the rightmost particle at time $t$ in a time-homogeneous one-dimensional branching diffusion process. Let $\gamma(\alpha,t)$ be the $\alpha$th quantile of $R_t$ under $P^0$, where $P^x$ denotes the probability measure of the branching diffusion process starting with a single particle at position $x$. We show that $\gamma(\alpha,t)$ is a limiting quantile of $R_t$ under $P^x$ in the sense that $\lim_{t \rightarrow\infty}P^x\{R_t \leq \gamma(\alpha,t)\}$ exists for all $\alpha \in (0,1)$ and all $x \in \mathbb{R}$. If the underlying diffusion is recurrent, we show that, after an appropriate rescaling of space, the $P^x$ distribution of $R_t - t$ converges weakly to a nontrivial limiting distribution $w_x$.