John B. Walsh

A STOCHASTIC MODEL OF NEURAL RESPONSE

We propose a model of a passive nerve cylinder undergoing random stimulus along its length. It is shown that this model is approximated by the solution of a stochastic partial differential equation. Numerous properties of the sample paths are derived, such as their modulus of continuity, quadratic and quartic variation, and it is shown that the solution exhibits the phenomenon of flicker noise. The first-passage problem is studied, and it is shown to be connected with a first-hitting time for an infinite-dimensional diffusion.

Markov processes, Brownian motion, and time symmetry

Markov Process.- Basic Properties.- Hunt Process.- Brownian Motion.- Potential Developments.- Generalities.- Markov Chains: a Fireside Chat.- Ray Processes.- Application to Markov Chains.- Time Reversal.- h-Transforms.- Death and Transfiguration: A Fireside Chat.- Processes in Duality.- The Martin Boundary.- The Basis of Duality: A Fireside Chat.

The rate of convergence of the binomial tree scheme

Abstract. We study the detailed convergence of the binomial tree scheme. It is known that the scheme is first order. We find the exact constants, and show it is possible to modify Richardson extrapolation to get a method of order three-halves. We see that the delta, used in hedging, converges at the same rate. We analyze this by first embedding the tree scheme in the Black-Scholes diffusion model by means of Skorokhod embedding. We remark that this technique applies to much more general cases.

Geography of the level sets of the Brownian sheet

SummaryWe describe geometric properties of {W>α}, whereW is a standard real-valued Brownian sheet, in the neighborhood of the first hitP of the level set {W>α} along a straight line or smooth monotone curveL. In such a neighborhood we use a decomposition of the formW(s, t)=α−b(s)+B(t)+x(s, t), whereb(s) andB(t) are particular diffusion processes andx(s, t) is comparatively small, to show thatP is not on the boundary of any connected component of {W>α}. Rather, components of this set form clusters nearP. An integral test for thorn-shaped neighborhoods ofL with tip atP that do not meet {W>α} is given. We then analyse the position and size of clusters and individual connected components of {W>α} near such a thorn, giving upper bounds on their height, width and the space between clusters. This provides a local picture of the level set. Our calculations are based on estimates of the length of excursions ofB andb and an accounting of the error termx.

Random currents through nerve membranes

The linear cable equation with uniform Poisson or white noise input current is employed as a model for the voltage across the membrane of a onedimensional nerve cylinder, which may sometimes represent the dendritic tree of a nerve cell. From the Greens function representation of the solutions, the mean, variance and covariance of the voltage are found. At large times, the voltage becomes asymptotically wide-sense stationary and we find the spectral density functions for various cable lengths and boundary conditions. For large frequencies the voltage exhibits “1/f3/2 noise”. Using the Fourier series representation of the voltage we study the moments of the firing times for the diffusion model with numerical techniques, employing a simplified threshold criterion. We also simulate the solution of the stochastic cable equation by two different methods in order to estimate the moments and density of the firing time.

The structure of a Brownian bubble

SummaryWe examine local geometric properties of level sets of the Brownian sheet, and in particular, we identify the asymptotic distribution of the area of sets which correspond to excursions of the sheet high above a given level in the neighborhood of a particular random point. It is equal to the area of certain individual connected components of the random set {(s, t):B(t)>b(s)}, whereB is a standard Brownian motion andb is (essentially) a Bessel process of dimension 3. This limit distribution is studied and, in particular, explicit formulas are given for the probability that a point belongs to a specific connected component, and for the expected area of a component given the height of the excursion ofB(t)-b(s) in this component. These formulas are evaluated numerically and compared with the results from direct simulations ofB andb.

The sharp Markov property of the Brownian sheet and related processes

Keywords: Brownian sheet ; sharp Markov property ; processes with independent planar increments Reference PROB-ARTICLE-1992-001doi:10.1007/BF02392978 Record created on 2008-12-01, modified on 2017-05-12

The intrinsic local time sheet of Brownian motion

SummaryMcGill showed that the intrinsic local time process

Stochastic Integration with Respect to Local Time

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