Tzuu-Shuh Chiang

Diffusion for global optimization in R n

We seek a global minimum of

Propagation of chaos and the McKean-Vlasov equation in duals of nuclear spaces

U:\mathbb{R}^n \to \mathbb{R}

On Eigenvalues and Annealing Rates

. The solution to

Large deviation of small perturbation of some unstable systems

( * )({d / {dt}})X(t) = - \nabla U(X(t))

Asymptotic behavior of eigenvalues and random updating schemes

will find local minima. Using the idea of simulated annealing, we consider th...

Small perturbation of diffusions in inhomogeneous media

An interacting system ofn stochastic differential equations taking values in the dual of a countable Hilbertian nuclear space is considered. The limit (in probability) of the sequence of empirical measures determined by the above systems asn tends to ∞ is identified with the law of the unique solution of the McKean-Vlasov equation. An application of our result to interacting neurons is briefly discussed. The propagation of chaos result obtained in this paper is shown to contain and improve the well-known finite-dimensional results.

The Asymptotic Behavior of Simulated Annealing Processes with Absorption

We evaluate asymptotically the eigenvalues of transition rate matrices Qijei,j=1n with Qije ∼ exp-Uj-Ui+/e for some function U using Ventcels graphic method. As a consequence, we can compare the “nearly optimal” annealing rate in Gidas, B. 1985. Global optimization via the Langevin equation. Proc. 24th IEEE Conf. Decision and Control, Ft. Lauderdale, FL, December. with the true optimal rate in Hajek, B. Cooling schedules for optimal annealing. Preprint.. A necessary and sufficient condition is given for the coincidence of those rates.

A Convexity Approach to Option Pricing with Transaction Costs in Discrete Models

In this paper, we consider the Ventcel-Freidlin theory of the following unstable I-dim stochastic differential equations: and where c≥0 is a non –negative constant and B t is a standatd Brownian motion. System (1) is unstable in the sense that its deterministic system (ϵ =0) may have more than one solutions. We establish the large deviation result of (1) with the rate function where sgn 0 is taken to be -1. System (2) has infinitely many nontrivial solutions and we shall show that the one which can be constructed through the random time change of a Brownian motion obeys a large deviation principle with the rate function .

Price systems for markets with transaction costs and control problems for some finance problems

For a stochastic matrix (QijT)i,j=1M withQijT∼ exp(−U(ij)/T) at the off-diagonal positions, we develop an algorithm to evaluate the asymptotic convergence rate of all eigenvalues ofQijT asT ↓ 0 using Ventcels optimal graphs. As an application we can compare the convergence rates of some random updating schemes used in image processing.

Large deviation of some infinite-dimensional markov processes ∗

Abstract For the system of d -dim stochastic differential equations, d X e (t)=b(X e (t)) d t+eσ(X e (t)) d W(t),t∈[0,1],X e (0)=x 0 ∈R d , where b ( x ) and σ ( x ) are smooth except possibly along the hyperplane {( x 1 ,…, x d ); x 1 =0}, we shall demonstrate that the natural setup of its large deviation principle is to consider the probability e 2 log P(‖X e −ϕ‖ ‖u e −ψ‖ ‖l e −η‖ of the triplet ( X e , u e ,l e ) simultaneously. Here, u e is the occupation time of X e 1 (·) in the positive half line and l e (·) is the local time of X e 1 (·) at 0. The explicit form of the rate function I (·,·,·) is obtained. The usual Wentzell–Friedlin theory concerns only probabilities of the form e 2 log P (‖ X e − ϕ ‖ δ ) and its limit is a consequence of the contraction principle of our result.