Probability Abstracts 18
This document contains abstracts 291-302.
They have been mailed on December 27, 1993.
Click here to see the
list of all abstract titles.
291. ON SIMULATING A MARKOV CHAIN STATIONARY DISTRIBUTION WHEN TRANSITION
PROBABILITIES ARE UNKNOWN
David Aldous
Consider an $n$-state Markov chain whose steps can be simulated but whose
transition probabilities are unknown. Asmussen-Glynn-Thorisson proved the
remarkable fact that there exists an procedure which outputs a random
state whose distribution is exactly the (unknown) stationary distribution,
regardless of the transition probabilities. But it seems difficult to
bound the running time of their procedure. We give a different procedure
which guarantees the output distribution is within $\epsilon$ of the stationary
distribution (total variation distance) and uses $O(\tau \epsilon^{-2})$ steps,
where $\tau$ is an average of the mean hitting times for the underlying chain.
aldous@stat.berkeley.edu (LaTeX file available)
292. THE GAP BETWEEN THE PAST SUPREMUM AND THE FUTURE INFIMUM OF A
TRANSIENT BESSEL PROCESS.
Davar Khoshnevisan
This paper consists of some remarks on an earlier
paper of the author with T.M. Lewis and W.V. Li
regarding some almost sure path properties of the
future infima of transient Bessel processes. In
particular, we study the asymptotic discrepancy
between sup_{s leq t}X(s) and inf_{s geq t} X(s).
davar@math.utah.edu
293. LIQUIDITY PREMIUM FOR CAPITAL ASSET PRICING WITH TRANSACTION COSTS
Steve Shreve
An agent solves the infinite-horizon consumption-investment problem when
the investment possibilities are a constant-interest-rate, risk-free
asset and a stock, modelled as a geometric Brownian motion. There are
proportional transaction costs associated with moving wealth between
these two assets. The direct utility for consumption is of the form
$\frac{1}{p} c^p$ for some $p \in (0,1)$. The sensitivity of the
indirect utility function (or value function) to small transaction costs
is found to be of the order of the transaction cost to the $2/3$ power.
This paper documents a talk presented at the Workshop on Mathematical
Finance, Institute for Mathematics and its Applications, Minneapolis,
June 14-18, 1993.
shreve+@cmu.edu
294. ON THE TIME A MARKOV CHAIN SPENDS IN A LUMPED STATE
Tommy Norberg
We want to know whether the time a Markov chain spends in a subset of the
state space is geometric or not. This note investigates the case of a
denumerable state space, the subset of interest having finite cardinality.
tommy@math.chalmers.se
295. THE LIE ALGEBRA OF THE ITERATED GRADIENTS OF A MARKOV
GENERATOR - VARIANCE AND ENTROPY EXPANSIONS
M. Ledoux
We define a natural Lie algebra structure associated to a Markov
generator which extends the ``carr\'e du champ" operator and its
iterated
gradients. While iterated gradients may be used in expansions of the
variance of a function of the domain, this structure is needed in formal
expansions of the entropy of a function.
ledoux@cict.fr (plain tex file available - in French)
296. OPTIMAL DRIFT ON [0,1]
Susan Lee
Consider one dimensional diffusions on the interval [0,1] of
the form dX(t) = dB(t) + b(X(t))dt, with 0 a reflecting boundary, b(x)
positive for all x, and the integral of b(x), as x goes from 0 to 1,
equal to 1. In this paper, we show that there is a unique drift which
minimizes the expected time for X(t) to hit 1, starting from X(0) = 0.
In the deterministic case dX(t) = b(X(t))dt, the optimal drift
is the function which is identically equal to 1. By contrast, if
dX(t)=dB(t)+b(X(t))dt, then the optimal drift is the step function
which is 2 on the interval [1/4, 3/4], and is 0 otherwise. We also
solve this problem for arbitrary starting point X(0)=x_0 and find that
the unique optimal drift depends on the starting point, x_0, in a
curious manner.
This work is a part of the author's Ph.D. dissertation, which
will be completed at Cornell University in May 1993.
slee@math.cornell.edu (Plain TEX file available.)
297. THE DISTRIBUTION OF BUBBLES OF BROWNIAN SHEET
Davar Khoshnevisan
Let W be real-valued, two parameter Brownian sheet.
Define N(t;h) to be the number of bubbles of W in [0,t]^2
whose maximum height exceeds h. Evidently, N(t;h) goes to
infinity either as t -> infinity or h -> 0. It is the goal of this
paper to provide fairly accurate estimates on N(t;h), both
as t -> infinity and as h -> 0. Loosely speaking, we prove that
there are t^3 and h^{-3} such bubbles, respectively.
davar@math.utah.edu
298. MAXIMUM ENTROPY PRINCIPLES FOR DISORDERED SPINS
Timo Seppalainen
We transform nonstationary independent random fields with exponential
Radon-Nikodym factors and study the asymptotics of the transformed
processes. The modifying Radon-Nikodym factors are functions of the
empirical distribution. Thus this is a way of introducing dependence
among the previously independent variables. The transformed
processes are tight in an appropriately weak sense, they satisfy
large deviation principles, and their limit points are mixtures of measures
characterized by variational principles called `maximum entropy principles'.
As applications we deduce conditional limit theorems for such random fields
and study a Curie-Weiss-type mean-field model of a quenched mixed
magnetic crystal. This model has quenched site disorder and
frustration but non-random coupling constants. We characterize the
limiting Gibbs measures and study the specific magnetization.
The model exhibits a continuous phase transition with nonzero spontaneous
magnetization at temperatures below critical, and with critical exponents
equal to those of the classical Curie-Weiss theory.
We find no ``spin glass'' phase where an overlap order parameter would be
strictly positive without the accompanying spontaneous magnetization.
seppalai@ima.umn.edu (Amstex file and hardcopy available)
299. PERMANENT SETS OF MEASURES CHARGING NO
EXCEPTIONAL SETS AND THE FEYNMAN-KAC FORMULA
Kazuhiro Kuwae
We consider the totality of Borel measures on locally compact
separable metric space X which charge no exceptional sets of symmetric Markov
process M on X. We constract the permanent set of such measures analytically
in terms of Dirichlet space. By using this set, we establish the one to one
correspondence between the equivalence class of such measures and the
the equivalence class of "extended positive continuous
additive functional"(PCAFe in abbreviation),
extending the results of J.Baxter-G.Dal.Maso-U.Mosco and K.T.Sturm,
who treat the case that M is the d-dimensional Brownian motion.
kuwae@gauss.ma.is.saga-u.ac.jp
300. CONDITIONED BROWNIAN MOTION IN PLANAR DOMAINS
Richard F. Bass and Krzysztof Burdzy
We give an upper bound for the Green functions of conditioned
Brownian motion in planar domains. A corollary is the conditional
gauge theorem in bounded planar domains.
bass@math.washington.edu
301. RANDOM TRAVELLING WAVES FOR THE KPP EQUATION WITH NOISE
C. Mueller and R. Sowers
Consider the stochastic partial differential equation
u_t&=u_{xx}+u-u^2+\epsilon\sqrt{u(1-u)}\dot W
u(0,x)&=u_0(x),
where $x\in\bold R$ and $ W=\dot W(t,x)$ is 2-parameter white noise.
Assume that $u_0$ is a continuous function taking values in $[0,1]$
such that for some constant $a>0$, we have
u_0(x)=1 for x<-a.
u_0(x)=0 for x>a.
Let the wavefront $b(t)=\sup\{x\in\bold R:u(t,x)>0\}$. We show that for
$\epsilon$ small enough, and with probability 1, $\lim_{t\to\infty}b(t)/t$
exists and lies in $(0,\infty)$. Also, the law of $v(t,x)\equiv u(t,b(t)+x)$
tends toward a stationary limit as $t\to\infty$. We also analyze the
length of the region $[a(t),b(t)]$, which is the smallest closed
interval containing the points $x$ at which $0<u(t,x)<1$. We show
that the length of this region tends toward a stationary distribution.
Thus, the wavefront does not degenerate.
Our methods depend on a careful analysis of the qualitative
behavior of the wavefront. When $[a(t),b(t)]$ is large, we show
that "supercritical blobs" tend to form which will fill in this
region and make it smaller. A "supercritical blob" is a growing
area on which $u(t,x)=1$. Some of our arguments depend on the
construction of a historical process. This idea comes from
superprocesses. Finally, a coupling argument shows that the limiting
speed is nonrandom and the stationary distribution of the shape of
the wavefront is unique.
cmlr@troi.cc.rochester.edu (amslatex file available)
302. CRITICAL DIMENSIONS FOR THE EXISTENCE OF SELF-INTERSECTION LOCAL TIMES
OF THE BROWNIAN SHEET IN $R^d$
Peter Imkeller and Ferenc Weisz
Fix two rectangles $A, B$ in $[0,1]^2$. Then the size of the random set of
double points of the Brownian sheet $(W_t)_{t \in [0,1]^2}$ in $R^d$, i.e.
the set of pairs $(s,t)$, where $s\in A, t \in B$, and $W_s = W_t$, can be
measured as usual by a self-intersection local time. If $A = B$, we show that
the critical dimension below which self-intersection local time exists as a
function, is given by $d=4$. If $A \cap B$ consists of an axial parallel line,
it is 6, if it consists of a point or is empty, 8. In all cases, we derive the
rate of explosion of canonical approximations of self-intersection local time
for dimensions above the critical one, and determine its smoothness in terms
of the canonical Dirichlet structure on Wiener space.
imkeller@grenet.fr
