113 research outputs found
On an effective solution of the optimal stopping problem for random walks
We find a solution of the optimal stopping problem for the case when a reward function is an integer power function of a random walk on an infinite time interval. It is shown that an optimal stopping time is a first crossing time through a level defined as the largest root of Appell's polynomial associated with the maximum of the random walk. It is also shown that a value function of the optimal stopping problem on the finite interval {0, 1, . . . , T} converges with an exponential rate as T ā ā to the limit under the assumption that jumps of the random walk are exponentially bounded
An Exact Formula for the Average Run Length to False Alarm of the Generalized Shiryaev-Roberts Procedure for Change-Point Detection under Exponential Observations
We derive analytically an exact closed-form formula for the standard minimax
Average Run Length (ARL) to false alarm delivered by the Generalized
Shiryaev-Roberts (GSR) change-point detection procedure devised to detect a
shift in the baseline mean of a sequence of independent exponentially
distributed observations. Specifically, the formula is found through direct
solution of the respective integral (renewal) equation, and is a general result
in that the GSR procedure's headstart is not restricted to a bounded range, nor
is there a "ceiling" value for the detection threshold. Apart from the
theoretical significance (in change-point detection, exact closed-form
performance formulae are typically either difficult or impossible to get,
especially for the GSR procedure), the obtained formula is also useful to a
practitioner: in cases of practical interest, the formula is a function linear
in both the detection threshold and the headstart, and, therefore, the ARL to
false alarm of the GSR procedure can be easily computed.Comment: 9 pages; Accepted for publication in Proceedings of the 12-th
German-Polish Workshop on Stochastic Models, Statistics and Their
Application
Bayesian switching multiple disorder problems
The switching multiple disorder problem seeks to determine an ordered infinite sequence of times of alarms which are as close as possible to the unknown times of disorders, or change-points, at which the observable process changes its probability characteristics. We study a Bayesian formulation of this problem for an observable Brownian motion with switching constant drift rates. The method of proof is based on the reduction of the initial problem to an associated optimal switching problem for a three-dimensional diffusion posterior probability process and the analysis of the equivalent coupled parabolic-type free-boundary problem. We derive analytic-form estimates for the Bayesian risk function and the optimal switching boundaries for the components of the posterior probability process
Lower Bounds for Bruss' Odds Problem with Multiple Stoppings
We give asymptotic lower bounds of the value for Bruss' optimal stopping
problem with multiple stopping chances. It interestingly consists of the
asymptotic threshold values in the optimal multiple stopping strategy. Another
interesting implication of the result is that the asymptotic value for each
secretary problem with multiple stoppings is in fact a typical lower bound in a
much more general class of multiple stopping problems as modifications of odds
problem.Comment: 41 pages, 4 figure
Rough paths in idealized financial markets
This paper considers possible price paths of a financial security in an
idealized market. Its main result is that the variation index of typical price
paths is at most 2, in this sense, typical price paths are not rougher than
typical paths of Brownian motion. We do not make any stochastic assumptions and
only assume that the price path is positive and right-continuous. The
qualification "typical" means that there is a trading strategy (constructed
explicitly in the proof) that risks only one monetary unit but brings infinite
capital when the variation index of the realized price path exceeds 2. The
paper also reviews some known results for continuous price paths and lists
several open problems.Comment: 21 pages, this version adds (in Appendix C) a reference to new
results in the foundations of game-theoretic probability based on Hardin and
Taylor's work on hat puzzle
The filtering equations revisited
The problem of nonlinear filtering has engendered a surprising number of
mathematical techniques for its treatment. A notable example is the
change-of--probability-measure method originally introduced by Kallianpur and
Striebel to derive the filtering equations and the Bayes-like formula that
bears their names. More recent work, however, has generally preferred other
methods. In this paper, we reconsider the change-of-measure approach to the
derivation of the filtering equations and show that many of the technical
conditions present in previous work can be relaxed. The filtering equations are
established for general Markov signal processes that can be described by a
martingale-problem formulation. Two specific applications are treated
On cubic difference equations with variable coefficients and fading stochastic perturbations
We consider the stochastically perturbed cubic difference equation with variable coefficients xn+1=xn(1āhnx2n)+Ļn+1Ī¾n+1,nāN,x0āR. Here (Ī¾n)nāN is a sequence of independent random variables, and (Ļn)nāN and (hn)nāN are sequences of nonnegative real numbers. We can stop the sequence (hn)nāN after some random time N so it becomes a constant sequence, where the common value is an FN -measurable random variable. We derive conditions on the sequences (hn)nāN , (Ļn)nāN and (Ī¾n)nāN , which guarantee that limnāāxn exists almost surely (a.s.), and that the limit is equal to zero a.s. for any initial value x0āR
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