We provide general conditions ensuring that the value functions of some nonlinear stopping problems with finite horizon converge to the value functions of the corresponding problems with infinite horizon. Our result can be formulated as result on stability, with respect to time horizon, of nonlinear $f$-expectations. We also study the rate of convergence. Many examples are given to illustrate our results. They include the analysis of time asymptotics of the fair prices of American options in a multidimensional exponential Lévy model.
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.
We consider optimal stopping problems, in which a sequence of independent random variables is drawn from a known continuous density. The objective of such problems is to find a procedure which maximizes the expected reward; this is often known as the "full information" problem. In this analysis, we obtain asymptotic expressions for the expectation and variance of the optimal stopping time as the number of drawn variables becomes large. In the case of distributions with infinite upper bound, the asymptotic behaviour of these statistics depends solely on the algebraic power of the probability di…
Suppose that i.i.d. random variables $X_{1}, X_{2}, \ldots$ are chosen uniformly from $[0,1]$, and let $f: [0,1] \rightarrow [0,1]$ be an increasing bijection. Define $μ_{f}$ to be the expected value of $f(X_{i})$ for each $i$. Define the random variable $K_{f}$ be to be minimal so that $\sum_{i = 1}^{K_{f}} f(X_{i}) > t$ and let $N_{f}(t)$ be the expected value of $K_{f}$. We prove that if $c_{f} = \frac{\int_{0}^{1} \int_{f^{-1}(u)}^{1} (f(x)-u) dx du}{μ_{f}}$, then $N_{f}(t) = \frac{t+c_{f}}{μ_{f}}+o(1)$. This generalizes a result of Ćurgus and Jewett (2007) on the case $f(x) = x$.
In this paper, we present a novel method for computing the asymptotic values of both the optimal threshold, and the probability of success in sequences of optimal stopping problems. This method, based on the resolution of a first-order linear differential equation, makes it possible to systematically obtain these values in many situations. As an example, we address nine variants of the well-known secretary problem, including the classical one, that appear in the literature on the subject, as well as four other unpublished ones.
We study randomly stopped sums via their asymptotic scales. First, finiteness of moments is considered. To generalise this study, asymptotic scales applicable to the class of all heavy-tailed random variables are used. The stopping is assumed to be independent of the underlying process, which is a random walk. The main result enables one to identify whether the asymptotic behaviour of a stopped sum is dominated by the increment, or the stopping variable. As a consequence of this result, new sufficient conditions for the moment determinacy of compounded sums are obtained.
Consider the problem of maximizing the probability of stopping with one of the two highest values in a Bernoulli random walk with arbitrary parameter $p$ and finite time horizon $n$. Allaart \cite{Allaart} proved that the optimal strategy is determined by an interesting sequence of constants $\{p_{n}\}$. He conjectured the asymptotic behavior to be $1/2$. In this work the best lower bound for this sequence is found and more of its properties are proven towards solving the conjecture.
In a classical optimal stopping problem in continuous time, the agent can choose any stopping time without constraint. Dupuis and Wang (Optimal stopping with random intervention times, Advances in Applied Probability, 34, 141--157, 2002) introduced a constraint on the class of admissible stopping times which was that they had to take values in the set of event times of an exogenous, time-homogeneous Poisson process. This can be thought of as a model of finite liquidity. In this article we extend the analysis of Dupuis and Wang to allow the agent to choose the rate of the Poisson process. Choos…
This paper analyses two-player nonzero-sum games of optimal stopping on a class of linear regular diffusions with not non-singular boundary behaviour (in the sense of Itô and McKean (1974), p.\ 108). We provide sufficient conditions under which Nash equilibria are realised by each player stopping the diffusion at one of the two boundary points of an interval. The boundaries of this interval solve a system of algebraic equations. We also provide conditions sufficient for the uniqueness of the equilibrium in this class.
We introduce a class of continuous planar processes, called "semimartingales on rays", and develop for them a change-of-variable formula involving quite general classes of test functions. Special cases of such planar processes are diffusions which choose, once they reach the origin, the rays for their subsequent voyage according to a fixed probability measure in the manner of Walsh (1978). We develop existence and uniqueness results up to an explosion time for these "Walsh diffusions", study their asymptotic behavior, and develop tests for explosions in finite time. We use these results to fin…
Let $X=(X_t)_{t\geq 0}$ be a known process and $T$ an unknown random time independent of $X$. Our goal is to derive the distribution of $T$ based on an iid sample of $X_T$. Belomestny and Schoenmakers (2015) propose a solution based the Mellin transform in case where $X$ is a Brownian motion. Applying their technique we construct a non-parametric estimator for the density of $T$ for a self-similar one-dimensional process $X$. We calculate the minimax convergence rate of our estimator in some examples with a particular focus on Bessel processes where we also show asymptotic normality.
We consider general transformations of random walks on groups determined by Markov stopping times and prove that the asymptotic entropy (resp., rate of escape) of the transformed random walks is equal to the asymptotic entropy (resp., rate of escape) of the original random walk multiplied by the expectation of the corresponding stopping time. This is an analogue of the well-known Abramov's formula from ergodic theory, its particular cases were established earlier by Kaimanovich [1983] and Hartman, Lima, Tamuz [2014].
A random walk on a $N$-dimensional hypercube is a discrete time stochastic process whose state space is the set $\{-1,+1\}^{N}$, which has uniform probability of reaching any neighbour state, and probability zero of reaching a non-neighbour state, in one step. This random walk is often studied as a process associated with the Ehrenfest Urn Model. This paper aims to present results about the time that such random walk takes to self-intersect and to return to a set of states. We also present results about the time that the random walk on a hypercube takes to visit a given set and a random set of…
We develop a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. This immediately leads to a formula for the distribution of the first exit times of diffusions, which is recently characterised by Karatzas and Ruf \cite{KR} as the minimal solution of an appropriate Cauchy problem under more stringent conditions. A particular limit of these transformations also turn out to be instrumental in characterising the stochastic solutions of Cauchy problems defined by the generators of strict loca…
An integral representation result for strictly positive subharmonic functions of a one-dimensional regular diffusion is established. More precisely, any such function can be written as a linear combination of an increasing and a decreasing subharmonic function that solve an integral equation \[ g(x)=a + \int v(x,y)μ_A(dy) + κs(x), \] where $a>0$, $κ\in \mathbb{R}$, $s$ is a scale function of the diffusion, $μ_A$ is a Radon measure, and $v$ is a kernel that is explicitly determined by the scale function. This integral equation in turn allows one construct a pair $(g,A)$ such that $g$ is a subha…
We develop a method to solve, theoretically and numerically, general optimal stopping problems. Our general setting allows for multiple exercise rights, i.e., optimal multiple stopping, for a robust evaluation that accounts for model uncertainty, and for general reward processes driven by multi-dimensional jump-diffusions. Our approach relies on first establishing robust martingale dual representation results for the multiple stopping problem that satisfy appealing pathwise optimality (i.e., almost sure) properties. Next, we exploit these theoretical results to develop upper and lower bounds t…
In this note, we study the asymptotic behaviour near extinction of (sub-) critical continuous state branching processes. In particular, we establish an analogue of Khintchin's law of the iterated logarithm near extinction time for a continuous state branching process whose branching mechanism satisfies a given condition and its reflected process at its infimum.
We investigate the asymptotic behavior of the a large class of reversible chemical reaction-diffusion equations with the same diffusion. In particular we prove the optimal rate in two cases : when there is no diffusion and in the classical "two-by-two" case.
We consider a Markov chain on $R^+$ with asymptotically zero drift and finite second moments of jumps which is positive recurrent. A power-like asymptotic behaviour of the invariant tail distribution is proven; such a heavy-tailed invariant measure happens even if the jumps of the chain are bounded. Our analysis is based on test functions technique and on construction of a harmonic function.
We study the asymptotic behaviour of a class of small-noise diffusions driven by fractional Brownian motion, with random starting points. Different scalings allow for different asymptotic properties of the process (small-time and tail behaviours in particular). In order to do so, we extend some results on sample path large deviations for such diffusions. As an application, we show how these results characterise the small-time and tail estimates of the implied volatility for rough volatility models, recently proposed in mathematical finance.
We investigate the distribution properties of the fractional Lévy motion. We consider separately the cases $0<H<1/2$ (short memory) and $1/2<H<1$ (long memory), where $H$ is the Hurst parameter, and present the asymptotic behaviour of the distribution density of the process. Some examples are provided, in which it is shown that the behaviour of the density in the cases $0<H<1/2$ and $1/2<H<1$ is completely different.
The present paper aims at describing in details the asymptotic composition of a class of d-colour Pólya urn: namely balanced, tenable and irreducible urns. We decompose the composition vector of such urns according to the Jordan decomposition of their replacement matrix. The projections of the composition vector onto the so-called small Jordan spaces are known to be asymptotically gaussian, but the asymptotic behaviour of the projections onto the large Jordan spaces are not known in full details up to now and are discribed by a limiting random variables called W, depending on the parameters of…
A random graph evolution based on the interactions of N vertices is studied. During the evolution both the preferential attachment method and the uniform choice of vertices are allowed. The weight of a vertex means the number of its interactions. The asymptotic behaviour of the weight and the degree of a fixed vertex, moreover the limit of the maximal weight and the maximal degree are described. The proofs are based on martingale methods.
Let $X_n,...,X_1$ be i.i.d. random variables with distribution function $F$. A statistician, knowing $F$, observes the $X$ values sequentially and is given two chances to choose $X$'s using stopping rules. The statistician's goal is to stop at a value of $X$ as small as possible. Let $V_n^2$ equal the expectation of the smaller of the two values chosen by the statistician when proceeding optimally. We obtain the asymptotic behavior of the sequence $V_n^2$ for a large class of $F$'s belonging to the domain of attraction (for the minimum) ${\cal D}(G^α)$, where $G^α(x)=[1-\exp(-x^α)]{\bf I}(x \g…
This paper considers the limiting distribution of $π_{λ,θ}$, the stationary distribution of the infinitely-many-alleles diffusion with symmetric overdominance \cite{MR1626158}. In \cite{MR2519357} the large deviation principle for $π_{λ,θ}$ indicates that there are countably many phase transitions for the limiting distribution of $π_{λ,θ}$, and the critical points are $λ=k(k+1), k\geq1$. The asymptotic behaviours at those critical points, however, are unclear. This article provides a definite description of the critical cases.
We consider the optimal stopping problem $v^{(\eps)}:=\sup_{τ\in\mathcal{T}_{0,T}}\mathbb{E}B_{(τ-\eps)^+}$ posed by Shiryaev at the International Conference on Advanced Stochastic Optimization Problems organized by the Steklov Institute of Mathematics in September 2012. Here $T>0$ is a fixed time horizon, $(B_t)_{0\leq t\leq T}$ is the Brownian motion, $\eps\in[0,T]$ is a constant, and $\mathcal{T}_{\eps,T}$ is the set of stopping times taking values in $[\eps,T]$. The solution of this problem is characterized by a path dependent reflected backward stochastic differential equations, from whic…
We give an analytic description for the infinitesimal generator constructed by Applebaum-Estrade for Lévy flights on a broad class of closed Riemannian manifolds including all negatively-curved manifolds, the flat torus and the sphere. Various properties of the associated semigroup and the asymptotics of the expected stopping time for Lévy flight based random searches for small targets, also known as the narrow capture problem, are then obtained using our newfound understanding of the infinitesimal generator. Our study also relates to the Lévy flight foraging hypothesis in the field of biology…
We revisit the product convolution semigroup of probability densities e_c(t),c>0 on the positive half-line with moments (n!)^c and determine the asymptotic behaviour of e_c(t) for large and small t>0. This shows that (n!)^c is indeterminate as Stieltjes moment sequence if and only if c>2
We study the asymptotic behaviour of a properly normalized time changed Wiener processes. The time change reflects the fact that we consider the Laplace operator (which generates a Wiener process) multiplied by a possibly degenerate state-space dependent intensity $λ(x)$. Applying a functional limit theorem for the superposition of stochastic processes, we prove functional limit theorems for the normalized time changed Wiener process. The normalization depends on the asymptotic behaviour of the intensity function $λ$. One of the possible limits is a skew Brownian motion.
We prove the asymptotic formulas for the transition densities of isotropic unimodal convolution semigroups of probability measures on $\mathbb{R} ^d$ under the assumption that its Lévy--Khintchine exponent varies slowly. We also derive some new estimates of the transition densities and Green functions.
This paper demonstrates a lower and upper solution method to investigate the asymptotic behaviour of the conservative reaction-diffusion systems associated with Markovian process algebra models. In particular, we have proved the uniform convergence of the solution to its constant equilibrium for a case study as time tends to infinity, together with experimental results illustrations.
University of Michigan（密歇根大学）；Florida State University（佛罗里达州立大学）
For the problem of prediction with expert advice in the adversarial setting with geometric stopping, we compute the exact leading order expansion for the long time behavior of the value function. Then, we use this expansion to prove that as conjectured in Gravin et al. [12], the comb strategies are indeed asymptotically optimal for the adversary in the case of 4 experts.
