Sequence Of Stopping Times Research Articles

A variation of the cube model for best–worst choice

In this paper, we propose a dynamical model for the best–worst choice task. The proposed model is a modification of the multi-episode Cube model proposed and studied in so-called (Mallahi-Karai and Diederich, 2019, 2021). This model postulates that best–worst choice (or more generally, ranking) task is the outcome of sequential choices made in a number of episodes. The underlying model is a multivariate Wiener process with drift issued from a point in the unit cube, where episodes are defined in terms of a sequence of stopping times. This model can also be extended to an attention-switching framework in a standard way.

Random attractors for rough stochastic partial differential equations

This paper is devoted to the existence of random attractors for rough partial differential equations driven by nonlinear multiplicative Hölder rough paths with exponents in (1/3,1/2]. Our approach relies upon rough paths theory and stopping times analysis in a suitable scale of interpolation spaces. The core step is to derive the adequate algebraic and analytical properties of a sequence of stopping times, which allows us to establish the required compact tempered absorbing set. The existence of a pullback attractor for the generated random dynamical system is straightforward. An illustrative example is presented by reaction-diffusion equations subjected to fractional Brownian rough paths.

Perturbed Skew Diffusion Processes

This work investigates whether there uniquely exists a solution to the perturbed skew diffusion process. We construct the solution by iteration and divide the whole time interval into parts on which we disperse the perturbed skew diffusion process into two tractable portions, one for perturbed diffusion process, the other for skew diffusion process. After this disposition, we only focus on the process in each time interval. Noticing the continuity on every time interval boundaries generalized by a sequence of stopping times, we acquire the main result of this paper as well as a time change for the perturbed skew process.

Two-time-scale stochastic differential delay equations driven by multiplicative fractional Brownian noise: Averaging principle

The main goal of this article is to study an averaging principle for a class of two-time-scale stochastic differential delay equations in which the slow-varying process includes a multiplicative fractional Brownian noise with Hurst parameter H∈(12,1) and the fast-varying process is a rapidly-changing diffusion. We would like to emphasize that the approach proposed in this paper is based on the fact that a stochastic integral with respect to fractional Brownian motion with Hurst parameter in (12,1) can be defined as a generalized Stieltjes integral. In particular, to prove a limit theorem for the averaging principle, we will introduce a sequence of stopping times to control the size of multiplicative fractional Brownian noise. Then, inspired by the Khasminskii's approach, an averaging principle is developed in the sense of convergence in the p-th moment uniformly in time.

Pricing Zero-Coupon CAT Bonds Using the Enlargement of Filtration Theory: A General Framework

The main goal of this paper is to use the enlargement of filtration framework for pricing zero-coupon CAT bonds. For this purpose, we develop two models where the trigger event time is perfectly covered by an increasing sequence of stopping times with respect to a reference filtration. Hence, depending on the nature of these stopping times the trigger event time can be either accessible or totally inaccessible. When some of these stopping times are not predictable, the trigger event time is totally inaccessible, and very nice mathematical computations can be derived. When the stopping times are predictable, the trigger event time is accessible, and this case would be a meaningful choice for Model 1 from a practical point of view since features like seasonality are already captured by some quantities such as the stochastic intensity of the Poisson process. We compute the main tools for pricing the zero-coupon CAT bond and show that our constructions are more general than some existing models in the literature. We obtain some closed-form prices of zero-coupon CAT bonds in Model 2 so we give a numerical illustrative example for this latter.

Open Markov Type Population Models: From Discrete to Continuous Time

We address the problem of finding a natural continuous time Markov type process—in open populations—that best captures the information provided by an open Markov chain in discrete time which is usually the sole possible observation from data. Given the open discrete time Markov chain, we single out two main approaches: In the first one, we consider a calibration procedure of a continuous time Markov process using a transition matrix of a discrete time Markov chain and we show that, when the discrete time transition matrix is embeddable in a continuous time one, the calibration problem has optimal solutions. In the second approach, we consider semi-Markov processes—and open Markov schemes—and we propose a direct extension from the discrete time theory to the continuous time one by using a known structure representation result for semi-Markov processes that decomposes the process as a sum of terms given by the products of the random variables of a discrete time Markov chain by time functions built from an adequate increasing sequence of stopping times.

Wasserstein convergence rates for random bit approximations of continuous Markov processes

We determine the convergence speed of a numerical scheme for approximating one-dimensional continuous strong Markov processes. The scheme is based on the construction of certain Markov chains whose laws can be embedded into the process with a sequence of stopping times. Under a mild condition on the process' speed measure we prove that the approximating Markov chains converge at fixed times at the rate of 1/4 with respect to every p-th Wasserstein distance. For the convergence of paths, we prove any rate strictly smaller than 1/4. Our results apply, in particular, to processes with irregular behavior such as solutions of SDEs with irregular coefficients and processes with sticky points.

An Analytical Approach to Portfolios of Real Options

A renewable energy asset manager can expand the power generation capacity of a particular site by an endogenous amount but may also want to shut down the location to save on fixed operating costs if the market prospects deteriorate. The site manager does not exercise these real options in a set sequence: the decision to exit is driven by the value of a capacity expansion option, while the decision to expand depends on the value of an exit option. From our analysis, we obtain three main economic insights. First, the presence of fixed costs renders the NPV threshold nonmonotonic in the firm's existing capacity. Second, the manager further delays shutting down production to avoid killing the expansion option. Finally, it is optimal to delay the timing \emph{and} the scale of the investment in capacity compared to the case without an exit option. Mathematically, we solve a nonstandard optimal stopping problem as the sequence of stopping times is at the discretion of the decision-maker and the objective of the capacity choice problem is not concave, which leads to a gain function that is not differentiable. We find a continuously differentiable solution to this optimal stopping problem and specify the corresponding continuation and stopping sets analytically.

A functional limit theorem for irregular SDEs

Soit $X_{1},X_{2},\ldots$ une suite de variables aleatoires independantes avec esperance $E(X_{i})=0$, et $Y^{N}_{k+1}=Y^{N}_{k}+a_{N}(Y^{N}_{k})X_{k+1}$ une marche aleatoire renormalisee avec une fonction $a_{N}:\mathbb{R}\to\mathbb{R}_{+}$. On montre, sous certaines conditions legeres sur la loi de $X_{i}$, que l’on peut choisir le facteur $a_{N}$ d’une facon que $(Y^{N}_{\lfloor Nt\rfloor})_{t\in\mathbb{R}_{+}}$ converge en loi, quand $N$ tend vers l’infini, vers une diffusion $(M_{t})_{t\in\mathbb{R}_{+}}$ etant la solution d’une equation differentielle stochastique avec des coefficients irreguliers. A cet effet, nous plongeons la marche aleatoire renormalisee dans la diffusion $M$ par une suite de temps d’arret ayant un pas de temps avec esperance $1/N$.

On one property of martingales with conditionally Gaussian increments and its application in the theory of nonasymptotic inference

A transformation of a discrete-time martingale with conditionally Gaussian increments into a sequence of i.i.d. standard Gaussian random variables is proposed as based on a sequence of stopping times constructed using the quadratic variation. It is shown that sequential estimators for the parameters in AR(1) and generalized first-order autoregressive models have a nonasymptotic normal distribution.

A versatile scheme for predicting renewal times

There are two kinds of universal schemes for estimating residual waiting times, those where the error tends to zero almost surely and those where the error tends to zero in some integral norm. Usually these schemes are different because different methods are used to prove their consistency. In this note we will give a single scheme where the average error is eventually small for all time instants, while the error itself tends to zero along a sequence of stopping times of density one.

Optimal Trend Following Trading Rules

This paper is concerned with the optimality of a trend following trading rule. The underlying market is modeled like a bull-bear switching market in which the drift of the stock price switches between two states: the uptrend (bull market) and the down trend (bear market). We consider the case when the market mode is not directly observable and model the switching process as a hidden Markov chain. This is a continuation of our earlier study reported in Dai et al. [Dai M, Zhang Q, Zhu Q (2010) Trend following trading under a regime-switching model. SIAM J. Fin. Math. 1:780–810] where a trend following rule is obtained in terms of a sequence of stopping times. Nevertheless, a severe restriction imposed in Dai et al. [Dai M, Zhang Q, Zhu Q (2010) trend following trading under a regime-switching model. SIAM J. Fin. Math. 1:780–810] is that only a single share can be traded over time. As a result, the corresponding wealth process is not self-financing. In this paper, we relax this restriction. Our objective is to maximize the expected log-utility of the terminal wealth. We show, via a thorough theoretical analysis, that the optimal trading strategy is trend following. Numerical simulations and backtesting, in support of our theoretical findings, are also reported.

Numerical approximation of irregular SDEs via Skorokhod embeddings

We provide a new algorithm for approximating the law of a one-dimensional diffusion M solving a stochastic differential equation with possibly irregular coefficients. The algorithm is based on the construction of Markov chains whose laws can be embedded into the diffusion M with a sequence of stopping times. The algorithm does not require any regularity or growth assumption; in particular it applies to SDEs with coefficients that are nowhere continuous and that grow superlinearly. We show that if the diffusion coefficient is bounded and bounded away from 0, then our algorithm has a weak convergence rate of order 1/4. Finally, we illustrate the algorithm's performance with several examples.

Brownian Motion in Minkowski Space

We construct a model of Brownian motion in Minkowski space. There are two aspects of the problem. The first is to define a sequence of stopping times associated with the Brownian “kicks” or impulses. The second is to define the dynamics of the particle along geodesics in between the Brownian kicks. When these two aspects are taken together, the Central Limit Theorem (CLT) leads to temperature dependent four dimensional distributions defined on Minkowski space, for distances and 4-velocities. In particular, our processes are characterized by two independent time variables defined with respect to the laboratory frame: a discrete one corresponding to the stopping times when the impulses take place and a continuous one corresponding to the geodesic motion in-between impulses. The subsequent distributions are solutions of a (covariant) pseudo-diffusion equation which involves derivatives with respect to both time variables, rather than solutions of the telegraph equation which has a single time variable. This approach simplifies some of the known problems in this context.

A new kind of augmentation of filtrations suitable for a change of probability measure by a strict local martingale

In this note we introduce a new kind of augmentation of filtrations along a sequence of stopping times. This augmentation is suitable for the construction of new probability measures associated to a positive strict local martingale as done in Kardaras et al. (2015), while it is on the other hand rich enough to make classical results from stochastic analysis hold true on some stochastic interval of interest.

On undiscounted non-linear optimal multiple stopping

We study and formulate an undiscounted non-linear optimal multiple stopping problem, with an application to the valuation of the perpetual American-style discretely monitored Asian options. When the reward process is continuous, we follow a vector-valued approach. Under the right-continuity of this process, the problem can be reduced to a sequence of ordinary optimal stopping problems. In the Markovian case, we characterise the value function of the problem in terms of excessive functions. Finally, in case of a regular diffusion, we provide an optimal sequence of stopping times. The results are illustrated by some examples, where the value function of the problem is given explicitly.

Adaptive Sampling for Linear State Estimation

When a sensor has continuous measurements but sends occasional messages over a data network to a supervisor which estimates the state, the available packet rate fixes the achievable quality of state estimation. When such rate limits turn stringent, the sensor's messaging policy should be designed anew. What are good causal messaging policies? What should message packets contain? What is the lowest possible distortion in a causal estimate at the supervisor? Is Delta sampling better than periodic sampling? We answer these questions for a Markov state process under an idealized model of the network and the assumption of perfect state measurements at the sensor. If the state is a scalar, or a vector of low dimension, then we can ignore sample quantization. If in addition we can ignore jitter in the transmission delays over the network, then our search for efficient messaging policies simplifies. First, each message packet should contain the value of the state at that time. Thus a bound on the number of data packets becomes a bound on the number of state samples. Second, the remaining choice in messaging is entirely about the times when samples are taken. For a scalar, linear diffusion process, we study the problem of choosing the causal sampling times that will give the lowest aggregate squared error distortion. We stick to finite horizons and impose a hard upper bound N on the number of allowed samples. We cast the design as a problem of choosing an optimal sequence of stopping times. We reduce this to a nested sequence of problems, each asking for a single optimal stopping time. Under an unproven but natural assumption about the least-square estimate at the supervisor, each of these single stopping problems are of standard form. The optimal stopping times are random times when the estimation error exceeds designed envelopes. For the case where the state is a Brownian motion, we give analytically: the shape of the optimal sampling envelopes, the shape of the envelopes under optimal Delta sampling, and their performances. Surprisingly, we find that Delta sampling performs badly. Hence, when the rate constraint is a hard limit on the number of samples over a finite horizon, we should not use Delta sampling.

Optimal Trend Following Trading Rules

We develop an optimal trend following trading rule in a bull-bear switching market, where the drift of the stock price switches between two parameters corresponding to an uptrend (bull market) and a downtrend (bear market) according to an unobservable Markov chain. We consider a finite horizon investment problem and aim to maximize the expected return of the terminal wealth. We start by restricting to allowing flat and long positions only and describe the trading decisions using a sequence of stopping times indicating the time of entering and exiting long positions. Assuming trading all available funds, we show that the optimal trading strategy is a trend following system characterized by the conditional probability in the uptrend crossing two threshold curves. The thresholds can be obtained by solving the associated HJB equations. In addition, we examine trading strategies with short selling in terms of an approximation. Simulations and empirical experiments are conducted and reported.

Intermittent Estimation for Gaussian Processes

Let {Xn}n=0 ? be a stationary real-valued Gaussian time series. We estimate the conditional expectation E(Xn+1|X0, ...,Xn) from a growing number of observations X0,..., Xn in a pointwise consistent way along a sequence of stopping times.

Trend Following Trading under a Regime Switching Model

This paper is concerned with the optimality of a trend following trading rule. The idea is to catch a bull market at its early stage, ride the trend, and liquidate the position at the first evidence of the subsequent bear market. We characterize the bull and bear phases of the markets mathematically using the conditional probabilities of the bull market given the up to date stock prices. The optimal buying and selling times are given in terms of a sequence of stopping times determined by two threshold curves. Numerical experiments are conducted to validate the theoretical results and demonstrate how they perform in a marketplace.