FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH STOPPING TIME

The existence and uniqueness results of fully coupled forward-backward stochastic differential equations with stopping time (unbounded) is obtained. One kind of comparison theorem for this kind of equations is also proved.

ON SOLUTIONS OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS,WITH UNBOUNDED STOPPING TIMES AS TERMINAL AND WITH NON-LIPSCHITZ COEFFICIENTS,AND PROBABILISTIC INTERPRETATION OF QUASI-LINEAR ELLIPTIC TYPE INTEGRO-DIFFERENTIAL EQUATIO

The existence and uniqueness of solutions to backward stochastic differential equations with jumps and with unbounded stopping time as terminal under the non_Lipschitz condition are obtained. The convergence of solutions and the continuous dependence of solutions on parameters are also derived. Then the probabilistic interpretation of solutions to some kinds of quasi_linear elliptic type integro_differential equations is obtained.

Optimal stopping time on discounted semi-Markov processes

This paper attempts to study the optimal stopping time for semi- Markov processes (SMPs) under the discount optimization criteria with unbounded cost rates. In our work, we introduce an explicit construction of the equivalent semi-Markov decision processes (SMDPs). The equivalence is embodied in the expected discounted cost functions of SMPs and SMDPs, that is, every stopping time of SMPs can induce a policy of SMDPs such that the value functions are equal, and vice versa. The existence of the optimal stopping time of SMPs is proved by this equivalence relation. Next, we give the optimality equation of the value function and develop an effective iterative algorithm for computing it. Moreover, we show that the optimal and ε-optimal stopping time can be characterized by the hitting time of the special sets. Finally, to illustrate the validity of our results, an example of a maintenance system is presented in the end.

Forward–Backward SDEs Driven by Levy Process in Stopping Time Duration

As the first part in the present paper,we study a class of backward stochastic differential equation(BSDE,for short)driven by Teugels martingales associated with some Levy processes having moment of all orders and an independent Brownian motion.We obtain an existence and uniqueness result for this type of BSDEs when the final time is allowed to be random.As the second part,we prove,under a monotonicity condition,an existence and uniqueness result for fully coupled forward-backward stochastic differential equation(FBSDE,for short)driven by Teugels martingales in stopping time duration.As an illustration of our theoretical results,we deal with a portfolio selection in Levy-type market.

Extended conditional G-expectations and related stopping times

In this paper,we extend the definition of conditional G-expectation to a larger space on which the dynamical consistency still holds.We can consistently define,by taking the limit,the conditional G-expectation for each random variable X,which is the downward limit(respectively,upward limit)of a monotone sequence (Xi) in L_(G)^(1)(Ω).To accomplish this procedure,some careful analysis is needed.Moreover,we present a suitable definition of stopping times and obtain the optional stopping theorem.We also provide some basic and interesting properties for the extended conditional G-expectation.

Zero-Sum Continuous-Time Markov Games with One-Side Stopping

The paper is concerned with a variant of the continuous-time finite state Markov game of control and stopping where both players can affect transition rates,while only one player can choose a stopping time.The dynamic programming principle reduces this problem to a system of ODEs with unilateral constraints.This system plays the role of the Bellman equation.We show that its solution provides the optimal strategies of the players.Additionally,the existence and uniqueness theorem for the deduced system of ODEs with unilateral constraints is derived.

Nothing Lasts Forever Time to stop pining for authentic hutongs

AS a foreigner anywhere, it is important to make an effort to experience the cultural heritage of the city you are in. In Paris you should visit Montmartre, in London you can＇t miss Buckingham Palace, and before you leave Sydney, a trip to the Blue Mountains is essential, For Beiiing, the unquestionable birthplace of Chinese culture is buried deep within the city＇s ancient hutongs.

Information-based approach:Pricing of a credit risky asset in the presence of default time

We extend the information-based asset-pricing framework by Brody,Hughston&Macrina to incorporate a stochastic bankruptcy time for the writer of the asset.Our model introduces a non-defaultable cash flow Zr to be made at time T,alongside the time T of a possible bankruptcy of the writer of the asset are in line with the filtration generated by a Brownian random bridge with length v=T^T and pinning point ZT,where is a constant.Quantities Z and T are not necessarily independent.The model does not depend crucially on the interpretation of as a bankruptcy time.We derived the price process of the asset and compute the prices of associated options.The dynamics of the price process satisfy a diffusion equation.Employing the approach of P.-A.Meyer,we provide the explicit computation of the compensator of v.Leveraging special properties of the bridge process,we also provide the explicit expression of the compensator of Zr I(v,+o).The resulting conclusion highlights the totally inaccessible property of the stopping time v.This characteristic is particularly suitable for financial markets where the time of default of a writer cannot be predictable from any other signal in the system until default happens.

Upper Bounds for the L_p-norms of the Maximal Functions of Martingales

Let 2≤p【∞ and let (f n) be a martingale. Using exponential bounds of the probabilities of the type P(|f n|】λ‖T(f n)‖ ∞) for some quasi-linear operators acting on martingales, we estimate upper bounds for the L p-norms of the maximal functions of martinglaes. Our result is the extension and improvements of the results obtained previously by HITCZENKO and ZENG .

UNIFORM PACKING DIMENSION RESULTS FOR MULTIPARAMETER STABLE PROCESSES

In this article, authors discuss the problem of uniform packing dimension of the image set of multiparameter stochastic processes without random uniform Holder condition, and obtain the uniform packing dimension of multiparameter stable processes. If Z is a stable （N, d, α）-process and αN ≤ d, then the following holds with probability 1 Dim Z（E）=α Dim E for any Borel setE ∈B（R ＋^N）, where Z（E）={x：E←t∈E,Z（t）=x}, Dim （E） denotes the packing dimension of E.

A Comparison Theorem for Solution of the Fully Coupled Backward Stochastic Differential Equations

The comparison theorems of solutions for BSDEs in fully coupled forward-backward stochastic differential equations (FBSDEs) are studied in this paper, here in the fully coupled FBSDEs the forward SDEs are the same structure.

MARKOWITZ STRATEGIES REVISED

Continuous-time Markowitz＇s by parameterizing a critical quantity. It mean-variance efficient strategies are modified is shown that these parameterized Markowitz strategies could reach the original mean target with arbitrarily high probabilities. This, in turn, motivates the introduction of certain stopped strategies where stock holdings are liquidated whenever the parameterized Markowitz strategies reach the present value of the mean target. The risk aspect of the revised Markowitz strategies are examined via expected discounted loss from the initial budget. A new portfolio selection model is suggested based on the results of the paper.

Ruin Distributions and Their Equations

In this paper, the ruin distributions were analyzed, including the distribution of surplus immediately before ruin, the distribution of claim at the time of ruin, the distribution of deficit, and the distribution of surplus at the beginning of the claim period before ruin. Several integral equations for the ruin distributions were derived and some solutions under special conditions were obtained.

Sequential Shrinkage Estimate for COX Regression Models with Uncertain Number of Effective Variables

In the applications of COX regression models, we always encounter data sets that contain too many variables that only a few of them contribute to the model. Therefore, it will waste much more samples to estimate the “noneffective” variables in the inference. In this paper, we use a sequential procedure for constructing the fixed size confidence set for the “effective” parameters to the model based on an adaptive shrinkage estimate such that the “effective” coefficients can be efficiently identified with the minimum sample size. Fixed design is considered for numerical simulation. The strong consistency, asymptotic distributions and convergence rates of estimates under the fixed design are obtained. In addition, the sequential procedure is shown to be asymptotically optimal in the sense of Chow and Robbins (1965).

Optimal stopping in predictable setting

In this study,we delve into the optimal stopping problem by examining the p(ϕ(τ),τ∈T_(0)^(p))case in which the reward is given by a family of nonnegative random variables indexed by predictable stopping times.We aim to elucidate various properties of the value function family within this context.We prove the existence of an optimal predictable stopping time,subject to specific assumptions regarding the reward functionϕ.

Ruin Probability in Linear Time Series Model

This paper analyzes a continuous time risk model with a linear model used to model the claim process. The time is discretized stochastically using the times when claims occur, using Doob’s stopping time theorem and martingale inequalities to obtain expressions for the ruin probability as well as both expo- nential and non-exponential upper bounds for the ruin probability for an infinite time horizon. Numerical re- sults are included to illustrate the accuracy of the non-exponential bound.

Relative Extreme Values and Extremum-Times of Jump Process:Upward Type

Relative extreme values are defined by the supremum and minimum of a general jump process before its first time quitting from some state set,and relative extremum-times are defined by the first times reaching relative extreme values.The main objective of this paper is to find out the exact distributions and moments of them as the maximum of the set is up or equal to the process initial state.As especial cases,these results are applied to a general birth-death process and generalized birth-death processes.

Optimization of transit total bus stop time models

Several factors influence bus transit reliability which includes bus stop conditions along the route, traffic conditions, route of travel and time of day. The overall transit bus reli- ability is generally affected by dwell time （DT）, the fare payment method, the bus stop location, and the number of passengers alighting or boarding. A new variable is defined in this study, total bus stop time （TBST）, which is the summation of DT and the time it takes a bus to effectively park at a bus stop and the re-entering the traffic stream. It is suggested that the overall bus transit reliability along routes could be improved if the TBST is mini- mized at bus stops. In this study, TBST models for bus stops located at mid-blocks and near intersections were developed based on multivariate regression analysis using ordinary least squares method. Data collection was conducted at 60 bus stops, 30 of which were near intersections and 30 at mid-blocks, in Washington DC during morning, mid-day and evening peak hours. The variables observed at each bus stop are as follows： number of passengers alighting or boarding, DT, TBST, bus stop type, bus pad, length number of lanes on approach to the bus stop, and permitted parking. Statistical inferences were based on 5% level of significance. From the results, it was inferred that the new variable, TBST, could potentially be used to improve scheduling and transit bus systems planning in a dense urban area.

An approach towards estimating critical value of waiting time at transit stops

Waiting time at transit stops is found to be an influential policy variable for a passenger’s decision on whether to undertake a given transit service. With regard to policy framework for improvement of operational service headway of a transit service and thereby its waiting time, the necessity to have knowledge on its critical value becomes inevitable. The critical value of waiting time for passengers waiting at transit stops is that duration beyond which passengers are found to be no more interested to wait for a that transit service. The paper demonstrates an approach for estimating the critical value of waiting time at urban transit stops with reference to public transport services such as city bus and shared-auto operational in Bhubaneswar, India. The critical value of waiting time is estimated from the point on cumulative distribution curve of waiting time frequency distribution, at which the maximum rate of change of the slope of curve occurs. The work assumes two positively skewed distributions such as gamma and log-normal for observed distributional pattern of waiting time. The work identifies that gamma distribution is comparatively fitting the observed data better than log-normal distribution. The study reveals that the critical value of waiting time for city bus passengers is about twice than that of shared auto passengers.Though, the study presents new information on critical values of waiting time with reference to an urban area of a developing country, it also demonstrates an experience on application of probability distribution functions for understanding distributional pattern of waiting time.

A methodology for rearranging transit stops for enhancing transit users generalized travel time

This study develops a methodology to consohdate transit stops. It develops a mathematical model and a program which takes stop consolidation decision（s） according to users gener- alized travel time savings and desired accessibility. The model iterates until the users generalized travel time savings are maximized. The study tests this mathematical model in different hypothetical scenarios. Six factors （distance between stops, passenger activity, average cruising speed, maximum walking distance, service frequency, and percentage of decreased passengers） with multiple levels were set to build the scenarios. Three responses {percentage of consolidated stops, percentages of travel time and operating time savings） were observed. The findings showed that the distance between the stops the passenger ac- tivity, and the probable demand change （or the percentage of decreased passengers） are the most influential factors. The frequency of service was found to be influential as well. The average cruising speed has very little influence on the response variables. Finally, the model is tested on two routes （route 900 and 930） ofAl Ain City public bus service. It shows that 22 and 32 out of 98 and 126 stops can be consolidated in route 900 and 930 respectively. This can save considerable amounts of users travel and operating times. In monetary values, the savings are about $329,827 and $491,094 per year for routes 900 and 930, respectively.