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.

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.

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.

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.

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.

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.

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.

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.

Existence of Solutions for Forward-Backward Stochastic Differential Equations with Jumps and Non-Lipschitzian Coefficients

This paper studies for ward-back ward differential equations with Poisson jumps and with stopping time as termination. Under some weak monotonicity conditions and for non-Lipschitzian coefficients, the existence and uniqueness of solutions are proved via a purely probabilistic approach, while a priori estimate is given. Here, we allow the forward equation to be degenerate.

On the Ratio Inequalities for Locally Square Integrable Martingales

In this paper, we establish some ratio inequalities for locally square inte-grable martingales, and give some extensions of the related results for continuous local martingales.

PEAK COVARIANCE STABILITY OF A RANDOM RICCATI EQUATION ARISING FROM KALMAN FILTERING WITH OBSERVATION LOSSES

We consider the stability of a random Riccati equation with a Markovian binary jump coefficient. More specifically, we are concerned with the boundedness of the solution of a random Riccati difference equation arising from Kalman filtering with measurement losses. A sufficient condition for the peak covariance stability is obtained which has a simpler form and is shown to be less conservative in some cases than a very recent result in existing literature. Furthermore, we show that a known sufficient condition is also necessary when the observability index equals one.

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.