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It is proved that the law of a possibly killed Levy process X , seen up to and including (resp. up to strictly before) a stopping time, determines already the law of X (resp. up to a compound Poisson component and killing).

There exists a simple, didactically useful one-to-one relationship between stopping times and adapted c\`agl\`ad (LCRL) processes that are non-increasing and take the values 0 and 1 only. As a consequence, stopping times are always hitting times.

In this paper we first prove a general representation theorem for generators of backward stochastic differential equations (BSDEs for short) by utilizing a localization method involved with stopping time tools and approximation techniques, where the generators only need to satisfy a weak monotonicity condition and a general growth condition in y and a Lipschitz condition in z . This result basically solves the problem of representation theorems for generators of BSDEs with general growth generators in y . Then, such representation theorem is adopted to prove a probabilistic formula, in viscosity sense, of semilinear parabolic PDEs of second order. The representation theorem approach seems to be a potential tool to the research of viscosity solutions of PDEs.

We establish the Default Barrier Intensity (DBI) model, based on the conditional survival probability (also called hazard function barrier), which allows the pricing of credit derivatives with stochastic parameters. Moreover, the DBI is an analytic model which combines the structural and the reduced form approaches. It deals with the impact of the default barrier intensity on the processes around the firm. Using this model we prove the Doob–Meyer decomposition of the default process associated with the random barrier. In this framework, we present the default barrier process as the sum of its compensator (which is a predictable process) and a martingale related to the smallest filtration making the random barrier a stopping time. Furthermore, the DBI as well as the Shifted Square Root Diffusion (SSRD) Alfonsi’s model emphasizes on the dependence between the stochastic default intensity and the interest rate. This model can be useful since it can be easily generalized to all the credit derivatives products such as Collateralized Debt Obligations (CDO) and Credit Default Swaps (CDS).

This paper provides a fixed-width confidence interval of a real-valued functional, which involves the comparison between the univariate distribution functions, under a response-adaptive design in which a Stein-type two-stage procedure is carried out.

We show a martingale solution for operators of the form L 1 1 { t τ } + L 2 1 { t ≥ τ } , where L 1 , L 2 are second order differential operators and τ is a stopping time. As applications, we study the existence of couplings.

Motivated by our earlier work on change-point analysis we prove a number of limit theorems for increments of renewal counting processes, or the corresponding first passage times. The starting point of the increments is deterministic as well as random, a typical example being the first stopping time to detect a change-point of some (continuously) observed process.

An optimal stopping time problem with a Poisson process, discount function and random starting time is considered. Generalizations to a problem with random horizon and to a multi-person stopping game with priorities are presented.

The cutoff phenomenon in the simulated methods has been widely investigated in recent years. An important question is to detect the stopping time after which one can obtain the convergence to equilibrium of an interesting Markov chain. We propose in this paper, a method for evaluating the cutoff instant using appropriate stopping times such as those found in [Lachaud, B. 2005. Cutoff and hitting times for a sample of Ornstein–Uhlenbeck processes and its everage, J. Appl. Probab. 42 (4), 1069–1080]. We give the conditions under which a Markov process has a cutoff time, in the sense of the generalized divergence measure. For illustration, we evaluate the effectiveness of our method on the Ornstein–Uhlenbeck process.

Let the random variables (r.v.’s) X 0 , X 1 , … be defined on the probability space ( X , A , P θ ) and take values in ( S , S ) , where S is a measurable subset of a Euclidean space and S is the σ -field of Borel subsets of S , and suppose that they form a general stochastic process. It is assumed that all finite dimensional joint distributions of the underlying r.v.’s have known functional form except that they depend on the parameter θ , a member of an open subset Θ of R k , k ≥ 1 . What is available to us is a random number of r.v.’s X 0 , X 1 , … , X ν n , where ν n is a stopping time as specified below. On the basis of these r.v.’s, a sequence of so-called regular estimates of θ is considered, which properly normalized converges in distribution to a probability measure L ( θ ) . Then the main theorem in this paper is the Hajek–Inagaki convolution representation of L ( θ ) . The proof of this theorem rests heavily on results previously established in the framework described here. These results include asymptotic expansions–in the probability sense–of log-likelihoods, their asymptotic distributions, the asymptotic distribution of a random vector closely related to the log-likelihoods, and a certain exponential approximation. Relevant references are given in the text.

This note deals with the question: what remains of the Burkholder–Davis–Gundy inequalities when stopping times T are replaced by arbitrary random times ρ? We prove that these inequalities still hold when T is a pseudo-stopping time and never holds for ends of predictable sets.

Let Q t = ( x t , y t ) be a two-dimensional geometric Brownian motion which is possibly correlated starting at ( x , y ) in the positive quadrant, and let τ be an F t Q -stopping time generated by the process Q t . Under certain conditions, we prove that E x , y [ ( x τ − y τ ) e − ∫ 0 τ Φ ( x s , y s ) d s ] ≤ C x n y 1 − n g ∗ , x μ y where Φ is a bounded Borel function, C > 0 , μ > 1 , n > 1 are constants and g ∗ is an explicit bound for a solution of a certain second order ordinary differential equation. The present result extends and supplements the explicit upper bound in Hu and Oksendal (1998) .

Dubins (1968) gave the first solution to the Skorokhod embedding problem (SEP) based solely on the underlying Brownian motion, and thus requiring no additional independent random variable. The Dubins solution to the SEP, can be expressed as τ ≔ sup { τ n } with τ n = inf { t ≥ τ n − 1 : W t ∈ support of μ n } . Since the measures μ n are defined recursively, in order to compute μ n , each of μ 0 , … , μ n − 1 must first be computed. In this note, we give a new solution to the SEP by showing how to construct a different sequence of measures { μ n } n ∈ N . The advantage of this solution is that for any given n , the measure μ n can be constructed directly without prior computation of the measures μ 0 , … , μ n − 1 .

Let X 1 , X 2 , … be i.i.d. mean zero random variables and S n = X 1 + ⋯ + X n . For any stopping time T w.r.t. { X n } let T denote a copy of T which is independent of { X n } . We exhibit a family of distributions and stopping times T such that E max 1 ≤ n ≤ T | ∑ 1 ≤ i ≠ j ≤ n X i X j | ∞ and yet E max 1 ≤ n ≤ T | ∑ 1 ≤ i ≠ j ≤ n X i X j | = ∞ .

We prove some analogs of results from renewal theory for random walks in the case when there is a drift, more precisely when the mean of the k th summand equals k γ μ , k ≥ 1 , for some μ > 0 and 0 γ ≤ 1 .

A problem of reduction of dimension of a plane Poisson process is faced here by considering this process along optional increasing paths. A suitable reparametrization of an optional increasing path makes the process along the path a one-parameter Poisson process. A convenient characterization of the spatial Poisson process is so obtained. The key idea developed in this paper relates to the corresponding result which holds for one-parameter martingales stopped by a bounded stopping time.

The stopping problem for max X 1 ,…, X n − en is treated when the distribution of X i is unknown. We give a general result on asymptotic optimality for a stopping time which incorporates an estimate for the unknown optimal level. This generalizes a result by Martinsek (1984) for exponential distributions with unknown mean.

Taking into consideration the relationship among the ‘optimal’ fixed sample size solutions of estimation (point, as well as, interval) and ranking and selection problems related to normal populations, a class of sequential procedures is developed. The confidence interval and ranking and selection problems are linked with the bounded risk point estimation problems under some suitable loss functions. Exploiting the common functional form of the risks associated with these problems, second-order approximations are obtained for the ‘regret’. It is illustrated that the results obtained under this general set-up provide second-order approximations to estimation and ranking and selection problems and, as such, no separate dealing is required. As an additional contribution, a much simple technique of obtaining the asymptotic distribution of stopping time is provided.

Let BtH, t⩾0 be a fractional Brownian motion with Hurst index 0 I n (t,H)= ∫ 0 t I n−1 (s,H) d B s H with I0(t,H)=1 and I1(t,H)=BtH. Then the inequalities c n,p ||τ nH || p ⩽ sup 0⩽t⩽τ |I n (t,H)| p ⩽C n,p ||τ nH || p , n=1,2,… are proved to hold for all stopping times τ of BH and 0 X t = ∫ 0 t u s d B s H , t⩾0, where u is a deterministic continuous function with some suitable conditions. We show that ∫ 0 τ u s 2 s 2H−1 d s 1/2 p and sup 0⩽t⩽τ |X t | p are equivalent for all 0

Let X1, X2,… be a sequence of independent random variables having finite second moments. Let T be a stopping time defined on the X's. If T is an integer-valued random variable independent of the X's but having the same distribution as T, then it is shown that 0 ⩽ ES2T ⩽ 2 ES2 T where Sn = X1 + ⋯ + Xn. Further, via examples it is shown that the above lower and upper bounds are sharp. Some different bounds when the variables are nonnegative are also given. For a sequence of independent random variables, upper and lower bounds for the expectation of a non-negative function of the randomly stopped sequence are obtained (see Result 4).

The aim of this paper is to show how decoupling techniques can be useful in proving domination results involving dependent random variables and arbitrary stopping times. Let {Xi} be a sequence of random variables adapted to { F i } , an increasing sequence of σ-fields and set G =σ(X 1 ,X 2 ,…) . Assume that for all t>0, (0.1) P(|X i |>t| F i−1 )⩽P(|Y i |>t| G ), where {Yi} is a sequence of conditionally independent given G . A case of particular importance is the one when the {Yi} is independent of G in which case the conditioning drops from (0.1). Then, for all stopping times T adapted to G , E max j⩽T |∑ i=1 j X i | p ⩽C p E|∑ i=1 T Y i | p whenever {Xi} is a mean-zero martingale difference sequence and p⩾1 or it is a sequence of non-negative or conditionally symmetric random variables and p>0. As a consequence of this, we obtain a sharp extension of Wald's equation for randomly stopped sums of i.i.d. random variables to the case of sums of the form ∑i=kTXi−k+1…Xi where the mean-zero martingale sequence {Xi} satisfies (0.1).

We consider the problem of optimal stopping of a finite sequence of dependent random variables. We explicitly determine the maximum of the stopping value within the Frechet class of all multivariate distributions with given continuous marginals. We show that the maximum is attained for a shuffle of min copula, and that it coincides with the value of a prophet.

Let W denote Brownian motion starting from the origin. The idea of this paper is give a computation of the expected exit time Eτ [ a , b ] from an interval [ a , b ], where a b , without the aid of Wald's Identity. Instead, the Strong Markov Property and other fundamental properties of Brownian motion are used directly to show that Eτ [ a , b ] is linear in both a and b , and then a limiting result about Brownian motion is used to compute the constant of linearity. As a part of the proof of the linearity of the expected exit time, we compute the distribution of W τ [ a , b ] .

We obtain Wald's equation for de-normalized U-statistics of degree m in the spirit of similar recent results by de la Pena and Lai. The proof uses Young's inequalities to obtain alternate conditions under which Wald's equation holds.

The expected ruin time and the probabilities of ruin are found for the n-player gambler's ruin problem with asymmetric play for certain cases of equal initial fortunes. Independence of the ruin time with which player is ruined is also shown.

The definition of a (discrete-time) martingale is generalized, allowing the process to be nonadapted, and it is shown that the optional stopping theorem still holds. This result was motivated by and is applied to the study of gambling systems, especially at games such as craps in which bets are not immediately resolved.

In this note we consider the gambler's ruin problem with four players, and we compute the expected ruin time when the initial fortunes of all four players are the same.

In this paper we extend Stroyan (1986) work regarding hyperfinite filtrations to the case of stopped σ-algebras.

The optimal sampling strategy for an urn, containing known numbers of plus and minus ones, can be simply described with the use of an empirically justified rule, based upon what appears to be a legitimate third-order asymptotic expansion of “the optimal stopping boundary” as the urn size goes to infinity performs exceedingly well. There is a known first-order asymptotic expansion due to Shepp. The reader is invited to try to justify a second-order asymptotic expansion of a type described by Chernoff and Petkau. The evidence presented in its support is very persuasive.

The asymptotic independence and exponentiality of ‘pure head run’ stopping times will be proved.

In the classical gambling problem (two players with initial capital a1 and a2, respectively) the expected time to ruin is a1a2. Here we extend this problem to three players and prove that the expected time to ruin equals (a1a2a3)/(a1 + a2 + a3 − 2).

Let {Xn; n ⩾ 1} be a sequence of i.i.d. random variables with partial sum sequence {Zn} and proper stopping time N. We show the following: For EX = 0, r ϵ (0, ∞), and α ϵ[1, 2]; ENr/α < ∞ and E|X|max(r,α) < ∞ together imply E|ZrN|< ∞. This result is known for r ⩾ 2, and improves known results for r < 2.