In this paper we study continuous-time two-player zero-sum optimal switching games on a finite horizon. Using the theory of doubly reflected backward stochastic differential equations with interconnected barriers, we show that this game has a value and an equilibrium in the players’ switching controls.

The classical SIR epidemic model is generalized to incorporate a detection process of infectives in the course of time. Our purpose is to determine the exact distribution of the population state at the first detection instant and the following ones. An extension is also discussed that allows the parameters to change with the number of detected cases. The followed approach relies on simple martingale arguments and uses a special family of Abel–Gontcharoff polynomials.

In this paper we consider a variation of the full-information secretary problem where the random variables to be observed are independent but not necessary identically distributed. The main result is a sharp lower bound for the optimal win probability. Precisely, if X 1,...,X n are independent random variables with known continuous distributions and V n (X 1,...,Xn ):=supτℙ(X τ=M n ), where M n ≔max{X 1,...,X n } and the supremum is over all stopping times adapted to X 1,...,X n then V n (X 1,...,X n )≥(1-1/n)n-1, and this bound is attained. The method of proof consists in reducing the problem to that of a sequence of random variables taking at most two possible values, and then applying Bruss' sum-the-odds theorem, Bruss (2000). In order to obtain a sharp bound for each n, we improve Bruss' lower bound, Bruss (2003), for the sum-the-odds problem.

In this paper we study a discrete-time optimal switching problem on a finite horizon. The underlying model has a running reward, terminal reward, and signed (positive and negative) switching costs. Using optimal stopping theory for discrete-parameter stochastic processes, we extend a well-known explicit dynamic programming method for computing the value function and the optimal strategy to the case of signed switching costs.

We analyse the survival time of a general duplex system sustained by an auxiliary cold standby unit and subjected to priority rules. The duplex system is attended by two general repairmen Rp and Rh . Repairman Rp has priority in repairing failed units with regard to repairman Rh provided that both repairmen are jointly idle. Otherwise, the priority is overruled. The auxiliary unit has its own repair facility. The duplex system has overall, break-in priority (often called pre-emptive priority) in operation and in standby with regard to the auxiliary unit. The analysis of the survival time is based on advanced complex function theory (sectionally holomorphic functions). The main problem is to convert a functional equation into a (parameter dependent) Sokhotski–Plemelj problem.

Let T be a stopping time associated with a sequence of independent and identically distributed or exchangeable random variables taking values in {0, 1, 2, …, m}, and let ST,i be the stopped sum denoting the number of appearances of outcome 'i' in X1, …, XT, 0 ≤ i ≤ m. In this paper we present results revealing that, if the distribution of T is known, then we can also derive the joint distribution of (T, ST,0, ST,1, …, ST,m). Two applications, which have independent interest, are offered to illustrate the applicability and the usefulness of the main results.

Consider a sequence of Markov-dependent trials where each trial produces a letter of a finite alphabet. Given a collection of patterns, we look at this sequence until one of these patterns appears as a run. We show how the method of gambling teams can be employed to compute the probability that a given pattern is the first pattern to occur.

Let (Xt)0 ≤ t ≤ T be a one-dimensional stochastic process with independent and stationary increments, either in discrete or continuous time. In this paper we consider the problem of stopping the process (Xt) ‘as close as possible’ to its eventual supremum MT := sup0 ≤ t ≤ TXt, when the reward for stopping at time τ ≤ T is a nonincreasing convex function of MT - Xτ. Under fairly general conditions on the process (Xt), it is shown that the optimal stopping time τ takes a trivial form: it is either optimal to stop at time 0 or at time T. For the case of a random walk, the rule τ ≡ T is optimal if the steps of the walk stochastically dominate their opposites, and the rule τ ≡ 0 is optimal if the reverse relationship holds. An analogous result is proved for Lévy processes with finite Lévy measure. The result is then extended to some processes with nonfinite Lévy measure, including stable processes, CGMY processes, and processes whose jump component is of finite variation.

Robbins' problem of optimal stopping is that of minimising the expected rank of an observation chosen by some nonanticipating stopping rule. We settle a conjecture regarding the value of the stopped variable under the rule that yields the minimal expected rank, by embedding the problem in a much more general context of selection problems with the nonanticipation constraint lifted, and with the payoff growing like a power function of the rank.

It is well know that the expected exponentially discounted total reward for a stochastic process can also be defined as the expected total undiscounted reward earned before an independent exponential stopping time (let us call this the stopped reward). Feinberg and Fei (2009) recently showed that the variance of the discounted reward is smaller than the variance of the stopped reward. We strengthen this result to show that the discounted reward is smaller than the stopped reward in the convex ordering sense.

Let (Bt)0≤t≤T be either a Bernoulli random walk or a Brownian motion with drift, and let Mt := max{Bs: 0 ≤ s ≤ t}, 0 ≤ t ≤ T. In this paper we solve the general optimal prediction problem sup0≤τ≤TE[f(MT − Bτ], where the supremum is over all stopping times τ adapted to the natural filtration of (Bt) and f is a nonincreasing convex function. The optimal stopping time τ* is shown to be of ‘bang-bang’ type: τ* ≡ 0 if the drift of the underlying process (Bt) is negative and τ* ≡ T if the drift is positive. This result generalizes recent findings of Toit and Peskir (2009) and Yam, Yung and Zhou (2009), and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good stocks as long as possible.

We consider the following two definitions of discounting: (i) multiplicative coefficient in front of the rewards, and (ii) probability that the process has not been stopped if the stopping time has an exponential distribution independent of the process. It is well known that the expected total discounted rewards corresponding to these definitions are the same. In this note we show that, the variance of the total discounted rewards is smaller for the first definition than for the second definition.

Markov and semi-Markov processes with block tridiagonal transition matrices for their embedded discrete-time Markov chains are underlying stochastic models in many applied probability problems. In particular, identity-by-descent (IBD) problems for uncle-type and cousin-type relationships fall into this class. More specifically, the exact distributions of relevant IBD statistics for two individuals in either an uncle-type or cousin-type relationship are of interest. Such statistics are the amount of genome shared IBD by the two related individuals on a chromosomal segment and the number of IBD pieces on such a segment. These lead to special reward distributions associated with block tridiagonal transition matrices for continuous-time Markov chains. A method is provided for calculating explicit, closed-form expressions for Laplace transforms of general reward functions for such Markov chains. Some calculation results on the cumulative probabilities of relevant IBD statistics via a numerical inversion of the Laplace transforms are also provided for uncle/nephew and first-cousin relationships.

In this paper we consider the class of Lévy processes that can be written as a Brownian motion time changed by an independent Lévy subordinator. Examples in this class include the variance-gamma (VG) model, the normal-inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call the first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that the standard first passage time is the almost-sure limit of iterations of the first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are led to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms.

Assume that an insurance company pays dividends to its shareholders whenever the surplus process is above a given threshold. In this paper we study the expected amount of dividends paid, and the expected time to ruin in the compound Poisson risk process perturbed by a Brownian motion. Two models are considered: In the first one the insurance company pays whatever amount exceeds a given level b as dividends to its shareholders. In the second model, the company starts to pay dividends at a given rate, smaller than the premium rate, whenever the surplus up-crosses the level b. The dividends are paid until the surplus down-crosses the level a, a < b . We assume that the claim sizes are phase-type distributed. In the analysis we apply the multidimensional Wald martingale, and the multidimensional Asmussesn and Kella martingale.

Let ξ, ξ1, ξ2,… be a sequence of independent and identically distributed random variables, and let Sn=ξ1+⋯+ξn and Mn=maxk≤nSk. Let τ=min{n≥1: Sn≤0}. We assume that ξ has a heavy-tailed distribution and negative, finite mean E(ξ)<0. We find the asymptotics of P{Mτ ∈ (x, x+T]} as x→∞, for a fixed, positive constant T.

The distribution theory for reward functions on semi-Markov processes has been of interest since the early 1960s. The relevant asymptotic distribution theory has been satisfactorily developed. On the other hand, it has been noticed that it is difficult to find exact distribution results which lead to the effective computation of such distributions. Note that there is no satisfactory exact distribution result for rewards accumulated over deterministic time intervals [0, t], even in the special case of continuous-time Markov chains. The present paper provides neat general results which lead to explicit closed-form expressions for the relevant Laplace transforms of general reward functions on semi-Markov and Markov additive processes.

Methods using gambling teams and martingales are developed and applied to find formulae for the expected value and the generating function of the waiting time to observation of an element of a finite collection of patterns in a sequence generated by a two-state Markov chain of first, or higher, order.

We study the expected time to ruin in a risk process in which dividends are paid when the surplus is above the barrier. We consider the case in which the dividend rate is smaller than the premium rate. We obtain results for the classical compound Poisson risk process with phase-type claim size. When the ruin probability is 1, we derive the expected time to ruin and the expected dividends paid. When the ruin probability is less than 1, these quantities are derived conditioning on the event that ruin occurs.