Showing papers in "Journal of Applied Probability in 1995"
TL;DR: In this paper, the authors show that many other probability measures can be defined in the same way to solve different asset-pricing problems, in particular option pricing, and this feature, besides providing a financial interpretation, permits efficient selection of the numeraire appropriate for the pricing of a given contingent claim and also permits exhibition of the hedging portfolio, which is in many respects more important than the valuation itself.
Abstract: The use of the risk-neutral probability measure has proved to be very powerful for computing the prices of contingent claims in the context of complete markets, or the prices of redundant securities when the assumption of complete markets is relaxed. We show here that many other probability measures can be defined in the same way to solve different asset-pricing problems, in particular option pricing. Moreover, these probability measure changes are in fact associated with numeraire changes, this feature, besides providing a financial interpretation, permits efficient selection of the numeraire appropriate for the pricing of a given contingent claim and also permits exhibition of the hedging portfolio, which is in many respects more important than the valuation itself. The key theorem of general numeraire change is illustrated by many examples, among which the extension to a stochastic interest rates framework of the Margrabe formula, Geske formula, etc.
726 citations
TL;DR: This paper approaches the problem of computing the price of an Asian option in two different ways, exploiting a scaling property and providing a lower bound which is so accurate that it is essentially the true price.
Abstract: This paper approaches the problem of computing the price of an Asian option in two different ways. Firstly, exploiting a scaling property, we reduce the problem to the problem of solving a parabolic PDE in two variables. Secondly, we provide a lower bound which is so accurate that it is essentially the true price.
596 citations
TL;DR: In this paper, the authors emphasize the central role played by the tail empirical process for the problem of consistency and show that Hill's estimator is consistent for infinite order moving averages of independent random variables.
Abstract: Consider a sequence of possibly dependent random variables having the same marginal distribution F, whose tail 1-F is regularly varying at infinity with an unknown index - a < 0 which is to be estimated. For i.i.d. data or for dependent sequences with the same marginal satisfying mixing conditions, it is well known that Hill's estimator is consistent for a -I and asymptotically normally distributed. The purpose of this paper is to emphasize the central role played by the tail empirical process for the problem of consistency. This approach allows us to easily prove Hill's estimator is consistent for infinite order moving averages of independent random variables. Our method also suffices to prove that, for the case of an AR model, the unknown index can be estimated using the residuals generated by the estimation of the autoregressive parameters.
145 citations
TL;DR: A new partial ordering among life distributions in terms of their uncertainties is introduced, and the notion of a ‘better system' is introduced based on this ordering and various existing orderings.
Abstract: A new partial ordering among life distributions in terms of their uncertainties is introduced. Our measure of uncertainty is Shannon information applied to the residual lifetime. The relationship between this ordering and various existing orderings of life distributions are discussed. Various properties of our proposed concept are examined. Based on our proposed ordering and various existing orderings, the notion of a ‘better system' is introduced.
130 citations
TL;DR: In this paper, the stability of open queueing systems under stationary ergodic assumptions is studied and a set of conditions, the monotone separable framework, ensuring that the stability region is given by the following saturation rule: saturate the queues which are fed by the external arrival stream, look ate the intensity [??] of the departure stream in this saturated system, then stability holds whenever the intensity of the arrival process, say l satisfies the condition [??], whereas the network is unstable if [??].
Abstract: This paper focuses on the stability of open queueing systems under stationary ergodic assumptions. It defines a set of conditions, the monotone separable framework, ensuring that the stability region is given by the following saturation rule : saturate the queues which are fed by the external arrival stream, look ate the intensity [??] of the departure stream in this saturated system, then stability holds whenever the intensity of the arrival process, say l satisfies the condition [??], whereas the network is unstable if [??]. Whenever the stability condition is satisfied, it is also shown that certain state variables associated with the network admit a finite stationary regime which is constructed pathwise using a Loynes type bacward argument. This framework involves two main pathwise properties, external monotonicity and separability, which are satisfied by several classical queueing networks. The main tool for the proof of this rule is sub-additive ergodic theory.
94 citations
TL;DR: In this paper, the authors consider the problem of conditioning a continuous-time Markov chain not to hit an absorbing barrier before time T; and the weak convergence of this conditional process as T → ∞.
Abstract: We consider the problem of conditioning a continuous-time Markov chain (on a countably infinite state space) not to hit an absorbing barrier before time T; and the weak convergence of this conditional process as T → ∞. We prove a characterization of convergence in terms of the distribution of the process at some arbitrary positive time, t, introduce a decay parameter for the time to absorption, give an example where weak convergence fails, and give sufficient conditions for weak convergence in terms of the existence of a quasi-stationary limit, and a recurrence property of the original process.
71 citations
TL;DR: In this paper, a unified presentation of stability results for stochastic vector difference equations Y n+1 = A n? Y n? B n based on various choices of binary operations was given.
Abstract: We give a unified presentation of stability results for stochastic vector difference equations Y n+1 = A n ? Y n ? B n based on various choices of binary operations ? and ?, assuming that {(A n , B n ), n ≥ 0} are stationary and ergodic. In the scalar case, under standard addition and multiplication, the key condition for stability is E[log |A 0 |]<0. In the generalizations, the condition takes the form γ <0, where γ is the limit of a subadditive process associated with {A(n), n≥0}. Under this and mild additional conditions, the process { Y n , n ≥ 0} has a unique finite stationary distribution to which it converges from all initial conditions The variants of standard matrix algebra we consider replace the operations + and x with (max,+), (max,x), (min,+), or (min,x). In each case, the appropriate stability condition parallels that for the standard recursions, involving certain subadditive limits. Since these limits are difficult to evaluate, we provide bounds, thus giving alternative, computable conditions for stability.
50 citations
TL;DR: In this paper, the authors studied the limit behavior of the distributions and the moments of the Bernoulli meander in the case where the number of steps in the random walk tends to infinity.
Abstract: This paper is concerned with the distibutions and the moments of the area and the local time of a random walk, called the Bernoulli meander. The limit behavior of the distributions and the moments is determined in the case where the number of steps in the random walk tends to infinity. The results of this paper yield explicit formulas for the distributions and the moments of the area and the local time for the Brownian meander.
49 citations
TL;DR: In this article, the distribution of Brownian quantiles is determined, simplifying related integral expressions obtained by Levy [9], [10] and more recently by Miura [11].
Abstract: The distribution of Brownian quantiles is determined, simplifying related integral expressions obtained by Levy [9], [10] and more recently by Miura [11]. Three proofs are given, two of them involving last-passage times of Brownian motion, before time 1, at a given level.
49 citations
TL;DR: In this article, a pure probabilistic approach is used, and the adjoint processes are characterized as solutions of related backward stochastic differential equations in finite-dimensional spaces.
Abstract: The partially observed control problem is considered for stochastic processes with control entering into the diffusion and the observation. The maximum principle is proved for the partially observable optimal control. A pure probabilistic approach is used, and the adjoint processes are characterized as solutions of related backward stochastic differential equations in finite-dimensional spaces. Most of the derivation is identified with that of the completely observable case.
47 citations
TL;DR: In this article, it is shown that γ(x) is the ruin probability in a risk process with initial reserve u, Poisson arrival rate β, claim size distribution B and premium rate p(x), at level x of the reserve.
Abstract: Let Ψ(u) be the ruin probability in a risk process with initial reserve u, Poisson arrival rate β, claim size distribution B and premium rate p(x) at level x of the reserve. Let γ(x) be the non-zero solution of the local Lundberg equation β(B [γ(x)] - 1) - γ(x)p(x) = 0 and I(u)= ∫ 0 u γ(x)dx. It is shown that Ψ(u)≤e -1(u) provided p(x) is non-decreasing and that log Ψ(u) -I(u) in a slow Markov walk limit. Though the results and conditions are of large deviations type, the proofs are elementary and utilize piecewise comparisons with standard risk processes with a constant p. Also simulation via importance sampling using local exponential change of measure defined in terms of the γ(x) is discussed and some numerical results are presented.
TL;DR: In this article, the Stein-Chen method was used to obtain compound Poisson approximations for the number of overlapping and non-overlapping occurrences of a fixed periodic k-letter word in a stationary Markov chain.
Abstract: Consider a stationary Markov chain {X j } j=1 n with state space consisting of the ξ-letter alphabet set Λ = {a 1 , a 2 ,...,a ξ }. We study the variables M=M(n,k) and N=N(n,k), defined, respectively, as the number of overlapping and non-overlapping occurrences of a fixed periodic k-letter word, and use the Stein-Chen method to obtain compound Poisson approximations for their distribution.
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TL;DR: In this article, the authors studied the number of occurrences of non-overlapping success runs of length k in a sequence of Bernoulli trials arranged on a circle and gave an exact formula for the probability function, along with some sharp bounds which turn out to be very useful in establishing limiting (Poisson convergence) results.
Abstract: We study the number of occurrences of non-overlapping success runs of length k in a sequence of (not necessarily identical) Bernoulli trials arranged on a circle. An exact formula is given for the probability function, along with some sharp bounds which turn out to be very useful in establishing limiting (Poisson convergence) results. Certain applications to statistical run tests and reliability theory are also discussed.
TL;DR: In this paper, the authors considered a triviariate stochastic process with random shocks at random intervals with random system states and derived the distribution of system lifetime, its moments and a related exponential limit theorem.
Abstract: A trivariate stochastic process is considered, describing a sequence of random shocks {Xn } at random intervals {Y n} with random system state {Jn }. The triviariate stochastic process satisfies a Markov renewal property in that the magnitude of shocks and the shock intervals are correlated pairwise and the corresponding joint distributions are affected by transitions of the system state which occur after each shock according to a Markov chain. Of interest is a system lifetime terminated whenever a shock magnitude exceeds a prespecified level z. The distribution of system lifetime, its moments and a related exponential limit theorem are derived explicitly. A similar transform analysis is conducted for a second type of system lifetime with system failures caused by the cumulative magnitude of shocks exceeding a fixed level z.
TL;DR: In this article, the integrability of continuous-time Markov chains in terms of their infinitesimal parameters has been studied, and sufficient conditions for integrinability have been given.
Abstract: We give simple sufficient conditions for integrability of continuous-time Markov chains in terms of their infinitesimal parameters Similar conditions for regularity are stated first, and a simple proof given
TL;DR: In this article, the Laplace transform of the end-to-end delay in tandem networks is obtained where the first and/or second queue are G-queue, i.e. they have negative arrivals.
Abstract: The Laplace transform of the probability distribution of the end-to-end delay in tandem networks is obtained where the first and/or second queue are G-queues, i.e. they have negative arrivals. For the most general case the method is based on the solution of a boundary value problem on a closed contour in the complex plane, which itself reduces to the solution of a Fredholm integral equation of the second kind. We also consider the dependence or independence of the sojourn times at each queue in the two special cases where only one of the queues is a G-queue, the other having no negative arrivals.
TL;DR: In this article, the authors prove the validity of the Volterra integral equation for the evaluation of first-passage-time probability densities through varying boundaries, given by Buonocore et al. [1], for the case of diffusion processes not necessarily time-homogeneous.
Abstract: In this paper we prove the validity of the Volterra integral equation for the evaluation of first-passage-time probability densities through varying boundaries, given by Buonocore et al. [1], for the case of diffusion processes not necessarily time-homogeneous. We study, specifically those processes that can be obtained from the Wiener process in the sense of [5]. A study of the kernel of the integral equation, in the same way as that by Buonocore et al. [1], is done. We obtain the boundaries for which closed-form solutions of the integral equation, without having to solve the equation, can be obtained. Finally, a few examples are given to indicate the actual use of our method.
TL;DR: A general method of estimating bounds for the state probabilities for X(t), based on inequalities for the solutions of the forward Kolmogorov equations is developed.
Abstract: Let X(t) be a non-homogeneous birth and death process. In this paper we develop a general method of estimating bounds for the state probabilities for X(t) based on inequalities for the solutions of the forward Kolmogorov equations. (EXCERPT)
TL;DR: In this article, it is shown how to explicitly compute the bundle strength survival distribution by using a new type of graph called the loading diagram, which is parallel in structure and recursive in nature and so would appear to lend itself to large scale computation.
Abstract: Harlow et al. (1983) have given a recursive formula which is fundamental for computing the bundle strength distribution under a general class of load sharing rules called monotone load sharing rules. As the bundle size increases, the formula becomes prohibitively complex and, by itself, does not give much insight into the relationship of the assumed load sharing rule to the overall strength distribution. In this paper, an algorithm is given which gives some additional insight into this relationship. Here it is shown how to explicitly compute the bundle strength survival distribution by using a new type of graph called the loading diagram. The graph is parallel in structure and recursive in nature and so would appear to lend itself to large-scale computation. In addition, the graph has an interesting property (which we refer to as the cancellation property) which is related to the asymptotics of the Weibull as a minimum stable law.
TL;DR: In this paper, the first-passage problem for the one-dimensional Wiener process with drift in the presence of elastic boundaries is considered and closed-form expressions for the expected value and the variance of the first passage time are derived.
Abstract: The first-passage problem for the one-dimensional Wiener process with drift in the presence of elastic boundaries is considered. We use the Kolmogorov backward equation with corresponding boundary conditions to derive explicit closed-form expressions for the expected value and the variance of the first-passage time. Special cases with pure absorbing and/or reflecting barriers arise for a certain choice of a parameter constellation.
TL;DR: In this paper, a stochastic model for the spread of an epidemic amongst a closed homogeneously mixing population, in which there are several different types of infective, each newly infected individual choosing its type at random from those available, is considered.
Abstract: We consider a stochastic model for the spread of an epidemic amongst a closed homogeneously mixing population, in which there are several different types of infective, each newly infected individual choosing its type at random from those available. The model is based on the carrier-borne model of Downton (1968), as extended by Picard and Lefevre (1990). The asymptotic distributions of final size and area under the trajectory of infectives are derived as the initial population becomes large, using arguments based on those of Scalia-Tomba (1985), (1990). We then use our limiting results to compare the asymptotic final size distribution of our model with that of a related multi-group model, in which the type of each infective is assigned deterministically.
TL;DR: In this paper, an optimal maintenance model for standby systems is studied, where the state of the system can only be determined through an inspection which may incorrectly identify the system state, and after each inspection, if the system is identified as in the down state, a repair action will be taken.
Abstract: In this paper, an optimal maintenance model for standby systems is studied. An inspection-repair-replacement policy is employed. Assume that the state of the system can only be determined through an inspection which may incorrectly identify the system state. After each inspection, if the system is identified as in the down state, a repair action will be taken. It will be replaced some time later by a new and identical one. The problem is to determine an optimal policy so that the availability of the system is high enough at any time and the long-run expected cost per unit time is minimized. An explicit expression for the long-run expected cost per unit time is derived. For a geometric model, a simple algorithm for the determination of an optimal solution is suggested.
TL;DR: In this paper, exact reliability formulas for linear and circular consecutive k-of-n : F systems were derived in the case of equal component reliabilities, where each component has equal component capabilities.
Abstract: Exact reliability formulas for linear and circular consecutive-k-of-n : F systems are derived in the case of equal component reliabilities. SYSTEM OF COMPONENTS; EQUAL COMPONENT RELIABILITIES
TL;DR: In this paper, a general approach for finding differential equations to evaluate probabilities of ruin in finite and infinite time is proposed, where the key point is to define a process of conditional ruin probabilities and identify this process stopped at the time of ruin as a martingale.
Abstract: The present paper proposes a general approach for finding differential equations to evaluate probabilities of ruin in finite and infinite time. Attention is given to reavalued non-diffusion processes where a Markov structure is obtainable. Ruin is allowed to occur upon a jump or between the jumps. The key point is to define a process of conditional ruin probabilities and identify this process stopped at the time of ruin as a martingale. Using the theory of marked point processes together with the change-of-variable formula or the martingale representation theorem for point processes, we obtain differential equations for evaluating the probability of ruin. Numerical illustrations are given by solving a partial differential equation numerically to obtain the probability of ruin over a finite time horizon
TL;DR: Given a sequence of independent identically distributed random variables, a moving-maximum sequence is derived (with random translations) and the extremal index of the derived sequence is computed and the limiting behaviour of clusters of high values is studied.
Abstract: Given a sequence of independent identically distributed random variables, we derive a moving-maximum sequence (with random translations). The extremal index of the derived sequence is computed and the limiting behaviour of clusters of high values is studied. We are then given two or more independent stationary sequences whose extremal indices are known. We derive a new stationary sequence by taking either a pointwise maximum or by a mixture of the original sequences. In each case, we compute the extremal index of the derived sequence.
TL;DR: In this paper, it was shown that every infinite-state stochastic matrix P say, that is irreducible and consists of positive-recurrrent states can be represented in the form I − P=(A − I)(B − S), where A is strictly upper-triangular, B is strictly lower-triagonal, and S is diagonal.
Abstract: We prove that every infinite-state stochastic matrix P say, that is irreducible and consists of positive-recurrrent states can be represented in the form I – P=(A – I)(B – S), where A is strictly upper-triangular, B is strictly lower-triangular, and S is diagonal. Moreover, the elements of A are expected values of random variables that we will specify, and the elements of B and S are probabilities of events that we will specify. The decomposition can be used to obtain steady-state probabilities, mean first-passage-times and the fundamental matrix.
TL;DR: In this paper, a sequence of bivariate random variables of lifetime and repair time are defined and properties of availability for the i.i.d. case are extended to this more general case.
Abstract: Availability is an important characteristic of a system. Different types of availability are defined. For the case when a sequence of bivariate random variables of lifetime and repair time are i.i.d. certain properties have been established previously. In practice, however, we need to consider the situation where these bivariate random variables are independent but not identically distributed. Properties of two measures of availability for the i.i.d. case are extended to this more general case.
TL;DR: In this paper, the authors investigated the limiting distribution of the completion time of the DNA replication process by considering related coverage problems investigated by Janson (1983) and Hall (1988).
Abstract: The DNA of higher animals replicates by an interesting mechanism. Enzymes recognise specific sites randomly scattered on the molecule and establish a bidirectional process of unwinding and replication from these sites. We investigate the limiting distribution of the completion time for this process by considering related coverage problems investigated by Janson (1983) and Hall (1988).
TL;DR: In this paper, it was shown that the critical two-level (2, d, 1, 1)-superprocess is persistent in dimensions d greater than 4. This complements the extinction result of Wu (1994) and implies that the Critical dimension is 4.
Abstract: It is shown that the critical two-level (2, d, 1, 1)-superprocess is persistent in dimensions d greater than 4. This complements the extinction result of Wu (1994) and implies that the critical dimension is 4. MEASURE-VALUED PROCESS; SUPERPROCESS; HIERARCHICAL BRANCHING; PERSISTENCE; LOCAL EXTINCTION
TL;DR: In this article, the authors consider the composition of random i.i.d. affine maps of a Hilbert space to itself and show convergence of the nth composition of these maps in the Wasserstein metric via a contraction argument.
Abstract: We consider the composition of random i.i.d. affine maps of a Hilbert space to itself. We show convergence of the nth composition of these maps in the Wasserstein metric via a contraction argument. The contraction condition involves the operator norm of the expectation of a bilinear form. This is contrasted with the usual contraction condition of a negative Lyapunov exponent. Our condition is stronger and easier to check. In addition, our condition allows us to conclude convergence of second moments as well as convergence in distribution.