CONTINUOUS-TIME FINANCE AND THE WAITING TIME DISTRIBUTION: MULTIPLE CHARACTERISTIC TIMES

2012 ◽  
Vol 26 (23) ◽  
pp. 1250151 ◽  
Author(s):  
KWOK SAU FA
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Waiting Time ◽  
Continuous Time ◽  
Probability Distribution Function ◽  
Time Distribution ◽  
Waiting Time Distribution ◽  
Exponential Functions ◽  
Multiple Characteristic ◽  
Sum Of Products

In this paper, we model the tick-by-tick dynamics of markets by using the continuous-time random walk (CTRW) model. We employ a sum of products of power law and stretched exponential functions for the waiting time probability distribution function; this function can fit well the waiting time distribution for BUND futures traded at LIFFE in 1997.

Subexponential asymptotics of a Markov-modulated random walk with queueing applications

1998 ◽  
Vol 35 (02) ◽  
pp. 325-347 ◽  
Author(s):  
Predrag R. Jelenković ◽  
Aurel A. Lazar
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Autocorrelation Function ◽  
Waiting Time ◽  
Time Distribution ◽  
Length Distribution ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Matrix Measure ◽  
Markov Modulated

Let {(X n ,J n )} be a stationary Markov-modulated random walk on ℝ x E (E is finite), defined by its probability transition matrix measure F = {F ij }, F ij (B) = ℙ[X 1 ∈ B, J 1 = j | J 0 = i], B ∈ B (ℝ), i, j ∈ E. If F ij ([x,∞))/(1-H(x)) → W ij ∈ [0,∞), as x → ∞, for some long-tailed distribution function H, then the ascending ladder heights matrix distribution G +(x) (right Wiener-Hopf factor) has long-tailed asymptotics. If 𝔼X n < 0, at least one W ij > 0, and H(x) is a subexponential distribution function, then the asymptotic behavior of the supremum of this random walk is the same as in the i.i.d. case, and it is given by ℙ[sup n≥0 S n > x] → (−𝔼X n )−1 ∫ x ∞ ℙ[X n > u]du as x → ∞, where S n = ∑1 n X k , S 0 = 0. Two general queueing applications of this result are given. First, if the same asymptotic conditions are imposed on a Markov-modulated G/G/1 queue, then the waiting time distribution has the same asymptotics as the waiting time distribution of a GI/GI/1 queue, i.e., it is given by the integrated tail of the service time distribution function divided by the negative drift of the queue increment process. Second, the autocorrelation function of a class of processes constructed by embedding a Markov chain into a subexponential renewal process, has a subexponential tail. When a fluid flow queue is fed by these processes, the queue-length distribution is asymptotically proportional to its autocorrelation function.

Subexponential asymptotics of a Markov-modulated random walk with queueing applications

1998 ◽  
Vol 35 (2) ◽  
pp. 325-347 ◽  
Author(s):  
Predrag R. Jelenković ◽  
Aurel A. Lazar
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Autocorrelation Function ◽  
Waiting Time ◽  
Time Distribution ◽  
Length Distribution ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Matrix Measure ◽  
Markov Modulated

Let {(Xn,Jn)} be a stationary Markov-modulated random walk on ℝ x E (E is finite), defined by its probability transition matrix measure F = {Fij}, Fij(B) = ℙ[X1 ∈ B, J1 = j | J0 = i], B ∈ B(ℝ), i, j ∈ E. If Fij([x,∞))/(1-H(x)) → Wij ∈ [0,∞), as x → ∞, for some long-tailed distribution function H, then the ascending ladder heights matrix distribution G+(x) (right Wiener-Hopf factor) has long-tailed asymptotics. If 𝔼Xn < 0, at least one Wij > 0, and H(x) is a subexponential distribution function, then the asymptotic behavior of the supremum of this random walk is the same as in the i.i.d. case, and it is given by ℙ[supn≥0Sn > x] → (−𝔼Xn)−1 ∫x∞ ℙ[Xn > u]du as x → ∞, where Sn = ∑1nXk, S0 = 0. Two general queueing applications of this result are given.First, if the same asymptotic conditions are imposed on a Markov-modulated G/G/1 queue, then the waiting time distribution has the same asymptotics as the waiting time distribution of a GI/GI/1 queue, i.e., it is given by the integrated tail of the service time distribution function divided by the negative drift of the queue increment process. Second, the autocorrelation function of a class of processes constructed by embedding a Markov chain into a subexponential renewal process, has a subexponential tail. When a fluid flow queue is fed by these processes, the queue-length distribution is asymptotically proportional to its autocorrelation function.

On exponential bounds for the waiting-time distribution function in GI/G/1

1976 ◽  
Vol 13 (2) ◽  
pp. 411-417 ◽  
Author(s):  
R. Bergmann ◽  
D. Stoyan
Keyword(s):  
Distribution Function ◽  
Waiting Time ◽  
Time Distribution ◽  
Arrival Time ◽  
Distribution Functions ◽  
Integral Inequalities ◽  
Arrival Time Distribution ◽  
Waiting Time Distribution ◽  
Exponential Bounds ◽  
Stationary Waiting Time

Exponential bounds for the stationary waiting-time distribution of the type ae–θt are considered. These bounds are obtained by the use of Kingman's method of ‘integral inequalities’. Approximations of Θ and a are given which are useful especially if the service and/or inter-arrival time distribution functions are NBUE or NWUE.

scholarly journals The use of Spitzer's identity in the investigation of the busy period and other quantities in the queue GI/G/1

1962 ◽  
Vol 2 (3) ◽  
pp. 345-356 ◽  
Author(s):  
J. F. C. Kingmán
Keyword(s):  
Random Walk ◽  
Characteristic Function ◽  
Waiting Time ◽  
Time Distribution ◽  
Busy Period ◽  
Waiting Time Distribution ◽  
Period Distribution ◽  
Stationary Waiting Time

As an illustration of the use of his identity [10], Spitzer [11] obtained the Pollaczek-Khintchine formula for the waiting time distribution of the queue M/G/1. The present paper develops this approach, using a generalised form of Spitzer's identity applied to a three-demensional random walk. This yields a number of results for the general queue GI/G/1, including Smith' solution for the stationary waiting time, which is established under less restrictive conditions that hitherto (§ 5). A soultion is obtained for the busy period distribution in GI/G/1 (§ 7) which can be evaluated when either of the distributions concerned has a rational characteristic function. This solution contains some recent results of Conolly on the quene GI/En/1, as well as well-known results for M/G/1.

Heavy traffic theory for queues with several servers. II

1979 ◽  
Vol 16 (2) ◽  
pp. 393-401 ◽  
Author(s):  
Julian Köllerström
Keyword(s):  
Distribution Function ◽  
Characteristic Function ◽  
Waiting Time ◽  
Error Bounds ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Asymptotic Properties ◽  
Traffic Intensity ◽  
Waiting Time Distribution ◽  
Traffic Theory

The queues being studied here are of the type GI/G/k in statistical equilibrium (with traffic intensity less than one). The exponential limiting formula for the waiting time distribution function in heavy traffic, conjectured by Kingman (1965) and established by Köllerström (1974), is extended here. The asymptotic properties of the moments are investigated as well as further approximations for the characteristic function and error bounds for the limiting foemulae.

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