Hostname: page-component-76dd75c94c-h9cmj Total loading time: 0 Render date: 2024-04-30T07:44:01.467Z Has data issue: false hasContentIssue false
  • English
  • Français

On Wald's equations in continuous time

Published online by Cambridge University Press:  14 July 2016

W. J. Hall
W. J. Hall*
Affiliation:
Stanford University and University of North Carolina
Get access Rights & Permissions [Opens in a new window]

Extract

Various formulas of Wald relating to randomly stopped sums have well known continuous-time analogs, holding in particular for Wiener processes. However, sufficiently general forms of most of these do not appear explicitly in the literature. Recent papers by Robbins and Samuel (1966) and by Brown (1969) provide general results on Wald's equations in discrete time and these are here extended (Theorems 2 and 3) to Wiener processes and other homogeneous additive processes, that is, continuous-time processes with stationary independent increments. We also give an inequality (Theorem 1) related to Wald's identity in continuous time, and we derive, as corollaries of Wald's equations, bounds on the variance of an arbitrary stopping time. The Wiener process versions of these results find application in a variety of stopping problems. Specifically, all are used in Hall ((1968), (1969)); see also Bechhofer, Kiefer, and Sobel (1968), Root (1969), and Shepp (1967).

Type
Research Papers
Information
Journal of Applied Probability , Volume 7 , Issue 1 , April 1970 , pp. 59 - 68
Copyright
Copyright © Applied Probability Trust 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bechhofer, R. E., Kiefer, J. and Sobel, M. (1968) Sequential Identification and Ranking Procedures. University of Chicago Press, Chicago.Google Scholar
Brown, B. M. (1969) Moments of a stopping rule related to the central limit theorem. Ann. Math. Statist. 40, 1236–1249.CrossRefGoogle Scholar
Burkholder, D. L. (1966) Martingale transforms. Ann. Math. Statist. 37, 1494–1504.CrossRefGoogle Scholar
Chow, Y. S., Robbins, H. and Teicher, H. (1965) Moments of randomly stopped sums. Ann. Math. Statist. 36, 789–799.CrossRefGoogle Scholar
Chow, Y. S. and Teicher, H. (1966) On second moments of stopping rules. Ann. Math. Statist. 37, 388–392.CrossRefGoogle Scholar
Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Hall, W. J. (1968) On the Skorokhod embedding theorem. Stanford Univ., Dept. of Stat. Techn. Report No. 33.Google Scholar
Hall, W. J. (1969) Embedding submartingales in Wiener processes with drift, with applications to sequential analysis. J. Appl. Prob. 6, 612–632.CrossRefGoogle Scholar
Millar, P. W. (1968) Martingale integrals. Trans. Amer. Math. Soc. 133, 145–166.CrossRefGoogle Scholar
Robbins, H. and Samuel, E. (1966) An extension of a lemma of Wald. J. Appl. Prob. 3, 272–273.CrossRefGoogle Scholar
Root, D. H. (1969) The existence of certain stopping times on Brownian motion. Ann. Math. Statist. 40, 715–718.CrossRefGoogle Scholar
Shepp, L. A. (1967) A first passage problem for the Wiener process. Ann. Math. Statist. 38, 1912–1914.CrossRefGoogle Scholar
Shepp, L. A. (1969) Explicit solutions to some problems of optimal stopping. Ann. Math. Statist. 40, 993–1010.CrossRefGoogle Scholar