On the construction and distribution of a local martingale with a given absolute value

Perkins, Edwin

doi:10.1090/S0002-9947-1982-0648092-2

On the construction and distribution of a local martingale with a given absolute value
HTML articles powered by AMS MathViewer

by Edwin Perkins

Trans. Amer. Math. Soc. 271 (1982), 261-281

DOI: https://doi.org/10.1090/S0002-9947-1982-0648092-2

PDF | Request permission

Abstract:

A local martingale is constructed on an appropriate Loeb space whose absolute value equals a given nonnegative local submartingale. Nonstandard analysis is used to reduce the problem to the discrete time setting where the original construction of D. Gilat is fairly simple. This approach has the advantage of allowing explicit computations. In particular, the distribution of the local martingale is described in terms of the Doob-Meyer decomposition of the original local submartingale.

References

A concept of weak convergence for stochastic processes viewed in the Strasbourg manner

Robert M. Anderson, A non-standard representation for Brownian motion and Itô integration, Israel J. Math. 25 (1976), no. 1-2, 15–46. MR 464380, DOI 10.1007/BF02756559
M. T. Barlow, Construction of a martingale with given absolute value, Ann. Probab. 9 (1981), no. 2, 314–320. MR 606994
M. T. Barlow and M. Yor, Sur la construction d’une martingale continue, de valeur absolue donnée, Seminar on Probability, XIV (Paris, 1978/1979) Lecture Notes in Math., vol. 784, Springer, Berlin, 1980, pp. 62–75 (French). MR 580109
Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
David Gilat, Every nonnegative submartingale is the absolute value of a martingale, Ann. Probability 5 (1977), no. 3, 475–481. MR 433586, DOI 10.1214/aop/1176995809
Douglas N. Hoover and Edwin Perkins, Nonstandard construction of the stochastic integral and applications to stochastic differential equations. I, II, Trans. Amer. Math. Soc. 275 (1983), no. 1, 1–36, 37–58. MR 678335, DOI 10.1090/S0002-9947-1983-0678335-1
H. Jerome Keisler, An infinitesimal approach to stochastic analysis, Mem. Amer. Math. Soc. 48 (1984), no. 297, x+184. MR 732752, DOI 10.1090/memo/0297
Peter A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113–122. MR 390154, DOI 10.1090/S0002-9947-1975-0390154-8
Peter A. Loeb, An introduction to nonstandard analysis and hyperfinite probability theory, Probabilistic analysis and related topics, Vol. 2, Academic Press, New York-London, 1979, pp. 105–142. MR 556680
B. Maisonneuve, Martingales de valeur absolue donnée, d’après Protter et Sharpe, Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78) Lecture Notes in Math., vol. 721, Springer, Berlin, 1979, pp. 642–645 (French, with English summary). MR 544834
Edwin Perkins, On the uniqueness of a local martingale with a given absolute value, Z. Wahrsch. Verw. Gebiete 56 (1981), no. 2, 255–281. MR 618275, DOI 10.1007/BF00535744
P. Protter and M. J. Sharpe, Martingales with given absolute value, Ann. Probab. 7 (1979), no. 6, 1056–1058. MR 548900
Charles Stone, Weak convergence of stochastic processes defined on semi-infinite time intervals, Proc. Amer. Math. Soc. 14 (1963), 694–696. MR 153046, DOI 10.1090/S0002-9939-1963-0153046-2
Douglas N. Hoover and H. Jerome Keisler, Adapted probability distributions, Trans. Amer. Math. Soc. 286 (1984), no. 1, 159–201. MR 756035, DOI 10.1090/S0002-9947-1984-0756035-8