|
Teoriya Veroyatnostei i ee Primeneniya, 1965, Volume 10, Issue 3, Pages 438–448
(Mi tvp540)
|
|
|
|
This article is cited in 69 scientific papers (total in 69 papers)
On decomposition of continuous submartingales
K. È. Dambis Moscow
Abstract:
Doob (see [2], p. 267) proved that every discrete parameter submartingale $X=(x_n,\mathfrak F_n)$, $1\le n<\infty$, may be decomposed as a sum $X=\Psi+\Gamma$ where $\Psi=(\psi_n,\mathfrak F_n)$ is a non-decreasing process and $\Gamma=(\gamma_n,\mathfrak F_n)$ is a martingale. Meyer (see [4], p. 199) found necessary and sufficient conditions for a right continuous submartingale $X=(x_t,\mathfrak F_t)$, $0\le t<\infty$ to have Doob's decomposition. In the present paper a generalisation of Doob's decomposition is obtained which is applicable to every continuous submartingale. The second main result of this paper consist in the fact that every continuous martingale $X=(x_t,\mathfrak F_t)$, $0\le t<\infty$, with $X_0=0$ has an equivalent one $X'=(x'_t,\mathfrak F'_t)$, $0\le t<\infty$, which may be obtained from some Wiener process by means of a continuous random time change. Finally we prove that sample functions of a continuous submartingale (martingale) either have infinite variation or nondecrease (are constant) on every interval.
Received: 04.02.1964
Citation:
K. È. Dambis, “On decomposition of continuous submartingales”, Teor. Veroyatnost. i Primenen., 10:3 (1965), 438–448; Theory Probab. Appl., 10:3 (1965), 401–410
Linking options:
https://www.mathnet.ru/eng/tvp540 https://www.mathnet.ru/eng/tvp/v10/i3/p438
|
Statistics & downloads: |
Abstract page: | 449 | Full-text PDF : | 328 | First page: | 4 |
|