On decomposition of continuous submartingales

K. È. Dambis

Teoriâ veroâtnostej i ee primeneniâ, Tome 10 (1965) no. 3, pp. 438-448

Résumé

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.