On invertibility of martingale time changes.(English)Zbl 0646.60051

Stochastic processes, Semin., Princeton/New Jersey 1987, Prog. Probab. Stat. 15, 193-221 (1988).
[For the entire collection see Zbl 0635.00011.]
A well-known theorem of the author states that if $$\{M_ k$$, $$k=1$$,..., $$N\}$$ $$(N=0$$, 1,...) is a set of continuous-path, orthogonal, square- integrable martingales with previsible, quadratic-variation processes $$\{<M_ k>$$, $$k=1$$,..., $$N\}$$ such that $$<M_ k>_ t\to \infty$$, then, defining $\tau_ k(t)=\inf \{s: <M_ k>_ s>t\},\quad B_ k(t)=M_ k(\tau_ k(t))$ is an N-vector of independent Brownian motions and $$M_ k(t)=B_ k(<M_ k>_ t)$$. The present paper considers the case $$<M_ 1>=...=<M_ N>$$ and the question of when $$<M_ k>$$ is a stopping time for the filtration generated by $$B_ 1$$,... $$B_ N$$. In such circumstances the time-transformation $$\tau =\tau_ 1=...=\tau_ N$$ is called invertible; the original process is then in a strong sense a time change of an N-dimensional Brownian motion.
A necessary condition found is that every square integrable martingale of the original filtraccount”.
Reviewer: A.Brezuleanu

MSC:

 60G44 Martingales with continuous parameter 60G17 Sample path properties 60H05 Stochastic integrals 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J65 Brownian motion 60G40 Stopping times; optimal stopping problems; gambling theory 60J25 Continuous-time Markov processes on general state spaces

Zbl 0635.00011