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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

Citations:

Zbl 0635.00011