Synonymity and processes with locally integrable variation. (English) Zbl 0647.60040

Licea, Gabriela

An. Univ. Bucur., Mat. 37, No. 1, 21-24 (1988).

It is proved that any process with locally integrable variation may be represented in a unique manner as \(X=X^ c+B+C+D+E\), where all processes in the right side are of locally integrable variation, \(X^ c\) is the continuous part of X, B is continuous, C and E are purely discontinuous, C being predictable, D and E are local martingales, the jumps of D being supported by a sequence of totally inaccessible stopping times and those of E by a sequence of predictable stopping times.
It is proved that if Y is a process synonymous to X [according to e.g.: D. N. Hoover, Ann. Probab. 12, 703-713 (1984; Zbl 0545.60040)] and the above decomposition for Y is \(Y=Y^ c+B'+B'C'+D'+E'\), then \((X^ c,B,C,D,E)\) and \((Y^ c,B',C',D',E')\) have the same probability law.

MSC:

60G07

General theory of stochastic processes

Keywords:

unique decomposition; sequence of totally inaccessible stopping times; predictable stopping times

Citations:

Zbl 0545.60040