Local characteristics of semi-martingales and changes of probabilities. (Caractéristiques locales des semi-martingales et changements de probabilités.) (French) Zbl 0779.60043

Arnaudon, Marc

Ann. Inst. Henri Poincaré, Probab. Stat. 29, No. 2, 251-267 (1993).

Manifold-valued semi-martingales and the transformations of their local characteristics by change of probabilities are studied. A general Girsanov theorem is given: Let \(F({\mathcal D}\tilde X)={\mathcal B}(X)\alpha\), where \(\alpha\) is a previsible locally bounded process with values in \(T^*V\) over \(X\). Let \(N_ t=\int^ t_ 0\langle\alpha,d X\rangle\), and \(Z={\mathcal E}(N)\) is the stochastic exponent of \(N\). Then \(N\) is a local martingale and if \(Z\) is a uniformly integrable martingale, \(X\) is a \(Q\)-martingale with \(Q=Z\cdot P\), \(Q\) is equivalent to \(P\). Then local characteristics of the product of two Lie group-valued semi-martingales \(X\) and \(X'\), and local characteristics of the projection of \(X\) in a homogeneous space are computed. As an application Girsanov theorem in symmetric space of noncompact type is proved: conditions are given on a diffusion \(X\), \(P\)-martingale in a symmetric space of noncompact type \(G/K\), and a path \(g\) in the Lie group \(G\), for \(g\cdot X\) to be a \(Q\)- martingale for some probability \(Q\) equivalent to \(P\).

Reviewer: M.N.Sviridenko (Moskva)

MSC:

60G44	Martingales with continuous parameter
58J65	Diffusion processes and stochastic analysis on manifolds
53C30	Differential geometry of homogeneous manifolds

Keywords:

semi-martingales; uniformly integrable martingale; local characteristics; Girsanov theorem; Lie group