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Manifold-valued martingales, changes of probabilities, and smoothness of finely harmonic maps. (English) Zbl 0946.60030

The main object of this paper is to study, on a manifold \(W\) (Riemannian or not), the variations of continuous martingales with prescribed limit at infinity under a change of probability. The latter is obtained via a real-valued local martingale \(M\) and the Doléans exponential \(E(M)\). If \(Y\) is a given martingale with fixed terminal value \(L\) and taking its values in a compact convex subset of \(W\) with \(p\)-convex geometry (i.e where a Riemannian distance is given, equivalent to the \(1/p\) exponent of a convex function, smooth outside the diagonal and vanishing precisely on the diagonal), if \(Y(M)\) is a semimartingale with the same terminal value \(L\) and obtained from \(Y\) via \(E(M)\), the authors roughly show that the \(L^{qp}\) norm of the uniform distance between \(Y\) and \(Y(M)\) can be controlled by the \(L^r\) norm of the predictable quadratic variation of \(M\), for any \(r\in(1,2)\) and \(q\in(1,r)\). This lemma (where the notion of \(p\)-convex geometry plays a key rôle) allows them to show their main (and difficult) theorem: the differentiability at \(M = 0\) of the initial-value map \(M\to Y_0(M)\). An expression of the derivative is also given in terms of the stochastic parallel transport along \(Y.(0)\). As a principal corollary, the authors give a different proof of the smoothness of continuous finely harmonic maps (which transform Brownian motion into a continuous martingale when the starting manifold is Riemannian) which was first obtained by W. S. Kendall [J. Funct. Anal. 126, No. 1, 228-257 (1994; Zbl 0808.60058)] via coupling methods.

MSC:

60G44 Martingales with continuous parameter
31C05 Harmonic, subharmonic, superharmonic functions on other spaces

Citations:

Zbl 0808.60058
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