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Stochastic calculus of order 2 and anticipative differential equations on a manifold. (Calcul stochastique d’ordre deux et équation différentielle anticipative sur une variété.) (French) Zbl 0849.60057

The paper is the sequel of the authors’ paper [Stochastics Stochastics Rep. 42, No. 1, 1-23 (1993; Zbl 0811.60041)]. The authors develop a stochastic calculus which allows to integrate anticipative processes taking their values in the space of second-order 1-forms that are above a manifold valued anticipating process. Let \(V\) be a Riemannian smooth manifold, let \(W\) be a \(d\)-dimensional Brownian motion, and \((\Omega, {\mathcal F}, ({\mathcal F}_t, t \in [0;1]), \mathbb{P})\), be the canonical filtered probability space. Let \({\mathcal S}\) be the set of simple real Wiener functionals, \(D\) be the usual stochastic gradient. For \(p > 1\) and \(q \in \mathbb{N}\), \(\mathbb{D}_{p,q}\) denotes the associated Sobolev space. Let \(X\) be a \(V\)-valued process, anticipative w.r.t. \(W\); the key tool is the bracket \(\{X,X\}\) which is a second order vector process (in the sense of P.-A. Meyer) above \(X\). \(\{X,X\}\) generalizes the bracket \([M,M]\) of an adapted semi-martingale \(M\). Using \(\{X,X\}\) the authors define a derivation process \((D_s^XF)_{s \in [0;1]}\) for \(F \in {\mathcal S}\), and, by a duality argument, a Skorokhod integral of second-order 1-forms process \(\alpha\) above \(X : \delta^X (\alpha) = \int^1_0 \alpha_s dX_s\). A Stratonovich integral is also built: \(S^X (\alpha) = \int^1_0 \alpha_s \circ dX_s\), which is used to prove the existence and uniqueness of solution of an anticipative stochastic differential equation \(dX_t = e(B_t, X_t) \circ dB_t\), \(t \in [0,1]\) and \(X_0 \in \mathbb{D}_{4,1} (V)\) a \(V\)-valued anticipative random variable.

MSC:

60H07 Stochastic calculus of variations and the Malliavin calculus
58J65 Diffusion processes and stochastic analysis on manifolds

Citations:

Zbl 0811.60041
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