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Grossissement d’une filtration et retournement du temps d’une diffusion. (Enlargement of a filtration and time reversal of a diffusion). (French) Zbl 0607.60042

Probabilités XX, Proc. Sémin., Strasbourg 1984/85, Lect. Notes Math. 1204, 48-55 (1986).
[For the entire collection see Zbl 0593.00014.]
Let \(\{X_ t\), \(t\in [0,1]\}\) be the solution of the following Itô stochastic differential equation: \[ (*)\quad X_ t=X_ 0+\int^{t}_{0}b(s,X_ s)ds+\int^{t}_{0}\sigma (s,X_ s)dW_ s \] and define \(\bar X{}_ t=X_{1-t}\), \(t\in [0,1]\). In this paper, sufficient conditions are given to ensure the existence of a Wiener process \(\{\bar W_ t\), \(t\in [0,1]\}\) and coefficients \(\bar b(t,x)\), \({\bar \sigma}\)(t,x), such that: \[ (**)\quad \bar X_ t=\bar X_ 0+\int^{t}_{0}\bar b(x,X_ s)ds+\int^{t}_{0}{\bar \sigma}(s,X_ s)d\bar W_ s. \] This is related to ”grossissement de filtration” as follows: define \({\mathcal F}^ t=\sigma \{W_ s-W_{1-t};1-t\leq s\leq 1\}\), and \({\mathcal H}^ t={\mathcal F}^ t\vee \sigma (X_ 1)\). Clearly, \(\tilde W_ t=W_{1-t}-W_ 1\) is an \({\mathcal F}^ t\)-Wiener process. If it is an \({\mathcal H}^ t\)-semi-martingale, and if we can give its semi- martingale decomposition, then (**) follows, and we can compute \(\bar b,\) \({\bar \sigma}\) and \(\bar W.\)
Related results by J. Picard appear in the same volume [Une classe de processus stable par retournement du temps pp 56-67 (1986)].

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G44 Martingales with continuous parameter
60J60 Diffusion processes

Citations:

Zbl 0593.00014
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