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On the Azéma martingales. (English) Zbl 0753.60045

Séminaire de probabilités XXIII, Lect. Notes Math. 1372, 66-87 (1989).
[For the entire collection see Zbl 0722.00030.]
The first part of this paper introduces what is called a structure equation \[ d[X,X]_ t=dt+\Phi_ tdx_ t, \tag{SE} \] where \(X\), the solution to the equation for given \(\Phi\), is a martingale, \(\Phi\) is a predictable process, and \(\int\Phi dX\) is a martingale. Properties of solutions \(X\) to SEs are studied, examples are given, and both existence and uniqueness questions are addressed. The second part of the paper analyzes solutions to a specific class of SEs, \(d[X,X]_ t=dt+(a+bX_{t-})dt\), where \(a\) and \(b\) are constants, \(b\neq 0\). When \(a=0\) and \(b=-1\), a solution is the martingale first obtained by Azéma, \(X_ t=(\text{sign} B_ t)\sqrt{2(t-G_ t)}\), where \(B_ t\) is Brownian motion and \(G_ t\) is the last time up until \(t\) when \(B_ t=0\). One major result (Proposition 6) is: a) If \(a=0\) and \(b\leq 0\), the solution is unique in law and is a strong Markov process; b) if \(-2\leq b\leq 0\), the solution has the chaotic representation property.

MSC:

60G44 Martingales with continuous parameter
60H05 Stochastic integrals

Citations:

Zbl 0722.00030
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