## 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

Zbl 0722.00030
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