Two-dimensional Azéma martingales. (Martingales d’Azéma bidimensionnelles.) (French) Zbl 0863.60045

Attal, S.; Émery, M.

Hommage à P. A. Meyer et J. Neveu. Paris: Société Mathématique de France, Astérisque. 236, 9-21 (1996).

The authors define two-dimensional Azéma martingales as processes \(X=(X^1,X^2)\) verifying a structure equation: \[ d[X^i,X^j]_t=\delta^{i,j}dt+\sum_{k=1}^2 f_k^{i,j}(X_{t-})dX_t^k, \] where \(f_k^{i,j}\) are affine functions such that almost surely, \(f_k^{i,j}(X_{t-})\) (resp. \(\displaystyle\sum_m f_m^{i,j}(X_{t-})f_l^{m,k}(X_{t-}))\) is symmetric with respect to \(i,j,k\) (resp. \(i,j,k,l\)). It is shown that: \[ f_1^{1,1}(X_{t-})=p(X_{t-}),\quad f_2^{2,2}(X_{t-})=q(X_{t-}),\quad f_2^{1,1}(X_{t-})=f_1 ^{1,2}(X_{t-})=r(X_{t-}), \]
\[ f_2^{1,2}(X_{t-})=f_1^{2,2}(X_{t-})=s(X_{t-}), \] \(p,q,r,s\) being affine functions verifying \(ps+qr=r^2+s^2\). Moreover, the authors characterize functions \(p,q,r,s\) of previous type. There exist exactly three explicit classes, roughly speaking, type (I) corresponds to \(r=s=0\), type (II) reduces to \(r-q=s=0\) and type (III) to \(p(x,y)=-x+ay\), \(q(x,y)=bx-y\), \(r(x,y)=-y\), \(s(x,y)=-x\). If \(r=s=0\), \(p(x,y)=ax+by+c\) and \(q(x,y)=\alpha x+\beta y+\gamma\), \(-2\leq a<0\), \(-2\leq \beta < 0\), \(0< b\alpha<a\beta\) and \(X_0\) is bounded, then the corresponding Azéma martingale is bounded on finite intervals and has the chaotic representation property.
For the entire collection see [Zbl 0847.00013].

Reviewer: P.Vallois (Nancy)

MSC:

60G44

Martingales with continuous parameter

Keywords:

two-dimensional Azéma martingales; chaotic representation property