A quick question : I want to know if the product of two continuous martingales under a brownian filtration could be a martingale ? I have particularly the product of an exponential martingale (come from a Girsanov changement of probability) and a general martingale (both under a common probability P).
Obviously, I have read carrefully the Did's answer here and the reference he gaves (Alexander Cherny : Some Particular Problems of Martingale Theory).
Again, since we are under a brownian filtration is it possible to get the product of two martingales (a priori non-independant) to be a martingale ? or it is not as stronger as stating the independance of the 2 processes ?
Maybe we can make use the martingale representation theorem and find another condition less stronger ?
Consider the martingales $X_t:=\int_0^t 1_{\{B_s>0\}}\,dB_s$ and $Y_t:=\int_0^t 1_{\{B_s<0\}}\,dB_s$, where $(B_s)$ is a Brownian motion. The covariation of these two martingales is $\int_0^t1_{\{B_s>0\}}1_{\{B_s<0\}}\,ds=0$, so the product $X_tY_t$ is a martingale.