I have the equation:

F(B,S)=F(0,0) + IFs(B,S) dS + IFb(B,S) dB,

where I denotes the integral between 0 and t, B is a standard brownian motion, and S is the maximum of the brownian motion between 0 and t, Fs and Fb are the derivatives of F wrt s and b.

I have to show that if Fs=0 for B=S, then F is a continuous local martingale.

any idea how to show this?