Let $B_t$ be a standard Wiener motion, $I_t=\int_0^t|B_s|^2\!\text{ds}\ $and $S_t=\max_{0\leq s\leq t}B_s$. Let also $F:\mathbb{R}^2_+\times\mathbb{R}\times\mathbb{R}_+\rightarrow\mathbb{R}$ a $C^{1,2,1}$ function.
Show that if:$F_t(t,a,x,s)+x^2F_a(t,a,x,s)+\frac{1}{2}F_{xx}(t,a,x,s)=0$ for every $(t,a,x,s)\in\mathbb{R}^2_+\times\mathbb{R}\times\mathbb{R}_+$ and $F_s(t,a,x,s)=0$ for $x=s$,
then $F(t,I_t,B_t,S_t)$ is a continuous local martingale for $t\geq0$.
What I have done:
Since $I_t$ and $S_t$ are increasing processes $\Longrightarrow\ I_t, S_t$ are processes of bounded variation. Hence, we can apply Ito's formula. We have that:
$F(t,I_t,B_t,S_t)=F(0,I_0,B_0,S_0)+\int_0^tF_t(r,I_r,B_r,S_r)\text{dr}+\int_0^tF_a(r,I_r,B_r,S_r)\text{d}I_r+\\ +\int_0^tF_x(r,I_r,B_r,S_r)\text{d}B_r+\int_0^tF_s(r,I_r,B_r,S_r)\text{d}S_r+ \frac{1}{2}\int_0^tF_{xx}(r,I_r,B_r,S_r)\text{d}\left_r=\\$
$=F(0,I_0,B_0,S_0)+\int_0^tF_t(r,I_r,B_r,S_r)\text{dr} +\int_0^tF_a(r,I_r,B_r,S_r)\text{d}I_r+\\ +\int_0^tF_x(r,I_r,B_r,S_r)\text{d}B_r+\int_0^tF_s(r,I_r,B_r,S_r)\text{d}S_r +\frac{1}{2}\int_0^tF_{xx}(r,I_r,B_r,S_r)\text{dr}$
In order to show that $F(t,I_t,B_t,S_t)$ is a continuous local martingale, we need to show that:
$\int_0^tF_t(r,I_r,B_r,S_r)\text{dr}+ \int_0^tF_a(r,I_r,B_r,S_r)\text{d}I_r+\int_0^tF_s(r,I_r,B_r,S_r)\text{d}S_r+\\ +\frac{1}{2}\int_0^tF_{xx}(r,I_r,B_r,S_r)\text{dr}=0$
What can we do with the terms $\int_0^tF_s(r,I_r,B_r,S_r)\text{d}S_r$ and $\int_0^tF_a(r,I_r,B_r,S_r)\text{d}I_r$? How will we relate them to the given relation?