Martingale and local martingales

Question

I have to show that $e^{B_t^1}\cos(B_t^2)$ is a martingale ($B=(B^1,B^2)$ is a two-dimensional Brownian Motion). I used Ito's formula and got $e^{B_t^1}\cos(B_t^2)=1+\int_0^t e^{B_s^1}\cos(B_s^2)dB_s^1-\int_0^t e^{B_s^1}\sin(B_t^2)dBs^2$. From the right side I know that these are local martingales, but are they martingales, and how can I show that? Do I have to show that it is bounded? Please help me!!!

Answer 1

Apply Ito's formula for: $$ f(x_1,x_2)=e^{x_1}\cos(x_2) $$ $$ f_{x_1}(x_1,x_2)=e^{x_1}\cos(x_2)\quad f_{x_2}(x_1,x_2)=-e^{x_1}\sin(x_2) $$ $$ f_{x_1x_1}(x_1,x_2)=e^{x_1}\cos(x_2)\quad f_{x_1x_2}(x_1,x_2)=-e^{x_1}\sin(x_2) $$ $$ f_{x_2x_2}(x_1,x_2)=e^{x_1}\cos(x_2) $$ Then: $$ \mathrm{d}f(B_1,B_2)=f_{x_1}(B_1,B_2)\mathrm{d}B_1+f_{x_2}(B_1,B_2)\mathrm{d}B_2+ $$ $$ +\frac{f_{x_1x_1}(B_1,B_2)+2{f_{x_1x_2}(B_1,B_2)}+{f_{x_2x_2}(B_1,B_2)}{}}{2}=... $$