Show that if
- \[I(t,\omega)=\int_0^t \sigma(s,\omega) dB(s,\omega)\]is a stochastic integral then \[I^2(t)-\int_0^t \sigma^2(s)ds\] is a martingale.
- What equation must \(u(t,x)\) satisfy so that
\[ t \mapsto u(t,B(t))e^{\int_0^t V(B(s))ds} \]
is a martingale? Here \(V\) is a bounded function. Hint: Set \(Y(t)=\int_0^t V(B(s))ds\) and apply It\^0’s formula to \(Z(t,B(t),Y(t))=u(t,B(t))\exp(Y(t))\).
