showing a process martingle from ito's lemma..

Question

**by ito formula, Ft: filtratoin

$M(t) = (aB(t) - t) \exp( 2B(t) - 2t ) $

find constant a for $M(t)$ to be a martingale

plz help!**

Answer 1

Let $ X_t=aB(t)-t$ and $Y_t=\exp(2B_t-2t)$, we have $$dX_t=-dt+adB_t$$ and $$dY_t=\underbrace{\left(-2e^{2B_t-2t}+\frac{1}{2}(2)^2e^{2B_t-2t}\right)}_{0}dt+2\,e^{2B_t-2t}\,dB_t=2\,e^{2B_t-2t}\,dB_t$$ $$d(X_tY_t)=Y_tdX_t+X_tdY_t+d[X_t,Y_t]$$ as a result $$d(X_tY_t)=e^{2B_t-2t}(-dt+adB_t)+(aB(t)-t)2e^{2B_t-2t}\,dB_t+2a\,e^{2B_t-2t}dt$$ thus $$d(X_tY_t)=(-1+2a)e^{2B_t-2t}dt+(a-2t+2a\,B_t)e^{2B_t-2t}dB_t$$ $$-1+2a=0\implies a=\frac{1}{2}$$