(The multivariate Itˆo formula and the Girsanov theorem). You are given an Itˆo process Xt,                          where Bt is a Brownian motion under measure P and η t is a process adapted to the...

(The multivariate Itˆo formula and the Girsanov theorem). You are given an Itˆo process Xt,

where Bt is a Brownian motion under measure P and η_t
is a process adapted to the filtration generated by B. There is also another Itˆo process m_t,

(a) Which of the three processes Xt, mt, Bt are martingales under P?

(b) Consider a new process Yt = X2_t
− t. Using the Itˆo formula find the SDE for the process

                           Yt: dYt =

(c) Use the multivariate Itˆo formula to find the SDE of the process mtYt. Is the process mtYt a martingale under P?

                d(mtYt) =

(d) Discuss the significance of your finding in part (c).