Ito unique SDE - Ch10

Hi,

Let us have two representations of dM_t

1st: dM_t = a(t,M_t) dt + b(t,M_t) dW_t
2nd: dM_t = c(t,M_t) dt + d(t,M_t) dW_t

It seems that in normal cases a=c, b=d should hold (I think this argument was used in Chapter 10, p 35 in the CMP, to argue as follows: the drift term is always zero for martingales hence the complicated formula at dt must equal to zero). My question is: why formulae a=c and b=d hold?

Note that this principle is used also in the solution of question 10.5.

Thanks in advance for your help.