Let S_t be the price of Stock and I_t be the index at timet, respectively.

Consider the following model:

lnS(i+1)t=a+blnS(i)t+(i+1)t(1)

lnI(i+1)t=c+dlnI(i)t+(i+1)t(2)

(i+1)t(1)N(0,12)i.i.d.

(i+1)t(2)N(0,22)i.i.d.

Corr(it(1),it(2))=

If b = d = 1 and(it(1),it(2)) follows bivariate normal distribution, what is the joint distribution of(lnS(i+1)t,lnI(i+1)t) conditional on (lnS(i)t,lnI(i)t). Write down the Stochastic Differential Equations that governs S_t and I_t jointly such thatlnS(i+1)t andlnI(i+1)t have the same joint distribution determined formerly. Use the parameters provided in the question and prove your claim.