Let St be the price at time t of a risky asset. Suppose that St obeys a stochastic differential equation of the form dSt= tStdt + tStdWt where tand tare continuous but nonrandom functions of time t. Suppose also that the market has a riskless asset whose price Bt obeys the ordinary differential equation dBt= rtBtdt where the short rate rt is again a continuous but nonrandom function of t. Solve the differential equations for Stand Bt Prove that, under any risk-neutral measure t= rt(Hint: Find the appropriate Radon-Nikodym process and use Girsanov theorem)

Problem 0.2. Let S be the price at time t of a risky asset. Suppose that St obeys a stochastic differential equation of the form dSt = pt Sidt+oSidW+ where it and Ot are continuous but nonrandom functions of time t. Suppose also that the market has a riskless asset whose price B obeys the ordinary differential equation dB4 =1+Bidt where the short rate rt is again a continuous but nonrandom function of t. Solve the differential equations for St and Bt Prove that, under any risk-neutral measure plz = (Hint: Find the ap- propriate Radon-Nikodym process and use Girsanov theorem) Problem 0.2. Let S be the price at time t of a risky asset. Suppose that St obeys a stochastic differential equation of the form dSt = pt Sidt+oSidW+ where it and Ot are continuous but nonrandom functions of time t. Suppose also that the market has a riskless asset whose price B obeys the ordinary differential equation dB4 =1+Bidt where the short rate rt is again a continuous but nonrandom function of t. Solve the differential equations for St and Bt Prove that, under any risk-neutral measure plz = (Hint: Find the ap- propriate Radon-Nikodym process and use Girsanov theorem)