2.6. Use Ito’s formula to show that the following are the solutions to the SDEs you solved numerically in Exercise 1.7:

1. X = (1+t)2(X0+t+W(t)) satisfies dX =

2X 1+t

+(1+t)2 dt+(1+t)2dW;

2. X = (1+W(t))2 satisfies dX = dt+2

√

XdW with X0 = 1 ;

3. X = log(1+W(t)) satisfies dX=−12 e−2Xdt+e−XdW with X0 = 0 ;

4. X = sinh(t+W(t)) satisfies dX =



12 X+



1+X2



dt+



1+X2dW with X0 = 0 ; and 5. Z = 1/(1+W(t)) satisfies dZ = Z3dt−Z2dW with Z0 = 1 .

What features that you saw in the numerical solutions are explained by these analytic solutions?