Solving an SDE by using Ito's formula

Question

I am having trouble starting this problem.

Consider the stochastic differential equation:

$$dX(t) = (1/4)dt + \sqrt{X(t)}dW(t) $$

1. Solve the equation by applying Ito’s lemma to the function $f(x)=\sqrt{x}$

Answer 1

We have that

$$f'(x)=\frac{1}{2}x^{-\frac{1}{2}}$$ $$f''(x)=-\frac{1}{4}x^{-\frac{3}{2}}$$

We define the process $Y_t=\sqrt{X_t}$, and we have $$dY_t=f'(X_t)dX_t+\frac{1}{2}f''(X_t)d\langle X,X\rangle_t$$

or $$dY_t=\frac{1}{2}{X_t}^{-\frac{1}{2}}dX_t-\frac{1}{2}\frac{1}{4}X_t^{-\frac{3}{2}}d\langle X,X\rangle_t$$ or $$dY_t=\frac{1}{2}{X_t}^{-\frac{1}{2}}dX_t-\frac{1}{2}\frac{1}{4}{X_t}^{-\frac{3}{2}}(X_tdt)$$

Finally ,

$$dY_t={\frac{1}{2}{X_t}^{-\frac{1}{2}}dX_t-\frac{1}{8}{X_t}^{-\frac{1}{2}}}$$ $$={\frac{1}{2}{X_t}^{-\frac{1}{2}}((1/4)dt + \sqrt{X(t)}dW(t))-\frac{1}{8}{X_t}^{-\frac{1}{2}}}$$

We have $$Y_t=Y_0+\frac{1}{2}W_t$$

thus $$\sqrt{X(t)}=\sqrt{X(0)}+\frac{1}{2}W_t$$