SDE two problems

Question

I have to solve these two SDEs: $$dX_t=-\frac{1}{1+t}X_tdt+\frac{1}{1+t}dW_t $$ $$dX_t=X_t^3dt+X_t^2dW_t $$ The notes that my professor gave me are pretty theoretic and they have no examples. The only lectures I had on SDE were today and the professor gave these problems for homework that is due tomorrow. Even if I pull an all-nighter I will have about 13-14 hours to learn brand new stuff from confusing notes and to solve these two problems, which is pretty much impossible for me. Any help will be greatly appreciated.

Answer 1

A standard approach is the following: Assume for the moment being that the solution is of the form $$X_t = f(W_t)$$

for some nice function $f$. Applying Itô's formula gives

$$X_t-X_0 = f(W_t)-f(0) = \int_0^t f'(W_s) \, dW_s + \frac{1}{2} \int_0^t f''(W_s) \, ds. \tag{1} $$

Now if we are interested in solving the SDE

$$dX_t = X_t^2 \, dW_t + X_t^3 \, dt \tag{2}$$

this means that we would like to find a function $f$ satisfying

$$f'(y) = f(y)^2 \quad \text{and} \quad \frac{1}{2} f''(y) = f(y)^3. \tag{3}$$

(If $f$ satisfies both equations, you can plug them into $(1)$ and you will easily see that $X_t = f(W_t)$ satisfies $(2)$.) Solve first the equation $f'(y) = f(y)^2$ and then verify that the solution $f$ satisfies also the second ODE.

---

In order to solve the SDE

$$dX_t = - \frac{1}{1+t} X_t \, dt + \frac{1}{1+t} \, dW_t$$

we can use a similar approach; however (because of the time dependence of the coefficients) we have to use the more general Ansatz

$$X_t = g(t,W_t).$$

If you apply Itô's formula, you will see that we have to find a function $g$ satisfying

$$\partial_x g(t,x) = \frac{1}{1+t} \quad \text{and} \quad - \frac{1}{1+t} g(t,x) = \frac{1}{2} \partial_{xx} g(t,x) + \partial_t g(t,x).$$

Solve first the equation on the left-hand side and then check whether the solution satisfies the ODE on the right hand side.

Answer 2

For the first equation consider that $$ d((1+t)X_t)=(1+t)dX_t+X_tdt $$ for the second apply the Ito formula for $Y=f(X)=X^{-2}$. This should reduce the problems to easily integrable terms.