Solving non linear SDE

Question

I am a little stuck in finding the solution of a non linear SDE. Hope you can help me out. The SDE has the form $dX_t = X_t^2 dt + dB_t$, where $B$ is a Brownian motion. Assuming $f(t,B_t) = X_t$ and using Ito doesnt do the trick. Thanks in advance!

Answer 1

We can re-write the SDE as $$ d(X(t)-B(t)) = X(t)^2 dt, $$ that is, letting $Y(t) \doteq X(t)-B(t)$, $$ dY(t) = (Y(t)+B(t))^2 \, dt $$ i.e., a random ODE that can be written as $$ Y'(t) = Y(t)^2 + 2 B(t) \, Y(t) + B(t)^2. $$ Now, you can find a solution to such equation (it is a Riccati equation) and thus find $X(t)$ by the relation $$ X(t) = Y(t) + B(t). $$