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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!

user2736062
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    Do you know that a solution exists? Actually, I believe that the solution explodes... – saz Mar 03 '17 at 20:21
  • Hello, I know that a solution exists. Does this already ensure we can write $X_t$ explicitly? I am searching for a systematic approach to solve this problem. Unfortunately i didnt find any SDE of this type in the literature. Why do you think the SDE explodes? – user2736062 Mar 04 '17 at 12:24

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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 $$ X(t) = Y(t) + B(t). $$

Arthur11
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  • May I ask you to complete the solution ? (or ) at least next step – Khosrotash Jun 19 '17 at 12:26
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    Is it correct to continue naively by writing $y'=(y+B)^2$ and treating this like a non-stochastic separable ODE, yielding $y(t) = -(B(t)-\frac{1}{t-t_o})$. This would indicate that indeed at $t=t_o$ the solution blows up as the comment by saz said. – Cyclone Nov 14 '17 at 22:52