非线性随机微分方程（nonlinear stochastic differential equation）求解咨询

Hey there! Let's tackle this nonlinear SDE together—looks like you're on the right track with Ito's formula, you just needed a strategic substitution to simplify those tricky nonlinear terms.

First, take a look at the structure of your equation: both the drift and diffusion terms share a 1 + X_t² factor. That's a huge clue! The arctangent function is perfect here because its derivative is 1/(1+x²)—exactly what we need to cancel out that common denominator-like term.

Let's define a new process:
Y_t = arctan(X_t)

Now apply Ito's formula to Y_t. First, compute the necessary derivatives:

• First derivative: $Y'(x) = \frac{1}{1+x²}$
• Second derivative: $Y''(x) = \frac{-2x}{(1+x²)^2}$

Plug these into the standard Ito formula:
$$dY_t = Y'(X_t)dX_t + \frac{1}{2}Y''(X_t)(dX_t)^2$$

Now substitute your original SDE $dX_t = X_t(1+X_t²)dt + (1+X_t²)dW_t$ into this equation, and break it down term by term:

1. The first term $Y'(X_t)dX_t$ simplifies nicely:
   $$\frac{1}{1+X_t²}\left[X_t(1+X_t²)dt + (1+X_t²)dW_t\right] = X_t dt + dW_t$$
2. Next, calculate $(dX_t)^2$—remember that in stochastic calculus, $(dW_t)^2 = dt$, while all other cross terms (like $dt \cdot dW_t$ or $dt^2$) vanish:
   $$(dX_t)^2 = (1+X_t²)^2 dt$$
3. The second term $\frac{1}{2}Y''(X_t)(dX_t)^2$ becomes:
   $$\frac{1}{2} \cdot \frac{-2X_t}{(1+X_t²)^2} \cdot (1+X_t²)^2 dt = -X_t dt$$

Now add these two results together—you'll notice the $X_t dt$ terms cancel each other out completely:
$$dY_t = X_t dt + dW_t - X_t dt = dW_t$$

That's a straightforward Brownian motion! Integrating both sides gives:
$$Y_t = Y_0 + W_t$$

Substitute back $Y_t = arctan(X_t)$ and solve for $X_t$ by taking the tangent of both sides:
$$X_t = \tan\left(\arctan(X_0) + W_t\right)$$

And that's your explicit solution! The key was spotting that substitution to turn the nonlinear equation into a simple one we already know how to solve. Hope this clears up where to start—happy solving!

备注：内容来源于stack exchange，提问作者bobby shmurda