Solve $dX_t = (\sqrt{1+X_t^2} + \frac{1}{2}X_t) \, dt + \sqrt{1+X_t^2} \, dW_t$ explicitly

Question

Solve explicitly the 1-dimensional equation:

$dX_t = (\sqrt{1+X_t^2} + \frac{1}{2}X_t)dt + \sqrt{1+X_t^2}dW_t$

I have hopelessly been guessing solutions to this. Does anyone know how to solve this or, more importantly, know a good general approach for attacking something like this?

Thanks

Answer 1

Hint: Apply Itô's formula to $$Z_t := \text{arsinh} \, X_t = \int_0^{X_t} \frac{1}{\sqrt{1+y^2}} \, dy.$$

General approach: The given SDE is an autonomous, i.e. the coefficients do not depend explicitly on the time:

$$dX_t = b(X_t) \, dt + \sigma(X_t) \, dW_t. \tag{1}$$

In some cases, these SDEs can be transformed into linear SDEs - and since linear SDEs can be solved analytically, this solves the problem. The only possible transformations are the following ones:

1. $Z_t := f(X_t)$ for $f(x) := \int^x \frac{1}{\sigma(y)} \, dy$.
2. $Z_t := e^{\delta f(x)}$ for $f(x) := \int^x \frac{1}{\sigma(y)} \, dy$ and a constant $c>0$.

In fact, there are conditions (in terms of the derivatives of $b,\sigma$) which can be checked in order to decide whether $(1)$ can be transformed into a linear SDE (and, if so, whether one should use the first or the second transformation). See e.g. René L. Schilling/Lothar Partzsch: Brownian Motion - An Introduction to Stochastic Processes.

Another approach is the following: First solve the determinstic ordinary differential equation

$$dx_t = b(x_t) \, dt$$

and then find a solution of $(1)$ by using a variation-of-constants-approach. See e.g. this answer for an example.