I would like to obtain an analytic solution to the following Stochastic Differential Equation (SDE) which looks like it could reduce to some standard form but I can't see it. The equation is
$$dX = -\frac{1}{2} X^2 k^2 \; dt \; + \; X k \; dW.\quad\quad\quad\quad (\star)$$ where $dW$ is a Wiener increment (standard Brownian motion) and $k$ is a positive constant. Also $X$ is positive in the circumstances I am considering.
This can be transformed with Ito's Lemma to give an alternative form $$d\log X = -\frac{1}{2} X k^2 \; dt \; + \; k \; dW\quad\quad\quad\quad (\dagger)$$
but neither of these SDEs are linear. Equation $(\dagger)$ has shades of Ornstein-Uhlenbeck except that the LHS is $d\log X$ and not $dX$.
Any analytic solutions or possible approaches to a solution for either of these SDEs would be very gratefully received. Thank you.
[As a side question, is there a source that tables analytic solutions for SDEs in the same way as Gradshteyn and Ryzhik contains tables of integrals? This would be very useful if it exists.]
This looks like the application of an Ito formula, as if $X=f(Y)$ where $dY=\sigma(Y) dW$ so that $$ dX = f'(Y)σ(Y) dW+\tfrac12f''(Y)σ(Y)^2dt. $$ So one would need $$ f'(Y)σ(Y)=kX=kf(Y)\\ f''(Y)σ(Y)^2=-k^2f(Y)^2=-f'(Y)^2σ(Y)^2 $$ This then implies $$ f'(y)=\frac1{c+y}, \\ f(y)=\ln(c+y)+d, \\ σ(y)=k(c+y)[\ln(c+y)+d] $$ So the solution for the given SDE is "reduced" to the "simplified" SDE (fixing the constants to $c=1$, $d=0$ as example) $$ dY=k(1+Y)\ln(1+Y) \, dW, ~~~ X=\ln(1+Y). $$