How to Solve a Truncated Lognormal PDF SDE?

[hydroviolante]

Hi,
I'm trying to solve this SDE;
the pdf of r is known (truncated lognormal)

\(\displaystyle -\d{z}{t}=\frac{r^2}{\alpha}\cdot\frac{\beta z-\theta/r}{z}\)

Please help me!

 

[Valkarie]

To solve this SDE, we can use the method of separation of variables. Let z(t) = xy(t). Then the SDE can be written as: -\frac{dx}{dt}=\frac{r^2}{\alpha}\cdot\frac{\beta xy -\theta/r}{xy}-\frac{dy}{dt}=\frac{r^2}{\alpha}\cdot\frac{\beta y -\theta/rx}{y}The solution to the first equation is:x = Ce^{-\frac{r^2}{\alpha}\cdot\left(\beta \int y~dt + \frac{\theta}{r}\right)}where C is an arbitrary constant. The solution to the second equation is:y = Ae^{-\frac{r^2}{\alpha}\cdot\beta t} + \frac{\theta}{r\alpha}where A is an arbitrary constant. Thus the solution of the SDE is:z(t) = Ae^{-\frac{r^2}{\alpha}\cdot\left(\beta t + \frac{\theta}{r}\right)} + \frac{\theta}{r\alpha}