I am reading through examples of linear filtering problems for SDE's, and the process first requires solving a (deterministic) riccati differential equation. In one of the examples, this is given by:

$$\frac{dS_t}{dt} = \frac{-1}{m^2}S^2_t + c^2$$

to solve it, they first rearrange it as

$$\frac{m^2 dS_t}{m^2 c^2 -S^2_t} = dt$$

and then state (without working or explanation) that this gives

$$\left| \frac{mc+S_t}{mc-S_t} \right|= \left| \frac{mc+a^2}{mc-a^2} \right| \exp \left( \frac{2ct}{m} \right)$$

And I'm unsure how to get from one to the other.

Clearly, since $t$ has emerged, it involves integrating both sides. But besides that I'm lost. Could someone explain this or give some steps/outline for me to do myself?


1 Answer 1


Don't see this as Riccati equation, see it as separable ODE with constant solutions at $\pm mc$. Then the last form has a partial fraction decomposition $$ \frac{mdS}{S+mc}-\frac{mdS}{S-mc} = 2cdt $$ which after integration and determination of the integration constant should give your solution form.


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