# Integrating factors for SDEs

Whenever I'm given a SDE problem that requires us to multiply both sides by an "integrating-factor", it's always given to us as a *Hint*. I would like to know how to come up with these integrating factors.

Here's some examples:

1) For the mean-reverting Ornstein-Uhlenbeck (OU) SDE $dX_t = (m-X_t)dt+\sigma X_tdB(t)$, the appropriate integrating factor is $e^t$.

2) For the non-mean-reverting OU SDE $dX_t = uX_tdt + \sigma dB_t$, the integrating factor is $e^{-ut}$.

3) For the SDE $dX_t = udt + \sigma X_t dB_t$, the integrating factor is $e^{-\sigma B_t + \frac12 \sigma^2 t}$.

Hi,
I suppose you are at the beginning of ODE course,so integrating factors will be discussed later on.
I hope you will find interest in the attached material here.

Last edited:
Hi,
I suppose you are at the beginning of ODE course,so integrating factors will be discussed later on.
I hope you will find interest in the attached material here.

Thanks,

1) What attached material?
2) I'm at the end of a financial mathematics course (stochastic calculus). Integrating factors are provided to us and we will never learn how to discover them. I want to learn how to do this -- they aren't going to teach this to me.

The attached material concern ordinary differential equations.I suupose it is the same for stochastic.

#### Attachments

• IFM.pdf
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