- #1
operationsres
- 103
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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 [itex]dX_t = (m-X_t)dt+\sigma X_tdB(t)[/itex], the appropriate integrating factor is [itex]e^t[/itex].
2) For the non-mean-reverting OU SDE [itex]dX_t = uX_tdt + \sigma dB_t[/itex], the integrating factor is [itex]e^{-ut}[/itex].
3) For the SDE [itex]dX_t = udt + \sigma X_t dB_t[/itex], the integrating factor is [itex]e^{-\sigma B_t + \frac12 \sigma^2 t}[/itex].
Here's some examples:
1) For the mean-reverting Ornstein-Uhlenbeck (OU) SDE [itex]dX_t = (m-X_t)dt+\sigma X_tdB(t)[/itex], the appropriate integrating factor is [itex]e^t[/itex].
2) For the non-mean-reverting OU SDE [itex]dX_t = uX_tdt + \sigma dB_t[/itex], the integrating factor is [itex]e^{-ut}[/itex].
3) For the SDE [itex]dX_t = udt + \sigma X_t dB_t[/itex], the integrating factor is [itex]e^{-\sigma B_t + \frac12 \sigma^2 t}[/itex].