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I have trouble deriving the state equations for a mixture of normal/lognormal stochastic differential, namely for its a) expected mean, (b) variance, and (c) drift adjustment for LMM - libor model

I have this equation : df = u * F * dt + sigma * dW(F)

I am having trouble getting the expected mean, variance, and it's final stochastic differential equation of the form : F(T) = F(0) * exp(...)

For example, given the geometric form df = u * F * dt + sigma * F * dW(F)

I know the

expected mean = enter image description here

expected variance = enter image description here

For example arithmetric form, given df = u * dt + sigma * dW(F)

expected mean = u * t

expected variance = sigma * T

But I am getting stumped on the equation which is a mixture of the two.

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  • $\begingroup$ Your notation is not clear. Do you mean $\text{d}F_t=uF_t\text{d}t+\sigma \text{d}W_t$? If so this is an Ornstein-Uhlenbeck process, you can check Wikipedia. $\endgroup$ – Daneel Olivaw May 24 at 8:50
  • $\begingroup$ hi Daneel, yes, that's what I meant. Thanks, I just checked, the Ornstein-Uhlenbeck is a mean-reverting formula of the form df = -u * F * dt + sigma dW $\endgroup$ – Kiann May 24 at 9:03
  • $\begingroup$ Well then that's it, you've got the solution. $\endgroup$ – Daneel Olivaw May 24 at 9:28
  • $\begingroup$ Thanks @Daneel Olivaw. The wikipedia page actually seems to give the derivation and solution to the Vasicek form of dr = b (a - r) dt + sigma * dW. I can see the expected mean term, and the variance term. It doesn't seem to show the form for the exponential form. $\endgroup$ – Kiann May 24 at 9:54
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Just use the integrating factor method.

$df=\mu f dt +\sigma dW_t$

Multiply by the integrating factor:

$e^{-\mu t} df =e^{-\mu t} \mu f dt +\sigma e^{-\mu t} dW_t$

$d\left( e^{-\mu t} f \right)=\sigma e^{-\mu t} dW_t$

Now just integrate from 0 to T:

$e^{-\mu T}f_T-f_0=\sigma \int_0^T{e^{-\mu t} dW_t}$

And you can then just isolate $f_T$ on the left hand side.

$f_T=f_0 e^{\mu T}+ \sigma \int_0^T{e^{\mu (T-t)} dW_t}$

And you see it’s like Ornstein Uhlenbeck with long term mean equal to zero, and just look up the mean and variance of that process, but you will have to replace $\mu$ with minus $\mu$ if you would like to make the comparison clearer.

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