Answer: **Very simple**.

There are two equivalent ways to do this:

- First method, like that showed for the Geometric Brownian case, is
just to substitute the real drift a of the
process by the
*risk-neutral drift***r - d**. The difference in mean-reversion case is that the dividend-yield d (or convenience yield for commodities) is not constant, it is a function of P. - Second method is to subtract a
*risk premium***ls**from the real drift of the process, where l is the__market price of risk__*for the underlying asset*.

Consider the following mean-reversion process, an arithmetic Ornstein-Uhlenbeck process, that has been largely used in financial and real options applications:

**dx = h (m -
x) dt + s dz**

For oil prices P, let** x = ln(P)**, as in Schwartz (1997, his "Model
1"). So, we can write the same equation for the logarithm of oil
prices following a mean-reverting process:

**d(lnP) = h (m -
lnP) dt + s dz**

The drift a here is the mean-reversion term,
**a = h (m - x)**.
For the risk-neutral process, the first method indicate that we can
substitute this for the *risk-neutralized drift*:

a* = r - d

But the risk-adjusted discount rate m is the
total expected return, which is the sum of capital gain rate with the
dividend rate, that is, **m = a + d**. So,
we can write the equation of d as:

d = m - a , hence: **d
= m - h (m - x) **

The risk-neutral drift becomes:

**a* = r -
m + h (m - x) **

The stochastic risk-neutral equation finally becomes:

**dx = [r - m + h (m
- x)] dt + s dz***

For the oil prices case:

**d(lnP) = [ h (m
- lnP) - (m -r) ]
dt + s dz*** ............. (Eq.1)

Where dz* is the Wiener increment under risk-neutral probability (or
under *martingale measure* as the mathematicians prefer).

This format of risk-neutral differential equation is found in Dixit &
Pindyck (e.g., see mean-reversion in chapters 5 and 12).

We can also rewrite the equation 1 by rearranging the terms:

**d(lnP) = h [m
- [(m
- r)/h] -
lnP] dt + s dz*** ............. (Eq.1a)

Note that **m - r** is the *risk-premium
for the underlying asset x*. The comparison between both drifts
indicates that the passage from the real process to the risk-neutral one,
can be viewed as subtracting the *normalized risk-premium* (**(
m - r )/h**)
from the long run mean level **m**.

Let us see the second method, which we can find other format for the
same Eq.1. The idea here is to subtract a risk premium **ls**
from the real drift of the process a. Of
course r - d = a - ls, but let us subtract the
premium risk directly from the stochastic equation, in order to get a
risk-neutral stochastic process:

dx = [h (m - x) - ls] dt + s dz*

So, is easy to see that:

dx = h [(m - ls/h)- x] dt + s dz*

Or, in oil prices terms:

**d(lnP) = h [(m
- ls/h) -
lnP] dt + s dz*** ..............(Eq.2)

This equation is equal to the Eqs.1 and 1a, but using other parameters.
The interpretation again is that the risk-neutral process difference from
the real process is just an adjustment in the equilibrium (or long-run
mean) level m, which the process reverts.

That is, by subtracting ls/h in the
equilibrium level m, we get the risk-neutralized process for
mean-reversion.

This second method interpretation is drawn from a similar case found in the Bjerksund & Ekern (1993) paper and Schwartz (1997, model1).

For additional information, see the page on simulation of stochastic processes.