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let define $$ \text{RP}_t = \sum_{u< t} \frac{dP_u}{P_u}$$ $$ \text{RQ}_t =\sum_{u<t} \frac{dP_u}{P_u}$$ $X$ is a mean reverting process so : $$ dX = \alpha (\mu - X)dt + \sigma dB $$ where $B$ is a brownian motion meanwhile using your relationship you get : $$ X_t = \text{RP}_t - b \text{RQ}_t - a t $$ you use $X$ dynamics with this and you get: ...



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