# Apply Girsanov theorem to a Ornstein-Uhlenbeck process

From the original paper by Schwartz (free access here, page 926):

Assume that the commodity spot price follows the stochastic process$$^5$$ $$\tag1 dS = \kappa(\mu-\log S)Sdt+\sigma SdW,$$ where $$dW$$ is an increment to a standard Brownian motion. Applying Ito lemma we can see that the log price $$X = \log S$$ is characterized by an Ornstein-Uhlenbeck process $$\tag{2,3} dX = \kappa(\gamma-X)dt+\sigma dW, \quad \text{ with } \gamma = \mu-\frac{\sigma^2}{2\kappa},$$ where $$\kappa > 0$$ measures the degree of mean reversion to the long run mean log price $$\gamma$$.

Now comes the part which I don't fully understand.

In this model, the commodity is not an asset in the usual sense$$^6$$ and the spot price, or equivalently the log of the spot price, plays the role of an underlying state variable upon which contingent claims can be written. Under standard assumptions, the dynamics of the Ornstein-Uhlenbeck process under the equivalent martingale measure can be written as$$^7$$ $$\tag4 dX = \kappa(\gamma^*-X)dt+\sigma dW^*,$$ where $$\gamma^* = \gamma-\lambda, \lambda$$ is the market price of risk (assumed constant)$$^8$$ and $$dW^*$$ is the increment to the Brownian motion under the equivalent martingale measure.

I'm trying to understand the steps to derive equation (4); in the article cited in $$^7$$ (see link below) it is written

With the stated assumptions on the economy (unfortunately, the page with the assumptions is missing from the preview), the appropriate transformation from the actual probability measure to the equivalent martingale measure is $$\tag{12.14} dZ(t) = dZ^*(t) - \lambda t.$$ where $$dZ^*(t)$$ has zero mean and variance $$dt$$ under the *-probability measure.

So it seems that they applied Girsanov theorem. In the following I will use this version of Girsanov theorem from Stochastic differential equations: an introduction with applications by Øksendal

Following the notation of the theorem, in our case $$\beta = \kappa(\gamma-X)$$ and $$\theta = \sigma$$. Moreover let $$u=\lambda$$ (a constant), then $$M_t = \exp\bigg(-\int_0^t \lambda \,dB_s + \frac12\int_0^t \lambda^2 \,ds\bigg)$$ so the Novikov sufficient condition holds (recall that $$\lambda$$ is a constant so it goes out of the integral) $$E\bigg[\exp\bigg(\frac12\int_0^T \lambda^2 \,ds\bigg)\bigg] = E\bigg[\exp\bigg(\frac12\lambda^2T\bigg)\bigg] = \exp\bigg(\frac12\lambda^2T\bigg) < \infty \text{ since }\lambda,T < \infty,$$ hence $$M_t$$ is a martingale and we can apply the theorem, which tells us that $$\widehat B = \lambda t+B$$, so we have recovered the equation (12.14) above. Now, we have to find $$\alpha$$ such that $$\theta u=\beta-\alpha$$, that is $$\sigma\lambda = \kappa(\gamma-X) - \alpha \ \implies\ \alpha = \kappa(\gamma-X)-\sigma\lambda = \kappa\Big(\gamma-\frac{\sigma\lambda}{\kappa}-X\Big).$$ So in terms of $$\widehat B$$ the process $$X$$ has the representation $$\tag5 dX = \kappa(\hat\gamma-X) dt + \sigma d\widehat B, \quad\text{with }\hat\gamma = \gamma-\frac{\sigma\lambda}{\kappa},$$ resembling equation (4), however notice that $$\hat\gamma \ne \gamma^* = \gamma-\lambda$$. Does this mean that the choice $$u=\lambda$$ is wrong? To obtain $$\hat\gamma \equiv \gamma^*$$ we have to choose $$u = \kappa\lambda/\sigma$$, however does the Novikov condition still hold with this choice for $$u$$?

$$^5$$ $$^6$$ Ross, S. A,, 1995, Hedging long run commitments: Exercises in incomplete market pricing

$$^7$$ Bjerksund, P., and S. Ekern, 1995, Contingent claims evaluation of mean-reverting cash flows in shipping, in L. Trigeorgis, Ed.: Real Options in Capital Investment: Models, Strategies, and Applications, Preager (partial preview here)

I think it might be possible that you got caught up in a sort of 'notation trap' and the vague notion of market price of risk (in this context) did the rest. The most plausible explanation is straightforward: $$dX_t=\kappa(\gamma-X_t)dt+\sigma dW_t$$ $$dX_t=\kappa(\gamma-X_t)dt+\sigma (dW^*_t-\nu dt)$$ $$dX_t=\kappa(\gamma-\underbrace{\frac{\sigma\nu}{\kappa}}_{:=\lambda}-X_t)dt+\sigma dW^*_t$$ $$dX_t=\kappa(\gamma^*-X_t)dt+\sigma dW^*_t$$
• Thank you for answering, could you explain more about what's happening from $dW_t$ to $dW_t^* - \nu dt$? Moreover, is the Girsanov theorem involved in your proof? Jul 6, 2021 at 12:16
• Yes it's involved in the change from $W_t$ to $W_t^*$. Now I'm trying to find the Bjerksund and Ekern's paper to be sure about the model itself. Jul 6, 2021 at 12:42