# If the spread between two assets is an OU process, what processes do the two assets follow?

Let $$(\Omega,\mathcal{F}, \mathbb{P}, (\mathcal{F}_{t})_{t\geq0})$$ be a filtered probability space. Furthemore, let $$(S_{t}^{1},S_{t}^{2})_{t\geq0}$$ be two assets (adapted to filtration, etc). Define $$X_{t}=S^{1}_{t}-S^{2}_{t}$$. If $$X_{t}$$ satisfies the SDE:
$$dX_{t}=\xi(\zeta-X_{t})dt+\sigma dW_{t}$$
($$W_{t}$$ is a $$\mathbb{P}$$ Brownian motion)
then what process does $$(S_{t}^{1},S^{2}_{t})$$ follow (assuming reasonable conditions like nonnegativity)?

• This is a complex question, and there is probably not a unique representation. As @Kermittfrog answer shows, one possible solution is that $S^1$ and $S^2$ follow OU processes themselves. May 25, 2022 at 12:46
• @Kermittfrog gave probably the answer you expected (+1) but note that your general question allows for silly examples like $\text{d}S_1=\alpha S_1\text{d}t+\sigma_1\text{d}W_1$ and $\text{d}S_2=\left(\alpha S_1-\xi(\zeta-(S_1-S_2))\right)\text{d}t+\sigma_2\text{d}W_2$. You can find infinitely many of these trivial examples unless you narrow down your a problem a little. May 25, 2022 at 13:08
• Good point @Kevin! May 25, 2022 at 13:25

\begin{align} dx_1&=\kappa_1(\theta_1-x_1)dt+\sigma_1dW_1\\ dx_2&=\kappa_2(\theta_2-x_2)dt+\sigma_2dW_2 \end{align} with $$E(dW_1dW_2)=\rho dt$$. Now let $$z=x_1-x_2$$. Then, if $$\kappa_1=\kappa_2=\kappa$$,
\begin{align} dz&=dx_1-dx_2\\ &=\kappa_1(\theta_1-x_1)dt+\sigma_1dW_1-\kappa_2(\theta_2-x_2)dt-\sigma_2dW_2\\ &=\kappa(\theta_1-x_1)dt+\sigma_1dW_1-\kappa(\theta_2-x_2)dt-\sigma_2dW_2\\ &=\kappa((\theta_1-\theta_2)-(x_1-x_2))dt+\sigma_1dW_1-\sigma_2dW_2\\ &\equiv\kappa(\theta_z-z)dt+\sigma_zdW_z \end{align}
where $$\theta_z=\theta_1-\theta_2$$ and $$\sigma_z^2=\sigma_1^2+\sigma_2^2-2\rho\sigma_1\sigma_2$$.