Cointegration in continuous time for factor models

Abstract

We develop cointegration for multivariate continuous-time stochastic processes, both in finite and infinite dimension. Our definition and analysis are based on factor processes and operators mapping to the space of prices and cointegration. The focus is on commodity markets, where both spot and forward prices are analysed in the context of cointegration. We provide many examples which include the most used continuous-time pricing models, including forward curve models in the Heath–Jarrow–Morton paradigm in Hilbert space.

Appendix A. The Filipović space

We present the Filipović space following Filipović [22]: Let $w:\mathbb {R}_+\rightarrow \mathbb {R}_+$ be a monotonely increasing function with $w(0)=1$ and $\int _0^{\infty }w^{-1}(x)\,dx<\infty$. Introduce the Filipović space, denoted $\mathsf H_w$, as the space of absolutely continuous functions $f:\mathbb {R}_+\rightarrow \mathbb {R}$ for which

$$\begin{aligned} |f|_w^2:=f^2(0)+\int _0^{\infty }w(x)(f'(x))^2\,dx<\infty , \end{aligned}$$

where $f'$ is the weak derivative of f. With the inner product

$$\begin{aligned} (f,g)_w=f(0)g(0)+\int _0^{\infty }w(x)f'(x)g'(x)\,dx \end{aligned}$$

for $f,g\in \mathsf H_w$, $\mathsf H_w$ becomes a separable Hilbert space. The shift operator ${\mathcal {S}}(t):f\mapsto f(t+\cdot )$ for $t\ge 0$ defines a $C_0$-semigroup on $\mathsf H_w$ which is uniformly bounded. Moreover, Benth and Krühner [11] show that the shift operator is quasi-contractive. The generator of ${\mathcal {S}}(t)$ is the derivative operator. The evaluation map $\delta _x:f\mapsto f(x)$ is a linear functional on $\mathsf H_w$. Finally, from Benth and Krühner [11], $\mathsf H_w$ becomes a Banach algebra after appropriate rescaling of the norm $|\cdot |_w$, that is, if $f,g\in H_w$, then $fg\in H_w$ and $\Vert fg\Vert _w\le \Vert f\Vert _w\Vert g\Vert _w$ with $\Vert \cdot \Vert _w:=c|\cdot |_w$ for a suitable constant $c>0$ depending on $\int _0^{\infty }w^{-1}(x)\,dx$.