# Skorohod metric

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Reference: [Bil99] ([Patrick Billingsley 1999])

Goal: RCLL space is the collection of all right continuous processes with left limit exists. In this below, we will give a definition of Skorohod metric on this space. More precisely, for a ${t\in (0, \infty)}$, we denote by ${\mathbb D^{d}_{t}}$ the collection of RCLL processes on ${[0, t]}$ taking values in ${\mathbb R^{d}}$. We also use ${\mathbb D^{d}_{\infty}}$ to denote the collection of RCLL processes on ${[0, \infty)}$.

For ${t\in [0, \infty)}$, most straightforward metric can be induced by ${\|x - y\|_{t}}$ in ${\mathbb D^{d}_{t}}$, where ${\|\cdot\|_{t}}$ is the sup norm

$\displaystyle \|x\|_{t} = \sup_{0\le s < t} |x(t)|. \ \ \ \ \ (1)$

Consider

$\displaystyle x_{n}(s) = I_{[n^{-1}, t]}(s), \quad x_{\infty}(s) = 1. \ \ \ \ \ (2)$

Obviously, we have ${\lim_{n\rightarrow \infty}\|x_{n} - x_{\infty} \|_{t} =1 > 0}$, and this motivates the Skorhod metric below.

• For ${t\in [0, \infty)}$, we denote by ${\Lambda_{t}}$ by the class of strictly increasing continuous mappings of ${[0, t]}$ onto itself. In particular, ${\lambda(0) = 0}$ and ${\lambda(t) = t}$ for all ${\lambda \in \Lambda}$. The identity ${I}$ on ${[0,t]}$ also belongs to ${\Lambda_{t}}$. We can define a functional in ${\Lambda_{t}}$ by

$\displaystyle \|\lambda\|^{o} = \sup_{0\le s< r\le t} \Big | \log \frac{\lambda\circ r - \lambda \circ s}{r - s} \Big|, \ \forall \lambda \in \Lambda_{t}.$

Note that ${\|\lambda\|^{o}}$ may not be necessarily finite in ${\Lambda_{t}}$.

• For ${t\in [0, \infty)}$, define the distance function ${d_{t}^{o}(x,y)}$ in ${\mathbb D^{d}_{t}}$ by

$\displaystyle d_{t}^{o}(x,y) = \inf_{\lambda\in \Lambda_{t}} \{\|\lambda \|^{o} \vee \|x - y \circ\lambda\|_{t} \}, \ \forall x,y \in \mathbb D^{d}_{t}.$

Example 1

Show that ${\lim_{n\rightarrow \infty}d^{o}_{t}(x_{n}, x_{\infty}) = 0}$ for ${x_{n}}$ and ${x}$ given by (2).
– QED –

By using the above definition ${d_{t}^{o}}$, we are ready for the definition of ${d_{\infty}^{o}}$:

• We define the distance function ${d_{\infty}^{o}(x,y)}$ in ${\mathbb D^{d}_{\infty}}$ by

$\displaystyle d^{o}_{\infty} (x,y) = \sum_{m=1}^{\infty} 2^{-m} (1 \wedge d^{o}_{m}(x^{m}, y^{m})) \ \forall x,y \in \mathbb D^{d}_{\infty},$

where ${x^{m}(t) = g_{m}(t) x(t)}$ for all ${t\ge 0}$ with a continuous function ${g_{m}}$ given by

$\displaystyle g_{m}(t) = \left\{ \begin{array} {ll} 1, & \hbox{ if } t\le m-1, \\ m-t, & \hbox{ if } m-1 \le t \le m, \\ 0, & \hbox{ otherwise. } \end{array} \right.$

Define a projector ${\Pi: \mathbb D^{d}_{\infty} \times [0, \infty) \mapsto \mathbb R^{d}}$ by

$\displaystyle \Pi(\omega, t) = \omega(t). \ \ \ \ \ (3)$

Proposition 1

The projection mapping ${\omega \mapsto \Pi(\omega, t)}$ is continuous at ${\omega_{0}}$ if ${t\mapsto \omega_{0}(t)}$ is continuous at ${t}$.

Proof: It’s a consequence of Theorem 12.5 of [Bil99]. $\Box$

Example 2

Consider ${O = (0, 1) \subset \mathbb R}$. Let ${\tau(\omega) = \inf\{t\ge 0: \omega(t) \in O^{c}\}}$.

• ${\tau}$ is not upper semicontinuous at ${\omega}$ given by

$\displaystyle \omega(t) = |t - 1/ 2 |$

since ${ \lim_{n} \tau(\omega_{n}) = 3/2 > 1/2 = \tau(\omega)}$ where ${\omega_{n} = \omega+ 1/n}$.

• ${\tau}$ is not lower semicontinuous at ${\omega}$ given by

$\displaystyle \omega(t) = (-t + 1 /3) I(t < 1 /3) + (-t + 2/ 3) I(t \ge 1/ 3).$

In fact, setting ${\omega_{n} = \omega - 1/n}$, we have ${\lim_{n} \tau(\omega_{n}) = 1/3 < 2/3 = \tau(\omega).}$

– QED –