Skorohod metric

PDF: here

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 –

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