The proof of Theorem 19 29 uses, implicitly, the following beautiful result due to Skorokhod 239 which is to be proved Lemma Let (b 0, infty) ightarrow mathbb R ) be a continuous function such that (b(0) geqslant 0 ) There exist unique functions (p, a 0, infty) ightarrow 0, infty) ) such that i) (b...

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The proof of Theorem 19.29 uses, implicitly, the following beautiful result due to Skorokhod [239] which is

Question:

The proof of Theorem 19.29 uses, implicitly, the following beautiful result due to Skorokhod [239] which is to be proved:

Lemma. Let \(b:[0, \infty) \rightarrow \mathbb{R}\) be a continuous function such that \(b(0) \geqslant 0\). There exist unique functions \(p, a:[0, \infty) \rightarrow[0, \infty)\) such that i) \(b(t)=p(t)-a(t)\);

ii) \(a(0)=0, a(t)\) is continuous and increasing;

iii) \(p(t) \geqslant 0\) and supp \([d a(\cdot)] \subset\{t: p(t)=0\}\).

Set \(a(t)=0 \vee \sup _{s \leqslant t}(-b(s))\) to show existence. For uniqueness, assume that there is a further pair \(\{\bar{a}, \bar{p}\}\) and assume that \(p\left(t_{1}\right)>\bar{p}\left(t_{1}\right)\) for some \(t_{1}\), hence on some suitable maximal interval \(\left(t_{0}, t_{1}\right]\). Now use iii) to see \(a\left(t_{0}\right)=a\left(t_{1}\right) \ldots\)

Remark. The third condition iii) can be equivalently expressed as \(a(t)=\int_{0}^{t} \mathbb{1}_{\{0\}}(p(s)) d a(s)\). This is the so-called Skorokhod (integral) equation.

Data From Theorem 19.29

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