Skorokhod reflection problem

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Problem 19.16 from Schilling R.L. "Brownian Motion: A Guide to Random Processes and Stochastic Calculus" [Sch21]More problems in Chapter 19 "Applications of Itô’s formula"

(a)(b)(c)

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

(a)

$b(t) = p(t) - a(t)$ ;

(b)

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

(c)

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

References

[Sch21] Schilling R.L. Brownian Motion: A Guide to Random Processes and Stochastic Calculus. 3rd ed., De Gruyter, 2021.