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Applied Mathematics & Optimization

, Volume 79, Issue 1, pp 179–180 | Cite as

Erratum to: A Verification Theorem for Optimal Stopping Problems with Expectation Constraints

  • Stefan AnkirchnerEmail author
  • Maike Klein
  • Thomas Kruse
Erratum
  • 431 Downloads

1 Erratum to: Appl Math Optim DOI 10.1007/s00245-017-9424-2

We correct the statement of Lemma 2.2 in the original article. The solution of the SDE (2.2) is, in general, not a martingale but only a supermartingale. The set of controls is restricted to those processes such that the solution of Eq. (2.2) is a martingale. The remaining results and examples are valid for the new set of controls.

We first correct the statement of Lemma 2.2 in the original article. For \(m\in \mathbb {R}_{+}\) the solution of the SDEis a supermartingale but not necessarily a martingale (see Example 2.3 below for a counterexample). To show that M is a martingale we conclude in the original article that \(\tau _n=\tau \) on \(\{M_\tau \le n\}\), which is not true in general. The corrected version of Lemma 2.2 reads as follows:

Lemma 2.2

Let \((\alpha _t)_{t\ge 0}=(\alpha ^1_t,\ldots ,\alpha ^d_t)_{t\ge 0}\in L^2_{loc}(W)\) and \(m\in \mathbb {R}_+\). Then there exists a unique strong solution M of (2.2). This solution is a non-negative supermartingale.

As a consequence, for the one-to-one-correspondence claimed in Proposition 2.3 to hold true, we need to require that the set of controls consists of processes \(\alpha \in L^2_{loc}(W)\) such that the solution of (2.2) is a true martingale. More precisely, let
$$\begin{aligned} \mathcal {A}&=\left\{ \alpha \in L^2_{loc}(W)\,|\, E[H_\tau ]=M_0,\, \text {where } M \text { solves }(2.2) \text { for }\alpha \text { and }\right. \\&\qquad \tau =\left. \inf \{t\ge 0 \,|\, M_t\le H_t\}\phantom {L^2_{loc}}\right\} \end{aligned}$$
and let \(\mathcal {M}(m)\) be the set of all solutions M of (2.2) with \((\alpha _t)_{t\ge 0} \in \mathcal {A}\). Observe that Lemma 2.2 implies that for \(\alpha \in \mathcal {A}\) the solution \((M_t)\) of (2.2) is a true martingale with \(M_t\rightarrow M_\infty \) in \(L^1(\Omega )\) for \(t\rightarrow \infty \). Moreover, \(M_\infty =M_\tau =H_\tau \) by the definition of \(\tau \). On the other hand, if for \(\alpha \in L^2_{loc}(W)\) the solution of (2.2) is a true martingale with \(M_t\rightarrow M_\infty \) in \(L^1(\Omega )\) for \(t\rightarrow \infty \), then \(E[H_\tau ]=E[M_\tau ]=M_0.\) Notice that \(\mathcal {A}\) is non-empty.

If \(L^2_{loc}(W)\) is replaced by \(\mathcal {A}\) in the subsequent statements, all results and arguments hold true. Moreover, observe that the processes \(\alpha \) and \(\alpha ^*\) in Example 2.6, 2.7, 4.5, 4.6 and 4.7 are contained in \(\mathcal {A}\). In the proof of the first part of Proposition 3.4 we now consider the control Open image in new window with \(a\in \mathbb {R}^d\). Then \(\alpha \in \mathcal {A}\). For applying Itô’s formula in (3.3) choose \(t\in (0,1)\). The remaining proof is unchanged.

The following example shows that \(\mathcal {A}\ne L^2_{loc}(W)\).

Example 2.3

Let \(d=1\) and \(h(y)=1\) for all \(y\in \mathbb {R}\). Let Open image in new window and \(m=2\). Then \(\tau _n=\inf \{t\ge 0\,|\, |\alpha _t|\ge n\}\) is a localizing sequence for \(\alpha \) and thus, \(\alpha \in L^2_{loc}(W)\). Moreover, the solution M of (2.2) is given by
$$\begin{aligned} M_t={\left\{ \begin{array}{ll} 1+\frac{1}{\sqrt{1-t}}e^{-\frac{W_t^2}{2(1-t)}},\quad &{}t<1,\\ 1, &{}t\ge 1. \end{array}\right. } \end{aligned}$$
Then \(M_t\ge 1\) for all \(t\ge 0\) and \(M_1=1=H_1\). Thus, \(\tau :=\inf \{t\ge 0\,|\, M_t\le t\}=1,\) a.s. Moreover, \((M_t)\) is a local martingale, but not a true martingale, because \(M_0=2\) and \( M_1=1\), a.s.

Notes

Acknowledgements

We are grateful to Song Yao for pointing out the error in the original article.

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Institute for MathematicsUniversity of JenaJenaGermany
  2. 2.Faculty of MathematicsUniversity of Duisburg-EssenEssenGermany

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