Applied Mathematics & Optimization

, Volume 79, Issue 1, pp 179–180

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

• Stefan Ankirchner
• Maike Klein
• Thomas Kruse
Erratum

## 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 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 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.