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

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## 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.

*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.

*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

*M*of (2.2) is given by

## Notes

### Acknowledgements

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