# Correction to: “Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting”

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The proof of (Geiss and Steinicke (2018), Theorem 3.5) needs an extra step addressing the problem that our conditions on the generator are not sufficient to guarantee the existence of the considered optional projection:

*f*

_{n}as the optional projection of

*y*,

*z*,

*u*). However, this optional projection does not always exist for generators

*f*satisfying (A1)–(A3).

*f*(

*ω*,

*s*,

*y*,

*z*,

*u*) by

*K*>0.

*f*

^{K}. Concerning (A3), one observes that only the cases where both factors of (

*y*−

*y*

^{′})(

*f*(

*s*,

*y*,

*z*,

*u*)−

*f*(

*s*,

*y*

^{′},

*z*

^{′},

*u*

^{′})) are either positive or negative are relevant. Since

*f*

^{K}. The above inequality implies that also (

*A*

*γ*) holds for

*f*

^{K}.

*f*

^{K}and

*f*

^{′K}and gets

*K*→

*∞*, so that \( \,\,\,Y_{t} \le Y^{\prime }_{t} \quad \mathbb {P}\text {-a.s.} \) follows. In the proof of Proposition 4.2 it was shown that for data (

*ξ*,

*f*) and (

*ξ*,

*f*

^{K}) it holds

The factor \(\sup _{t \in [0,T]} \|Y_{t}-Y^{K}_{t}\|\) is bounded according to Proposition 4.1, and the integral goes to zero by monotone convergence. Since \({\lim }_{x\to 0} h(a,b,x) =0,\) one derives that \({\lim }_{K \to \infty }\|Y_{t} - Y^{K}_{t} \| =0,\) and in the same way it follows \({\lim }_{K \to \infty }\|Y^{\prime }_{t} - Y^{\prime {K}}_{t} \| =0.\)

Moreover, Theorem 3.4 and Lemma 5.1 in Geiss and Steinicke (2018) are only valid, if *f*_{n} in Definition 3.3 exists. For the proof of Theorem 3.5 this does not cause a problem since we need these results for \(f^{K}_{n}\) only.

For more general conditions for the existence of an optional projection than non-negativity or boundedness we refer to (Dellacherie and Meyer (1982), Remarks VI.44.(f)) and (He et al. 1992).

## Notes

### Authors’ contributions

Both authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

## References

- 1.Dellacherie, C., Meyer, P. -A.: Probabilities and potential. B. North-Holland Publishing Co., Amsterdam (1982).zbMATHGoogle Scholar
- 2.Geiss, C., Steinicke, A.: Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting. Probab. Uncertain. Quant. Risk. 3(9) (2018).Google Scholar
- 3.He, S., Wang, J., Yan, J.: Semimartingale Theory and Stochastic Calculus. Science Press, CRC Press, New York (1992).zbMATHGoogle Scholar

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