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

  • Christel Geiss
  • Alexander SteinickeEmail author
Open Access
Correction
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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:

In Definition 3.3 we defined fn as the optional projection of
$$(\omega,t,y,z,u)\mapsto \phantom{I}^{\!\!\!o,\mathbbm{J}\!\!} f(n,\omega,t,y,z,u)$$
with respect to \(\mathbbm {F}^{n} \) (given by \(\mathcal {F}_{t}^{n}:=\mathcal {F}_{t}\cap \mathcal {J}^{n}\)), with parameters (y,z,u). However, this optional projection does not always exist for generators f satisfying (A1)–(A3).
Sufficient for the existence of the optional projection of a process is boundedness or non-negativity. To guarantee the existence one can replace first f(ω,s,y,z,u) by
$$f^{K} (\omega,s,y,z,u) = (-K) \vee f(\omega,s,y,z,u) \wedge K $$
for some K>0.
Clearly, (A1) and (A2) are satisfied for fK. Concerning (A3), one observes that only the cases where both factors of (yy)(f(s,y,z,u)−f(s,y,z,u)) are either positive or negative are relevant. Since
$$\begin{aligned} \min\left\{f(s,y,z,u)-f(s,y^{\prime},z^{\prime},u^{\prime}), 0\right\} & \le f^{K}(s,y,z,u)-f^{K}\left(s,y^{\prime},z^{\prime},u^{\prime}\right) \\ &\le \max\left\{f(s,y,z,u)-f(s,y^{\prime},z^{\prime},u^{\prime}),0 \right\}, \end{aligned} $$
(A3) is satisfied for fK. The above inequality implies that also (Aγ) holds for fK.
Hence in order to prove Theorem 3.5, one first starts with fK and fK and gets
$$Y^{K}_{t} \le Y^{\prime {K}}_{t} \quad \mathbb{P}\text{-a.s.} $$
Next we will see that \(\|Y_{t} - Y^{K}_{t} \|\) and \(\|Y^{\prime }_{t} - Y^{\prime {K}}_{t} \|\) converge to zero for 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,fK) it holds
$$\begin{aligned} & \sup_{t \in [0,T]} \|Y_{t}-Y^{K}_{t} \|^{2} \\ &\leq h\left(a,b,2\mathbb{E}\!\int_{0}^{T}\!|Y_{t}-Y^{K}_{t}|\left|f(t,Y_{t},Z_{t},U_{t})-f^{K}(t,Y_{t},Z_{t},U_{t})\right|dt\right). \end{aligned} $$
To see that the r.h.s. goes to zero, one can use that
$$\begin{array}{@{}rcl@{}} && \mathbb{E}\!\int_{0}^{T}\!|Y_t-Y^{K}_t|\left|f(t,Y_t,Z_t,U_t)-f^{K}(t,Y_t,Z_t,U_t)\right|dt \\ & \le& \sqrt{T} \sup_{t \in [0,T]} \|Y_t-Y^{K}_{t}\| \left (\mathbb{E}\!\int_{0}^{T} \left |f(t,Y_t,Z_t,U_t)-f^{K}(t,Y_t,Z_t,U_t)\right|^2dt \right)^{1/2}. \end{array} $$

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 fn 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. 1.
    Dellacherie, C., Meyer, P. -A.: Probabilities and potential. B. North-Holland Publishing Co., Amsterdam (1982).zbMATHGoogle Scholar
  2. 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. 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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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.University of Jyvaskyla, Department of Mathematics and StatisticsJyvaskylaFinland
  2. 2.Department of Applied Mathematics and Information Technology, Montanuniversitaet Leoben, Peter Tunner-Straße 25/I, A-8700LeobenAustria

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