# American Institute of Mathematical Sciences

September  2021, 6(3): 159-188. doi: 10.3934/puqr.2021008

## Reduced-form setting under model uncertainty with non-linear affine intensities

 1 Department of Mathematics, Workgroup Financial and Insurance Mathematics, University of Munich (LMU), Theresienstraße 39, 80333 Munich, Germany 2 Department of Mathematics of Natural, Social and Life Sciences, Gran Sasso Science Institute (GSSI), Viale F. Crispi 7, 67100 L’Aquila, Italy

Email: francesca.biagini@math.lmu.de

Received  December 23, 2020 Accepted  July 06, 2021 Published  September 2021

In this paper we extend the reduced-form setting under model uncertainty introduced in [5] to include intensities following an affine process under parameter uncertainty, as defined in [15]. This framework allows us to introduce a longevity bond under model uncertainty in a way consistent with the classical case under one prior and to compute its valuation numerically. Moreover, we price a contingent claim with the sublinear conditional operator such that the extended market is still arbitrage-free in the sense of “no arbitrage of the first kind” as in [6].

Citation: Francesca Biagini, Katharina Oberpriller. Reduced-form setting under model uncertainty with non-linear affine intensities. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 159-188. doi: 10.3934/puqr.2021008
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1Galmarino’s Test [42, Exercise 4.21]: Let $\Omega = C({\mathbb{R}}_+, {\mathbb{R}})$ , ${\cal{F}}$ the Borel $\sigma$ -algebra with respect to the topology of locally uniform convergence, and ${\mathbb{F}}$ be the raw filtration generated by the canonical process $B$ on $\Omega$ . Then, an ${\cal{F}}$ -measurable function $\tau: \Omega \to {\mathbb{R}}_+$ is an ${\mathbb{F}}$ -stopping time if and only if $\tau(\omega) \leq t$ and $\omega\vert_{[0, t]} = \omega'\vert_{[0, t]}$ imply $\tau(\omega) = \tau(\omega')$ . Furthermore, given an ${\mathbb{F}}$ -stopping time $\tau$ , an ${\cal{F}}$ -measurable function $f$ is ${\cal{F}}_{\tau}$ -measurable if and only if $f = f \circ \iota_{\tau}$ , where $\iota_{\tau}: \Omega \to \Omega$ is the stopping map $(\iota_{\tau}(\omega))_t = \omega_{t \wedge \tau(\omega)}$ .

2By the same arguments regarding the filtration as in Remark 5.2, $S$ is also a $(\tilde{P}, {\mathbb{G}}^{*, {\tilde{\cal{P}}}}_+)$ -semimartingale for all $\tilde{P} \in \tilde{\cal{P}}$ .

3The sigma-martingale property holds with respect to the filtration ${\mathbb{G}}^{*, {\tilde{\cal{P}}}}_+$ for all $\tilde{P} \in \tilde{\cal{P}}$ .

4Note, the assumption $\sup_{P \in {{\cal{Z}}}} E^{{P}}[e^{-\int_0^T B_s^{\mu} {\rm{d}}s }] < \infty$ is always satisfied for $B^{\mu} > 0$ which is the case for a mortality intensity.