# 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
##### References:
 [1] Beatrice Acciaio, Mathias Beiglböck, Friedrich Penkner and Walter Schachermayer, A modelfree version of the fundamental theorem of asset pricing and the super-replication theorem, Mathematical Finance, 2021, 26(2): 233-251. [2] Bahar Akhtari, Francesca Biagini, Andrea Mazzon, and Katharina Oberpriller, Generalized Feynman-Kac Formula under volatility uncertainty, arXiv: 2012.08163, 2012. [3] Anna Aksamit and Monique Jeanblanc, Enlargement of Filtration with Finance in View, Springer, 2017. [4] Erhan Bayraktar, Yuchong Zhang, and Zhou Zhou, A note on the fundamental theorem of asset pricing under model uncertainty, Risks, 2014, 2(4): 425-433. [5] Francesca Biagini and Yinglin Zhang, Reduced-form framework under model uncertainty, The Annals of Applied Probability, 2019, 29(4): 2481-2522. [6] Sara Biagini, Bruno Bouchard, Constantinos Kardaras, and Marcel Nutz, Robust fundamental theorem for continuous processes, Mathematical Finance, 2017, 27(4): 963-987. [7] Thomasz R. Bielecki and Marek Rutkowski, Credit Risk: Modeling, Valuation and Hedging, Springer, 2004. [8] Enrico Biffis, Affine processes for dynamic mortality and actuarial valuations, SSRN Electronic Journal, 2004, https://dx.doi.org/10.2139/ssrn.647421. [9] Bruno Bouchard and Marcel Nutz, Arbitrage and duality in nondominated discrete-time models, The Annals of Applied Probability, 2015, 25(2): 823-859. [10] Andrew Cairns, David Blake, and Kevin Dowd, Pricing death: Frameworks for the valuation and securitization of mortality risk, ASTIN Bulletin, 2006, 36(1): 79-120. [11] Giorgia Callegaro, Monique Jeanblanc, and Behnaz Zargari, Carthaginian enlargement of filtrations, arXiv: 1111.3073v1, 2018. [12] Mikkel Dahl, Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts, Insurance: Mathematics and Economics, 2004, 35(1): 113-136. doi: 10.1016/j.insmatheco.2004.05.003. [13] Nicole El Karoui and Xiaolu Tan, Capacities, measurable selection & dynamic programming, Part I: Abstract framework, arXiv: 1310.3363v1, 2013. [14] Nicole El Karoui and Xiaolu Tan, Capacities, measurable selection & dynamic programming, Part Ⅱ: Application in stochastic control problems, arXiv: 1310.3364v2, 2015. [15] Tolulope Fadina, Ariel Neufeld, and Thorsten Schmidt, Affine processes under parameter uncertainty, Probability, Uncertainty and Quantitative Risk, 2019, 4: 5. doi: 10.1186/s41546-019-0039-1. [16] Tolulope Fadina and Thorsten Schmidt, Default ambiguity, Risks, 2019, 7(2): 64. [17] Damir Filipovic, Term-Structure Models: A Graduate Course, Springer, 2009. [18] Hans Föllmer and Philip Protter, Local martingales and filtration shrinkage, ESAIM: Probability and Statistics, 2011, 15: S25-S28. [19] Hans Föllmer and Alexander Schied, Stochastic Finance: An Introduction in Discrete Time, De Gruyter, 2016. [20] Djibril Gueye, Monique Jeanblanc, and Libo Li, Models of default times and Cox model revisited, Freiburg FRIAS: Finance and Insurance, November 2019. [21] Daniel Hollender, Lèvy-type processes under uncertainty and related nonlocal equations, PhD Thesis, TU Dresden, 2016. [22] Julian Hölzermann, The Hull-White model under Knightian uncertainty about the volatility, arXiv: 1808.03463v2, 2019. [23] Julian Hölzermann, Pricing interest rate derivatives under volatility uncertainty, arXiv: 2003.04606v1, 2020. [24] Julian Hölzermann and Lian Quian, Term structure modeling under volatility uncertainty, arXiv: 1904.02930, 2020. [25] Jean Jacod and Albert N. Shiryaev, Limit Theorems for Stochastic Processes, Springer, 2013. [26] Yuri Kabanov, Constantinos Kardaras, and Shiqi Song, No arbitrage of the first kind and local martingale numèraires, Finance and Stochastics, 2016, 20(4): 1097-1108. doi: 10.1007/s00780-016-0310-6. [27] Constantinos Kardaras, Finitely additive probabilities and the fundamental theorem of asset pricing, In: Contemporary Quantitative Finance, Springer, 2010: 19-34. [28] Dimitri O. Kramkov, Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets, Probability Theory and Related Fields, 1996, 105(4): 459-479. doi: 10.1007/BF01191909. [29] David Lando, On Cox processes and credit risky securities, Review of Derivatives Research, 1998, 2(2-3): 99-120. [30] Elisa Luciano and Elena Vigna, Non-mean reverting affine processes for stochastic mortality, SSRN Electronic Journal, 2005, https://ssrn.com/abstract=724706. [31] Ariel Neufeld and Marcel Nutz, Superreplication under volatility uncertainty for measurable claims, Electronic Journal of Probability, 2013, 18(48): 1-14. [32] Ariel Neufeld and Marcel Nutz, Measurability of semimartingale characteristics with respect to the probability law, Stochastic Processes and their Applications, 2014, 124(11): 3819-3845. doi: 10.1016/j.spa.2014.07.006. [33] Ariel Neufeld and Marcel Nutz, Nonlinear Lèvy processes and their characteristics, Transactions of the American Mathematical Society, 2017, 369(1): 69-95. doi: 10.1090/tran/6656. [34] Ariel Neufeld and Marcel Nutz, Robust utility maximization with Lévy processes, Mathematical Finance, 2018, 28(1): 82-105. [35] Marcel Nutz, Random G-expectations, The Annals of Applied Probability, 2013, 23(5): 1755-1777. [36] Marcel Nutz, Superreplication under model uncertainty in discrete time, Finance and Stochastics, 2014, 18(4): 791-803. doi: 10.1007/s00780-014-0238-7. [37] Marcel Nutz, Robust superhedging with jumps and diffusion, Stochastic Processes and their Applications, 2015, 125(12): 4543-4555. doi: 10.1016/j.spa.2015.07.008. [38] Marcel Nutz and Roman Van Handel, Constructing sublinear expectations on path space, Stochastic Processes and their Applications, 2013, 123(8): 3100-3121. doi: 10.1016/j.spa.2013.03.022. [39] Marcel Nutz and Mete Soner, Superhedging and dynamic risk measures under volatility uncertainty, SIAM Journal on Control and Optimization (SICON), 2012, 50(4): 2065-2089. doi: 10.1137/100814925. [40] Shige Peng, Nonlinear expectations and stochastic calculus under uncertainty, arXiv: 1002.456v1, 2010. [41] Philip E, Protter, Stochastic Integration and Differential Equations, Springer, 2005. [42] Daniel Revuz and Marc Yor, Continuous Martingales and Brownian Motion, Springer, 2005. [43] Thorsten Rheinländer and Jenny Sexton, Hedging Derivatives, World Scientific, 2011. [44] David F. Schrager, Affine stochastic mortality, Insurance: Mathematics and Economics, 2006, 38(1): 81-97. doi: 10.1016/j.insmatheco.2005.06.013. [45] Jörg Vorbrink, Financial markets with volatility uncertainty, Journal of Mathematics Economics, 2014, 53: 64-78. doi: 10.1016/j.jmateco.2014.05.008. [46] Yinglin Zhang, Insurance modeling in continuous time, PhD Thesis, Ludwig-Maximilians University Munich, 2018.

show all references

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.

##### References:
 [1] Beatrice Acciaio, Mathias Beiglböck, Friedrich Penkner and Walter Schachermayer, A modelfree version of the fundamental theorem of asset pricing and the super-replication theorem, Mathematical Finance, 2021, 26(2): 233-251. [2] Bahar Akhtari, Francesca Biagini, Andrea Mazzon, and Katharina Oberpriller, Generalized Feynman-Kac Formula under volatility uncertainty, arXiv: 2012.08163, 2012. [3] Anna Aksamit and Monique Jeanblanc, Enlargement of Filtration with Finance in View, Springer, 2017. [4] Erhan Bayraktar, Yuchong Zhang, and Zhou Zhou, A note on the fundamental theorem of asset pricing under model uncertainty, Risks, 2014, 2(4): 425-433. [5] Francesca Biagini and Yinglin Zhang, Reduced-form framework under model uncertainty, The Annals of Applied Probability, 2019, 29(4): 2481-2522. [6] Sara Biagini, Bruno Bouchard, Constantinos Kardaras, and Marcel Nutz, Robust fundamental theorem for continuous processes, Mathematical Finance, 2017, 27(4): 963-987. [7] Thomasz R. Bielecki and Marek Rutkowski, Credit Risk: Modeling, Valuation and Hedging, Springer, 2004. [8] Enrico Biffis, Affine processes for dynamic mortality and actuarial valuations, SSRN Electronic Journal, 2004, https://dx.doi.org/10.2139/ssrn.647421. [9] Bruno Bouchard and Marcel Nutz, Arbitrage and duality in nondominated discrete-time models, The Annals of Applied Probability, 2015, 25(2): 823-859. [10] Andrew Cairns, David Blake, and Kevin Dowd, Pricing death: Frameworks for the valuation and securitization of mortality risk, ASTIN Bulletin, 2006, 36(1): 79-120. [11] Giorgia Callegaro, Monique Jeanblanc, and Behnaz Zargari, Carthaginian enlargement of filtrations, arXiv: 1111.3073v1, 2018. [12] Mikkel Dahl, Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts, Insurance: Mathematics and Economics, 2004, 35(1): 113-136. doi: 10.1016/j.insmatheco.2004.05.003. [13] Nicole El Karoui and Xiaolu Tan, Capacities, measurable selection & dynamic programming, Part I: Abstract framework, arXiv: 1310.3363v1, 2013. [14] Nicole El Karoui and Xiaolu Tan, Capacities, measurable selection & dynamic programming, Part Ⅱ: Application in stochastic control problems, arXiv: 1310.3364v2, 2015. [15] Tolulope Fadina, Ariel Neufeld, and Thorsten Schmidt, Affine processes under parameter uncertainty, Probability, Uncertainty and Quantitative Risk, 2019, 4: 5. doi: 10.1186/s41546-019-0039-1. [16] Tolulope Fadina and Thorsten Schmidt, Default ambiguity, Risks, 2019, 7(2): 64. [17] Damir Filipovic, Term-Structure Models: A Graduate Course, Springer, 2009. [18] Hans Föllmer and Philip Protter, Local martingales and filtration shrinkage, ESAIM: Probability and Statistics, 2011, 15: S25-S28. [19] Hans Föllmer and Alexander Schied, Stochastic Finance: An Introduction in Discrete Time, De Gruyter, 2016. [20] Djibril Gueye, Monique Jeanblanc, and Libo Li, Models of default times and Cox model revisited, Freiburg FRIAS: Finance and Insurance, November 2019. [21] Daniel Hollender, Lèvy-type processes under uncertainty and related nonlocal equations, PhD Thesis, TU Dresden, 2016. [22] Julian Hölzermann, The Hull-White model under Knightian uncertainty about the volatility, arXiv: 1808.03463v2, 2019. [23] Julian Hölzermann, Pricing interest rate derivatives under volatility uncertainty, arXiv: 2003.04606v1, 2020. [24] Julian Hölzermann and Lian Quian, Term structure modeling under volatility uncertainty, arXiv: 1904.02930, 2020. [25] Jean Jacod and Albert N. Shiryaev, Limit Theorems for Stochastic Processes, Springer, 2013. [26] Yuri Kabanov, Constantinos Kardaras, and Shiqi Song, No arbitrage of the first kind and local martingale numèraires, Finance and Stochastics, 2016, 20(4): 1097-1108. doi: 10.1007/s00780-016-0310-6. [27] Constantinos Kardaras, Finitely additive probabilities and the fundamental theorem of asset pricing, In: Contemporary Quantitative Finance, Springer, 2010: 19-34. [28] Dimitri O. Kramkov, Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets, Probability Theory and Related Fields, 1996, 105(4): 459-479. doi: 10.1007/BF01191909. [29] David Lando, On Cox processes and credit risky securities, Review of Derivatives Research, 1998, 2(2-3): 99-120. [30] Elisa Luciano and Elena Vigna, Non-mean reverting affine processes for stochastic mortality, SSRN Electronic Journal, 2005, https://ssrn.com/abstract=724706. [31] Ariel Neufeld and Marcel Nutz, Superreplication under volatility uncertainty for measurable claims, Electronic Journal of Probability, 2013, 18(48): 1-14. [32] Ariel Neufeld and Marcel Nutz, Measurability of semimartingale characteristics with respect to the probability law, Stochastic Processes and their Applications, 2014, 124(11): 3819-3845. doi: 10.1016/j.spa.2014.07.006. [33] Ariel Neufeld and Marcel Nutz, Nonlinear Lèvy processes and their characteristics, Transactions of the American Mathematical Society, 2017, 369(1): 69-95. doi: 10.1090/tran/6656. [34] Ariel Neufeld and Marcel Nutz, Robust utility maximization with Lévy processes, Mathematical Finance, 2018, 28(1): 82-105. [35] Marcel Nutz, Random G-expectations, The Annals of Applied Probability, 2013, 23(5): 1755-1777. [36] Marcel Nutz, Superreplication under model uncertainty in discrete time, Finance and Stochastics, 2014, 18(4): 791-803. doi: 10.1007/s00780-014-0238-7. [37] Marcel Nutz, Robust superhedging with jumps and diffusion, Stochastic Processes and their Applications, 2015, 125(12): 4543-4555. doi: 10.1016/j.spa.2015.07.008. [38] Marcel Nutz and Roman Van Handel, Constructing sublinear expectations on path space, Stochastic Processes and their Applications, 2013, 123(8): 3100-3121. doi: 10.1016/j.spa.2013.03.022. [39] Marcel Nutz and Mete Soner, Superhedging and dynamic risk measures under volatility uncertainty, SIAM Journal on Control and Optimization (SICON), 2012, 50(4): 2065-2089. doi: 10.1137/100814925. [40] Shige Peng, Nonlinear expectations and stochastic calculus under uncertainty, arXiv: 1002.456v1, 2010. [41] Philip E, Protter, Stochastic Integration and Differential Equations, Springer, 2005. [42] Daniel Revuz and Marc Yor, Continuous Martingales and Brownian Motion, Springer, 2005. [43] Thorsten Rheinländer and Jenny Sexton, Hedging Derivatives, World Scientific, 2011. [44] David F. Schrager, Affine stochastic mortality, Insurance: Mathematics and Economics, 2006, 38(1): 81-97. doi: 10.1016/j.insmatheco.2005.06.013. [45] Jörg Vorbrink, Financial markets with volatility uncertainty, Journal of Mathematics Economics, 2014, 53: 64-78. doi: 10.1016/j.jmateco.2014.05.008. [46] Yinglin Zhang, Insurance modeling in continuous time, PhD Thesis, Ludwig-Maximilians University Munich, 2018.
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