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arxiv: 2606.30070 · v1 · pith:5AM4MQT7new · submitted 2026-06-29 · 💱 q-fin.MF · math.PR

Financial Resilience Evaluation: From Conditional Expectations to Dynamic Convex Risk Measures

Matteo Ferrari , Roger J. A. Laeven , Emanuela Rosazza Gianin , Marco Zullino This is my paper

Pith reviewed 2026-06-30 03:38 UTC · model grok-4.3

classification 💱 q-fin.MF math.PR
keywords financial resiliencedynamic convex risk measuresbackward stochastic differential equationsdual representationItô processesconditional expectationvolatility adjustment
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The pith

For convex dynamic risk measures induced by BSDEs with Lipschitz or quadratic drivers, the resilience evaluation of an Itô process equals the worst-case conditional expectation of an effective drift that adds the process drift to the risk a

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper replaces the earlier resilience rate, which only tracked conditional expected change, with a version that applies a full dynamic convex risk measure to small increments of a price or risk process. When the risk measure arises from a backward stochastic differential equation whose driver satisfies a Lipschitz or quadratic condition, the resulting resilience evaluation exists as an L1 limit and equals a dual object: the infimum over a zero-penalty family of measure changes of the conditional expectation of an adjusted instantaneous drift. This representation distinguishes portfolios that share the same mean recovery rate yet differ in how their volatility is penalized by the risk measure.

Core claim

For a normalized, cash-additive convex dynamic risk measure ρ induced by a BSDE with Lipschitz or quadratic driver, and for an Itô process π, the resilience evaluation defined by the L1-limit as ε→0+ of (1/ε) ρ_s(π_{t+ε}−π_t) admits the dual representation inf E^Q[μ_t + adjustment term involving the volatility of π and the driver of ρ | F_s], where the infimum runs over the zero-penalty class of equivalent measures Q.

What carries the argument

The resilience evaluation D_s^ρ π_t, defined as the L1-limit of scaled risk-measure evaluations of increments, together with its dual representation obtained by changing measure over the zero-penalty class associated with the BSDE driver.

If this is right

  • The limit defining resilience evaluation exists and is finite under the stated driver conditions.
  • The dual representation is attained by at least one equivalent measure in the zero-penalty class.
  • The effective drift inside the conditional expectation combines the original drift of π with an explicit risk-adjustment term that depends on the volatility of π and on the BSDE driver.
  • The construction recovers the classical conditional-expectation resilience rate when the risk measure reduces to expectation.
  • Examples and counterexamples confirm the necessity of the Lipschitz or quadratic driver assumption.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical schemes could approximate the resilience evaluation by solving the associated BSDE and then optimizing the dual problem over the zero-penalty measures.
  • The same limit construction might be examined in discrete-time settings if the continuous-time BSDE can be approximated by a recursive sequence of one-step risk measures.
  • Different choices of the underlying dynamic risk measure would produce different orderings of resilience among otherwise identical price processes.

Load-bearing premise

The dynamic risk measure must be induced by a BSDE whose driver is Lipschitz continuous or quadratic.

What would settle it

A concrete Itô process together with a convex dynamic risk measure induced by a BSDE driver that is neither Lipschitz nor quadratic, for which either the L1-limit fails to exist or the limit does not coincide with the stated worst-case conditional expectation of the effective drift.

read the original abstract

Financial resilience concerns the rate at which a position recovers, or further deteriorates, in response to adverse conditions. As a first step, Laeven, Ferrari, Rosazza Gianin, and Zullino (arXiv:2505.07502) introduced the resilience rate, defined as the expected instantaneous rate of (favorable) change of a price or risk-assessment process. Since this quantity captures only the conditional mean of future increments, it cannot distinguish between positions having the same expected recovery but different conditional risk profiles. We obtain a richer characterization by evaluating such increments through a genuine, possibly nonlinear, dynamic risk measure. More precisely, for an It\^o process $\pi$ and a normalized, cash-additive dynamic risk measure $\rho$, we define the resilience evaluation by \[\mathcal D_s^\rho\pi_t := L^1\text{-}\lim_{\varepsilon\to0^+} \frac{1}{\varepsilon}\rho_s(\pi_{t+\varepsilon}-\pi_t), \qquad 0\leq s\leq t<T,\] whenever the limit exists. When $\rho$ is a convex dynamic risk measure induced by a BSDE with a Lipschitz or quadratic driver, we prove that this limit is well-posed and admits an explicit dual representation. It is given by the worst-case conditional expectation, over a zero-penalty class of measure changes, of an effective drift combining the drift of $\pi$ with the risk adjustment assigned by $\rho$ to its volatility. We further establish attainment of the optimal scenario and illustrate the scope of the construction, as well as the role of the assumptions, through examples and counterexamples.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper defines the resilience evaluation D_s^ρ π_t as the L^1-limit as ε→0+ of (1/ε) ρ_s(π_{t+ε}-π_t) for an Itô process π and a normalized cash-additive dynamic convex risk measure ρ. When ρ is induced by a BSDE with Lipschitz or quadratic driver, the limit is shown to exist and equal an explicit dual representation: the worst-case conditional expectation, over a zero-penalty class of measure changes, of an effective drift combining the drift of π with the risk adjustment assigned by ρ to its volatility. Attainment of the optimal scenario is established, and examples/counterexamples delineate the scope of the driver assumptions.

Significance. If the proofs hold, the work meaningfully extends the resilience-rate concept (from conditional expectations) to nonlinear dynamic risk measures, enabling distinction among positions with identical mean recovery but differing conditional risk profiles. The explicit dual form, attainment result, and counterexamples clarifying necessity of the Lipschitz/quadratic driver condition are concrete strengths that connect BSDE theory to resilience evaluation in a falsifiable way.

minor comments (2)
  1. [Abstract] Abstract, definition of D_s^ρ π_t: the L^1-limit is taken in an unspecified space; clarify whether it is L^1(Ω, F_s, P) or a conditional version to avoid ambiguity in the well-posedness statement.
  2. [Abstract] The dual representation is stated in terms of a 'zero-penalty class'; a brief reminder of how this class is identified from the BSDE driver (e.g., via the convex conjugate) would improve readability even if it appears later in the text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the accurate summary of our contribution and the positive assessment of its significance. The recommendation for minor revision is noted. No specific major comments were provided in the report, so there are no technical points requiring rebuttal or manuscript changes at this time.

Circularity Check

0 steps flagged

Minor self-citation to prior resilience-rate definition; derivation rests on standard BSDE theory

full rationale

The central object D_s^ρ π_t is newly defined in this paper as the L1-limit of (1/ε)ρ_s(π_{t+ε}-π_t). The main theorem establishes well-posedness and the explicit dual representation precisely when ρ is induced by a BSDE with Lipschitz or quadratic driver; this is proved using standard BSDE results rather than reducing to any fitted quantity or prior result from the same authors. The sole self-citation (to arXiv:2505.07502) merely recalls the linear case as motivation and is not load-bearing for the existence, dual form, or necessity of the driver assumption. No step equates a claimed prediction to its inputs by construction, and the paper supplies examples/counterexamples to delineate the assumption's role. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

The construction rests on standard properties of dynamic risk measures and Itô processes; the new entity is the resilience evaluation itself. No free parameters are introduced or fitted.

axioms (3)
  • domain assumption ρ is a normalized, cash-additive dynamic convex risk measure.
    Explicitly required in the definition of D_s^ρ π_t.
  • domain assumption The underlying process π is an Itô process on [0,T).
    Stated as the setting in which the limit is taken.
  • domain assumption When ρ arises from a BSDE, the driver is Lipschitz or quadratic.
    Required for the limit to exist and for the dual representation to hold.
invented entities (1)
  • resilience evaluation D_s^ρ π_t no independent evidence
    purpose: To quantify the instantaneous rate of favorable change of a position under a dynamic risk measure rather than conditional expectation.
    Newly defined via the scaled L1-limit; no independent evidence outside the paper is supplied.

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discussion (0)

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Reference graph

Works this paper leans on

300 extracted references · 175 canonical work pages · 1 internal anchor

  1. [1]

    2015 , issn =

    Modeling financial contagion using mutually exciting jump processes , journal =. 2015 , issn =. doi:https://doi.org/10.1016/j.jfineco.2015.03.002 , url =

    work page doi:10.1016/j.jfineco.2015.03.002 2015

  2. [2]

    2018 , url =

    Resilience , booktitle =. 2018 , url =

    2018

  3. [3]

    2024 , eprint=

    SIG-BSDE for Dynamic Risk Measures , author=. 2024 , eprint=

    2024

  4. [4]

    Artzner, Philippe and Delbaen, Freddy and Eber, Jean-Marc and Heath, David and Ku, Hyejin , TITLE =. Ann. Oper. Res. , FJOURNAL =. 2007 , PAGES =. doi:10.1007/s10479-006-0132-6 , URL =

    work page doi:10.1007/s10479-006-0132-6 2007

  5. [5]

    Artzner, Philippe and Delbaen, Freddy and Eber, Jean-Marc and Heath, David , TITLE =. Math. Finance , FJOURNAL =. 1999 , NUMBER =. doi:10.1111/1467-9965.00068 , URL =

    work page doi:10.1111/1467-9965.00068 1999

  6. [6]

    Oxford Economic Papers , volume =

    Bakas, Dimitrios and Chortareas, Georgios and Magkonis, Georgios , title =. Oxford Economic Papers , volume =. 2018 , month =. doi:10.1093/oep/gpy065 , url =

    work page doi:10.1093/oep/gpy065 2018

  7. [7]

    2023 , eprint=

    Elicitability of Return Risk Measures , author=. 2023 , eprint=

    2023

  8. [8]

    Finance Stoch

    Barrieu, Pauline and El Karoui, Nicole , TITLE =. Finance Stoch. , FJOURNAL =. 2005 , NUMBER =. doi:10.1007/s00780-005-0152-0 , URL =

    work page doi:10.1007/s00780-005-0152-0 2005

  9. [9]

    Indifference Pricing: Theory and Applications , editor =

    Barrieu, Pauline and El Karoui, Nicole , title =. Indifference Pricing: Theory and Applications , editor =. 2009 , isbn =

    2009

  10. [10]

    Ma, Jin and Yao, Song , TITLE =. Stoch. Anal. Appl. , FJOURNAL =. 2010 , NUMBER =. doi:10.1080/07362994.2010.482827 , URL =

    work page doi:10.1080/07362994.2010.482827 2010

  11. [11]

    2025 , eprint=

    Unbounded Dynamic Concave Utilities via BSDEs , author=. 2025 , eprint=

    2025

  12. [12]

    Management Sci

    Ben-Tal, Aharon and Teboulle, Marc , TITLE =. Management Sci. , FJOURNAL =. 1986 , NUMBER =. doi:10.1287/mnsc.32.11.1445 , URL =

    work page doi:10.1287/mnsc.32.11.1445 1986

  13. [13]

    Bellini, Fabio and Koch-Medina, Pablo and Munari, Cosimo and Svindland, Gregor , TITLE =. SIAM J. Financial Math. , FJOURNAL =. 2021 , NUMBER =. doi:10.1137/20M1341258 , URL =

    work page doi:10.1137/20m1341258 2021

  14. [14]

    Bellini, Fabio and Laeven, Roger J. A. and Rosazza Gianin, Emanuela , TITLE =. European J. Oper. Res. , FJOURNAL =. 2021 , NUMBER =. doi:10.1016/j.ejor.2019.08.049 , URL =

    work page doi:10.1016/j.ejor.2019.08.049 2021

  15. [15]

    Bellini, Fabio and Laeven, Roger J. A. and Rosazza Gianin, Emanuela , TITLE =. Math. Financ. Econ. , FJOURNAL =. 2018 , NUMBER =. doi:10.1007/s11579-017-0188-x , URL =

    work page doi:10.1007/s11579-017-0188-x 2018

  16. [16]

    Bellini, Fabio and Bignozzi, Valeria , TITLE =. Quant. Finance , FJOURNAL =. 2015 , NUMBER =. doi:10.1080/14697688.2014.946955 , URL =

    work page doi:10.1080/14697688.2014.946955 2015

  17. [17]

    Insurance Math

    Bellini, Fabio and Rosazza Gianin, Emanuela , TITLE =. Insurance Math. Econom. , FJOURNAL =. 2012 , NUMBER =. doi:10.1016/j.insmatheco.2012.03.005 , URL =

    work page doi:10.1016/j.insmatheco.2012.03.005 2012

  18. [18]

    Bellini, Fabio and Rosazza Gianin, Emanuela , TITLE =. Statist. Decisions , FJOURNAL =. 2008 , NUMBER =. doi:10.1524/stnd.2008.0915 , URL =

    work page doi:10.1524/stnd.2008.0915 2008

  19. [19]

    2008 , issn =

    On Haezendonck risk measures , journal =. 2008 , issn =. doi:https://doi.org/10.1016/j.jbankfin.2007.07.007 , url =

    work page doi:10.1016/j.jbankfin.2007.07.007 2008

  20. [20]

    Bellini, Fabio and Frittelli, Marco , TITLE =. Math. Finance , FJOURNAL =. 2002 , NUMBER =. doi:10.1111/1467-9965.00001 , URL =

    work page doi:10.1111/1467-9965.00001 2002

  21. [21]

    Stochastic Process

    Bion-Nadal, Jocelyne , TITLE =. Stochastic Process. Appl. , FJOURNAL =. 2009 , NUMBER =. doi:10.1016/j.spa.2008.02.011 , URL =

    work page doi:10.1016/j.spa.2008.02.011 2009

  22. [22]

    Finance Stoch

    Bion-Nadal, Jocelyne , TITLE =. Finance Stoch. , FJOURNAL =. 2008 , NUMBER =. doi:10.1007/s00780-007-0057-1 , URL =

    work page doi:10.1007/s00780-007-0057-1 2008

  23. [23]

    Optimality and risk---modern trends in mathematical finance , PAGES =

    Biagini, Sara and Frittelli, Marco , TITLE =. Optimality and risk---modern trends in mathematical finance , PAGES =. 2009 , ISBN =. doi:10.1007/978-3-642-02608-9\_1 , URL =

    work page doi:10.1007/978-3-642-02608-9 2009

  24. [24]

    Biagini, Sara and Frittelli, Marco , TITLE =. Ann. Appl. Probab. , FJOURNAL =. 2008 , NUMBER =. doi:10.1214/07-AAP469 , URL =

    work page doi:10.1214/07-aap469 2008

  25. [25]

    Econometrica , FJOURNAL =

    Bernoulli, Daniel , TITLE =. Econometrica , FJOURNAL =. 1954 , PAGES =. doi:10.2307/1909829 , URL =

    work page doi:10.2307/1909829 1954

  26. [26]

    Ben-Tal, Aharon and Teboulle, Marc , TITLE =. Math. Finance , FJOURNAL =. 2007 , NUMBER =. doi:10.1111/j.1467-9965.2007.00311.x , URL =

    work page doi:10.1111/j.1467-9965.2007.00311.x 2007

  27. [27]

    , title =

    Brunnermeier, Markus K. , title =. 2021 , isbn =

    2021

  28. [28]

    Equilibrium in a Reinsurance Market , urldate =

    Borch, Karl , journal =. Equilibrium in a Reinsurance Market , urldate =

  29. [29]

    Castagnoli, Erio and Cattelan, Giacomo and Maccheroni, Fabio and Tebaldi, Claudio and Wang, Ruodu , TITLE =. Oper. Res. , FJOURNAL =. 2022 , NUMBER =. doi:10.1287/opre.2022.2303 , URL =

    work page doi:10.1287/opre.2022.2303 2022

  30. [30]

    Calvia, Alessandro and Rosazza Gianin, Emanuela , TITLE =. SIAM J. Financial Math. , FJOURNAL =. 2020 , NUMBER =. doi:10.1137/19M1259134 , URL =

    work page doi:10.1137/19m1259134 2020

  31. [31]

    Journal of Financial Risk Management , volume =

    Castagnoli, Erio and De Donno, Marzia and Favero, Gino and Modesti, Paola , title =. Journal of Financial Risk Management , volume =. 2015 , doi =

    2015

  32. [32]

    , TITLE =

    B\"uhlmann, Hans and Delbaen, Freddy and Embrechts, Paul and Shiryaev, Albert N. , TITLE =. CWI Quarterly , FJOURNAL =. 1996 , NUMBER =

    1996

  33. [33]

    ASTIN Bulletin , volume =

    Bühlmann, Hans , title =. ASTIN Bulletin , volume =. 1984 , doi =

    1984

  34. [34]

    Astin Bull

    B\"uhlmann, Hans , TITLE =. Astin Bull. , FJOURNAL =. 1980/81 , NUMBER =. doi:10.1017/S0515036100006619 , URL =

    work page doi:10.1017/s0515036100006619 1980

  35. [35]

    1996 , PAGES =

    B\"uhlmann, Hans , TITLE =. 1996 , PAGES =

    1996

  36. [36]

    Cheridito, Patrick and Kupper, Michael , TITLE =. Int. J. Theor. Appl. Finance , FJOURNAL =. 2011 , NUMBER =. doi:10.1142/S0219024911006292 , URL =

    work page doi:10.1142/s0219024911006292 2011

  37. [37]

    Finance Research Letters , author=

    Time-inconsistency of VaR and time-consistent alternatives , year=. Finance Research Letters , author=

  38. [38]

    Cheridito, Patrick and Kupper, Michael , TITLE =. Math. Financ. Econ. , FJOURNAL =. 2009 , NUMBER =. doi:10.1007/s11579-009-0020-3 , URL =

    work page doi:10.1007/s11579-009-0020-3 2009

  39. [39]

    Cheridito, Patrick and Li, Tianhui , TITLE =. Math. Finance , FJOURNAL =. 2009 , NUMBER =. doi:10.1111/j.1467-9965.2009.00364.x , URL =

    work page doi:10.1111/j.1467-9965.2009.00364.x 2009

  40. [40]

    Cheridito, Patrick and Li, Tianhui , TITLE =. Math. Financ. Econ. , FJOURNAL =. 2008 , NUMBER =. doi:10.1007/s11579-008-0013-7 , URL =

    work page doi:10.1007/s11579-008-0013-7 2008

  41. [41]

    Electron

    Cheridito, Patrick and Delbaen, Freddy and Kupper, Michael , TITLE =. Electron. J. Probab. , FJOURNAL =. 2006 , PAGES =. doi:10.1214/EJP.v11-302 , URL =

    work page doi:10.1214/ejp.v11-302 2006

  42. [42]

    Finance Stoch

    Cheridito, Patrick and Delbaen, Freddy and Kupper, Michael , TITLE =. Finance Stoch. , FJOURNAL =. 2005 , NUMBER =. doi:10.1007/s00780-004-0150-7 , URL =

    work page doi:10.1007/s00780-004-0150-7 2005

  43. [43]

    Stochastic Process

    Cheridito, Patrick and Delbaen, Freddy and Kupper, Michael , TITLE =. Stochastic Process. Appl. , FJOURNAL =. 2004 , NUMBER =. doi:10.1016/j.spa.2004.01.009 , URL =

    work page doi:10.1016/j.spa.2004.01.009 2004

  44. [44]

    Econometrica , FJOURNAL =

    Chen, Zengjing and Epstein, Larry , TITLE =. Econometrica , FJOURNAL =. 2002 , NUMBER =. doi:10.1111/1468-0262.00337 , URL =

    work page doi:10.1111/1468-0262.00337 2002

  45. [45]

    Ambiguity through confidence functions , JOURNAL =

    Chateauneuf, Alain and Faro, Jos\'e. Ambiguity through confidence functions , JOURNAL =. 2009 , NUMBER =. doi:10.1016/j.jmateco.2009.05.001 , URL =

    work page doi:10.1016/j.jmateco.2009.05.001 2009

  46. [46]

    Cerreia-Vioglio, Simone and Maccheroni, Fabio and Marinacci, Massimo and Montrucchio, Luigi , TITLE =. Math. Finance , FJOURNAL =. 2011 , NUMBER =. doi:10.1111/j.1467-9965.2010.00450.x , URL =

    work page doi:10.1111/j.1467-9965.2010.00450.x 2011

  47. [47]

    European J

    Centrone, Francesca and Rosazza Gianin, Emanuela , TITLE =. European J. Oper. Res. , FJOURNAL =. 2018 , NUMBER =. doi:10.1016/j.ejor.2017.11.051 , URL =

    work page doi:10.1016/j.ejor.2017.11.051 2018

  48. [48]

    Mathematical Finance , volume =

    Dana, Rose-Anne , title =. Mathematical Finance , volume =. doi:https://doi.org/10.1111/j.1467-9965.2005.00253.x , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9965.2005.00253.x , year =

    work page doi:10.1111/j.1467-9965.2005.00253.x 2005

  49. [49]

    , TITLE =

    Csisz\'ar, I. , TITLE =. Ann. Probability , FJOURNAL =. 1975 , PAGES =. doi:10.1214/aop/1176996454 , URL =

    work page doi:10.1214/aop/1176996454 1975

  50. [50]

    2015 , month = mar, howpublished =

    Cheridito, Patrick and Delbaen, Freddy and Drapeau, Samuel and Kupper, Michael , title =. 2015 , month = mar, howpublished =. doi:10.2139/ssrn.2572745 , url =

    work page doi:10.2139/ssrn.2572745 2015

  51. [51]

    Risk measures and attitudes , SERIES =

    Cheridito, Patrick and Drapeau, Samuel and Kupper, Michael , TITLE =. Risk measures and attitudes , SERIES =. 2013 , ISBN =. doi:10.1007/978-1-4471-4926-2\_1 , URL =

    work page doi:10.1007/978-1-4471-4926-2 2013

  52. [52]

    In memoriam

    Delbaen, Freddy , TITLE =. In memoriam. 2006 , ISBN =. doi:10.1007/978-3-540-35513-7\_17 , URL =

    work page doi:10.1007/978-3-540-35513-7 2006

  53. [53]

    Advances in finance and stochastics , PAGES =

    Delbaen, Freddy , TITLE =. Advances in finance and stochastics , PAGES =. 2002 , ISBN =

    2002

  54. [54]

    1989 , issn =

    A martingale approach to premium calculation principles in an arbitrage free market , journal =. 1989 , issn =. doi:https://doi.org/10.1016/0167-6687(89)90002-4 , url =

    work page doi:10.1016/0167-6687(89)90002-4 1989

  55. [55]

    Journal of Risk , year=

    Coherent allocation of risk capital , author=. Journal of Risk , year=

  56. [56]

    Finance Stoch

    Detlefsen, Kai and Scandolo, Giacomo , TITLE =. Finance Stoch. , FJOURNAL =. 2005 , NUMBER =. doi:10.1007/s00780-005-0159-6 , URL =

    work page doi:10.1007/s00780-005-0159-6 2005

  57. [57]

    Dynamic Asset Allocation: Portfolio Decomposition Formula and Applications , urldate =

    Detemple, Jérôme and Rindisbacher, Marcel , journal =. Dynamic Asset Allocation: Portfolio Decomposition Formula and Applications , urldate =

  58. [58]

    , TITLE =

    Deprez, Olivier and Gerber, Hans U. , TITLE =. Insurance Math. Econom. , FJOURNAL =. 1985 , NUMBER =. doi:10.1016/0167-6687(85)90014-9 , URL =

    work page doi:10.1016/0167-6687(85)90014-9 1985

  59. [59]

    Denuit, Michel and Dhaene, Jan and Goovaerts, Marc and Kaas, Rob and Laeven, Roger , TITLE =. Statist. Decisions , FJOURNAL =. 2006 , NUMBER =. doi:10.1524/stnd.2006.24.1.1 , URL =

    work page doi:10.1524/stnd.2006.24.1.1 2006

  60. [60]

    1994 , PAGES =

    Denneberg, Dieter , TITLE =. 1994 , PAGES =. doi:10.1007/978-94-017-2434-0 , URL =

    work page doi:10.1007/978-94-017-2434-0 1994

  61. [61]

    , TITLE =

    Delbaen, Freddy and Bellini, Fabio and Bignozzi, Valeria and Ziegel, Johanna F. , TITLE =. Finance Stoch. , FJOURNAL =. 2016 , NUMBER =. doi:10.1007/s00780-015-0279-6 , URL =

    work page doi:10.1007/s00780-015-0279-6 2016

  62. [62]

    Finance Stoch

    Delbaen, Freddy and Peng, Shi-Ge and Rosazza Gianin, Emanuela , TITLE =. Finance Stoch. , FJOURNAL =. 2010 , NUMBER =. doi:10.1007/s00780-009-0119-7 , URL =

    work page doi:10.1007/s00780-009-0119-7 2010

  63. [63]

    2012 , publisher=

    Monetary Utility Functions , author=. 2012 , publisher=

    2012

  64. [64]

    Drapeau, Samuel and Kupper, Michael , TITLE =. Math. Oper. Res. , FJOURNAL =. 2013 , NUMBER =. doi:10.1287/moor.1120.0560 , URL =

    work page doi:10.1287/moor.1120.0560 2013

  65. [65]

    2024 , eprint=

    Cash non-additive risk measures: horizon risk and generalized entropy , author=. 2024 , eprint=

    2024

  66. [66]

    Di Nunno, Giulia and Rosazza Gianin, Emanuela , TITLE =. SIAM J. Financial Math. , FJOURNAL =. 2024 , NUMBER =. doi:10.1137/23M1546804 , URL =

    work page doi:10.1137/23m1546804 2024

  67. [67]

    El Karoui, Nicole and Ravanelli, Claudia , TITLE =. Math. Finance , FJOURNAL =. 2009 , NUMBER =. doi:10.1111/j.1467-9965.2009.00380.x , URL =

    work page doi:10.1111/j.1467-9965.2009.00380.x 2009

  68. [68]

    and Peng, S.-G and Quenez, M

    El Karoui, N. and Peng, S.-G and Quenez, M. C. , TITLE =. Math. Finance , FJOURNAL =. 1997 , NUMBER =. doi:10.1111/1467-9965.00022 , URL =

    work page doi:10.1111/1467-9965.00022 1997

  69. [69]

    2016 , eprint=

    Quadratic Exponential Semimartingales and Application to BSDEs with Jumps , author=. 2016 , eprint=

    2016

  70. [70]

    Eiling, Esther and Laeven, Roger J. A. and Xu, Danjun , title =. 2024 , month =

    2024

  71. [71]

    , TITLE =

    Duffie, Darrell and Epstein, Larry G. , TITLE =. Econometrica , FJOURNAL =. 1992 , NUMBER =. doi:10.2307/2951600 , URL =

    work page doi:10.2307/2951600 1992

  72. [72]

    , volume =

    Esscher, F. , volume =. On the probability function in the collective theory of risk , journal =. 1932 , publisher =. doi:10.1080/03461238.1932.10405883 , URL =

    work page doi:10.1080/03461238.1932.10405883 1932

  73. [73]

    and Schneider, Martin , TITLE =

    Epstein, Larry G. and Schneider, Martin , TITLE =. J. Econom. Theory , FJOURNAL =. 2003 , NUMBER =. doi:10.1016/S0022-0531(03)00097-8 , URL =

    work page doi:10.1016/s0022-0531(03)00097-8 2003

  74. [74]

    , TITLE =

    Hansen, Lars Peter and Sargent, Thomas J. , TITLE =. 2008 , PAGES =. doi:10.1515/9781400829385 , URL =

    work page doi:10.1515/9781400829385 2008

  75. [75]

    Hamad\`ene, Said and Jeanblanc, Monique , TITLE =. Math. Oper. Res. , FJOURNAL =. 2007 , NUMBER =. doi:10.1287/moor.1060.0228 , URL =

    work page doi:10.1287/moor.1060.0228 2007

  76. [76]

    and Goovaerts, M

    Haezendonck, J. and Goovaerts, M. , TITLE =. Insurance Math. Econom. , FJOURNAL =. 1982 , NUMBER =. doi:10.1016/0167-6687(82)90020-8 , URL =

    work page doi:10.1016/0167-6687(82)90020-8 1982

  77. [77]

    and Linders, Dani\"el and Van Weert, Koen and Tank, Fatih , TITLE =

    Goovaerts, Marc J. and Linders, Dani\"el and Van Weert, Koen and Tank, Fatih , TITLE =. Insurance Math. Econom. , FJOURNAL =. 2012 , NUMBER =. doi:10.1016/j.insmatheco.2012.02.012 , URL =

    work page doi:10.1016/j.insmatheco.2012.02.012 2012

  78. [78]

    and Kaas, Rob and Laeven, Roger J

    Goovaerts, Marc J. and Kaas, Rob and Laeven, Roger J. A. , TITLE =. Insurance Math. Econom. , FJOURNAL =. 2010 , NUMBER =. doi:10.1016/j.insmatheco.2010.05.003 , URL =

    work page doi:10.1016/j.insmatheco.2010.05.003 2010

  79. [79]

    and Kaas, Rob and Laeven, Roger J

    Goovaerts, Marc J. and Kaas, Rob and Laeven, Roger J. A. , TITLE =. Insurance Math. Econom. , FJOURNAL =. 2010 , NUMBER =. doi:10.1016/j.insmatheco.2010.07.004 , URL =

    work page doi:10.1016/j.insmatheco.2010.07.004 2010

  80. [80]

    and Laeven, Roger J

    Goovaerts, Marc J. and Laeven, Roger J. A. , TITLE =. Insurance Math. Econom. , FJOURNAL =. 2008 , NUMBER =. doi:10.1016/j.insmatheco.2007.04.001 , URL =

    work page doi:10.1016/j.insmatheco.2007.04.001 2008

Showing first 80 references.