Mean-field approximations in insurance

Philipp C. Hornung

arxiv: 2511.04198 · v2 · submitted 2025-11-06 · 💱 q-fin.RM · math.PR

Mean-field approximations in insurance

Philipp C. Hornung This is my paper

Pith reviewed 2026-05-18 00:46 UTC · model grok-4.3

classification 💱 q-fin.RM math.PR

keywords mean-field approximationinsurance liabilitiesjump processesconvergencecohortintegro-differential equationslife insurancenon-life insurance

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The pith

Insurance liability for a large dependent cohort converges to its mean-field counterpart as the number of individuals tends to infinity.

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

Calculating insurance liabilities for many interdependent individuals normally requires solving a high-dimensional system of coupled linear forward integro-differential equations that quickly becomes intractable. The paper replaces this with a mean-field model consisting of a low-dimensional system of nonlinear forward integro-differential equations. Under regularity conditions, the true liability expressed as a conditional expectation of a functional of an underlying jump process converges to the mean-field version when cohort size grows without bound. This matters because it renders exact liability calculations feasible for realistic portfolio sizes containing thousands of policyholders. Concrete examples drawn from both life insurance and non-life insurance illustrate the computational gain.

Core claim

Subject to certain regularity conditions, the insurance liability viewed as a (conditional) expectation of a functional of an underlying jump process converges to its mean-field counterpart as the number of individuals in the cohort goes to infinity.

What carries the argument

The mean-field limit obtained by replacing the high-dimensional linear system with a low-dimensional nonlinear system of forward integro-differential equations for the jump-process functional.

If this is right

The original high-dimensional system of coupled linear equations can be replaced by a low-dimensional nonlinear system for any sufficiently large cohort.
Insurance liability calculations become computationally tractable for portfolios with thousands of interdependent policyholders.
The same mean-field replacement applies equally to life-insurance and non-life-insurance models.
Exact high-dimensional solutions remain infeasible while the approximating nonlinear system stays solvable.

Where Pith is reading between the lines

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

Portfolio-level risk measures such as value-at-risk could be computed directly from the low-dimensional mean-field equations rather than from the full high-dimensional system.
The convergence result may extend to related interacting-agent models in credit risk or systemic-risk settings where many contracts depend on a common jump process.
Empirical checks of convergence speed for finite but large cohorts would indicate how many policyholders are needed before the approximation becomes accurate enough for pricing.
Relaxing the regularity conditions to allow for more realistic dependence structures could be tested by constructing counter-examples where convergence fails.

Load-bearing premise

The regularity conditions placed on the jump process and on the functional are strong enough to guarantee convergence of the liability to the mean-field limit.

What would settle it

A numerical experiment on a concrete jump process satisfying the regularity conditions in which the computed liability for successively larger finite cohorts fails to approach the value given by the corresponding mean-field equations.

Figures

Figures reproduced from arXiv: 2511.04198 by Philipp C. Hornung.

**Figure 1.** Figure 1: State space E “ t1, 2, 3, 4u for the SIRD model. The arrows represent the possible transitions. The state of the individual can be modelled as a jump process X with intensity kernel µtpdy, x, ρq “ γ y t px, ρqνpdyq, where ν is the counting measure on E and γ y t px, ρq are transition intiensities satisfying the conditions in Proposition 6.6. The only non-zero transition intensities are γ 2 t p1, ρq, γ 3 t… view at source ↗

read the original abstract

The calculation of the insurance liabilities of a cohort of dependent individuals in general requires the solution of a high-dimensional system of coupled linear forward integro-differential equations, which is infeasible for a larger cohort. However, by using a mean-field model, the high dimensional system of linear forward equations can be replaced by a low-dimensional system of non-linear forward integro-differential equations. We show that, subject to certain regularity conditions, the insurance liability viewed as a (conditional) expectation of a functional of an underlying jump process converges to its mean-field counterpart, as the number of individuals in the cohort goes to infinity. Examples from both life- and non-life insurance illuminate the practical importance of mean-field approximations.

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.

Desk Editor's Note private letter to a colleague

The paper proves convergence of insurance liabilities to a mean-field limit for large jump-process cohorts under regularity conditions, but the conditions may not cover common discontinuous functionals such as ruin indicators.

read the letter

The main point is that for a cohort of dependent individuals whose risks follow jump processes, the expected liability converges to the solution of a lower-dimensional nonlinear forward integro-differential equation as the number of people goes to infinity. This replaces an intractable high-dimensional linear system with something computable, which matters for large portfolios in both life and non-life insurance. The paper supplies examples to show where the approximation would actually be used in practice, and that is the useful part of the work. It takes existing mean-field ideas for interacting particles and applies them directly to the insurance liability as a conditional expectation, which is a reasonable incremental step in this setting. The derivation itself looks formally grounded on the model assumptions given in the abstract. The soft spot is the handling of the regularity conditions. The claim requires the functional of the jump process to behave well under convergence of the empirical measure, yet many standard insurance quantities are discontinuous, such as indicators that aggregate loss exceeds a threshold or first-passage times. If the paper only states generic conditions without checking continuity or providing explicit verification for the examples, the interchange of limit and expectation can fail even when the particle system converges in law. That is a concrete gap rather than a minor quibble. This paper is for actuarial researchers and risk modelers who work on scalable approximations for dependent portfolios and want theoretical support for mean-field methods. A reader focused on stochastic processes in insurance would find the setup and limit result worth examining. It deserves a serious referee because the core convergence statement is stated clearly and the application area is practically relevant, even if revisions will be needed to tighten the conditions and address the continuity issue for typical functionals. I would send it to peer review.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes mean-field approximations to compute insurance liabilities for large cohorts of dependent individuals, replacing intractable high-dimensional systems of coupled linear forward integro-differential equations with low-dimensional nonlinear forward equations. The central claim is that, subject to certain regularity conditions, the insurance liability expressed as a conditional expectation of a functional of an underlying jump process converges to its mean-field counterpart as the number of individuals tends to infinity. Examples from life and non-life insurance are used to illustrate practical relevance.

Significance. If the convergence holds under regularity conditions that are verified for standard insurance functionals, the result would provide a theoretically grounded method to reduce computational complexity for large-portfolio risk calculations, with direct implications for actuarial pricing and solvency assessment. The approach leverages stochastic process convergence and mean-field limits, offering a clear path from particle systems to tractable approximations.

major comments (2)

[Abstract] Abstract: the convergence result E[f(X^N) | filtration] → E[f(X^∞)] is stated to hold under 'certain regularity conditions,' but these conditions are not specified in detail. Insurance functionals such as ruin indicators or aggregate-loss thresholds are typically discontinuous with respect to the Skorokhod or Wasserstein topology in which the empirical measure converges; without explicit verification that f is continuous (or bounded and continuous a.e.) or an exclusion of such cases, the interchange of limit and conditional expectation may fail even when the particle system converges in law.
[Examples section] Examples section (life- and non-life insurance): the illuminated examples invoke the convergence for practical insurance liabilities, yet do not check or state that the specific functionals (e.g., first-passage times or max-loss indicators) satisfy the required continuity or boundedness properties with respect to the mean-field limit topology. This leaves the practical importance of the approximation dependent on an unverified assumption.

minor comments (1)

[Introduction] The notation for the underlying filtration and the precise definition of the jump process X^N could be introduced earlier to improve readability of the conditional-expectation statement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript on mean-field approximations for insurance liabilities and for the constructive comments emphasizing the need for explicit regularity conditions and their verification in examples. We respond to each major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the convergence result E[f(X^N) | filtration] → E[f(X^∞)] is stated to hold under 'certain regularity conditions,' but these conditions are not specified in detail. Insurance functionals such as ruin indicators or aggregate-loss thresholds are typically discontinuous with respect to the Skorokhod or Wasserstein topology in which the empirical measure converges; without explicit verification that f is continuous (or bounded and continuous a.e.) or an exclusion of such cases, the interchange of limit and conditional expectation may fail even when the particle system converges in law.

Authors: We agree that the abstract is concise and does not elaborate on the regularity conditions. The main theorem in the manuscript states the result under the assumption that the functional f is bounded and continuous with respect to the topology of weak convergence of empirical measures (or, more generally, continuous almost everywhere with respect to the law of the limiting process). This is sufficient for the interchange of limit and expectation via the portmanteau theorem. We acknowledge that many insurance functionals, including ruin indicators and threshold exceedance indicators, are discontinuous. The result continues to hold provided the discontinuity set has measure zero under the limiting measure, which is a standard condition in the mean-field literature. In the revision we will expand the abstract to mention these conditions explicitly and add a short clarifying paragraph after the theorem statement. revision: yes
Referee: [Examples section] Examples section (life- and non-life insurance): the illuminated examples invoke the convergence for practical insurance liabilities, yet do not check or state that the specific functionals (e.g., first-passage times or max-loss indicators) satisfy the required continuity or boundedness properties with respect to the mean-field limit topology. This leaves the practical importance of the approximation dependent on an unverified assumption.

Authors: The examples are presented to illustrate computational tractability rather than to serve as complete case studies. We accept that the manuscript does not explicitly verify continuity or boundedness for the concrete functionals (first-passage times in the life-insurance example and maximum-loss indicators in the non-life example). In the revised version we will insert a brief discussion in the examples section (or a short appendix) stating the precise assumptions under which these functionals meet the required regularity conditions. For the first-passage-time functional we note that it is bounded and continuous almost everywhere when the limiting process has no atoms at the barrier; for the max-loss indicator we invoke the same almost-everywhere continuity argument with respect to the Wasserstein topology. These additions will make the practical applicability fully rigorous. revision: yes

Circularity Check

0 steps flagged

Mean-field convergence is a standard limit theorem, self-contained and non-circular

full rationale

The paper derives an asymptotic convergence result: under regularity conditions, the conditional expectation of a functional of an N-particle jump process converges to the mean-field limit as N tends to infinity. This follows directly from the model setup of interacting jump processes and standard propagation-of-chaos arguments for empirical measures. No parameters are fitted to data and then relabeled as predictions, no self-citations form the load-bearing justification, and the functional is not defined in terms of the limit it is supposed to approximate. The derivation chain is therefore independent of the claimed result and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on regularity conditions for the jump process and functional; these are standard domain assumptions in stochastic analysis but are not detailed here.

axioms (1)

domain assumption Regularity conditions on the underlying jump process and the functional of interest
Invoked in the abstract to ensure the convergence of the expectation to the mean-field limit holds.

pith-pipeline@v0.9.0 · 5401 in / 1237 out tokens · 45202 ms · 2026-05-18T00:46:15.975732+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Disability insurance with collective health claims: A mean-field approach
q-fin.RM 2025-12 unverdicted novelty 6.0

A mean-field approximation reduces collective health claims in a semi-Markov disability model to a lower-dimensional system of non-linear forward integro-differential equations for pricing.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · cited by 1 Pith paper

[1]

Luisa Andreis, Paolo Dai Pra, and Markus Fischer,McKean-Vlasov limit for interacting systems with simultaneous jumps, Stochastic Analysis and Applications36(2018), 960–995

work page 2018
[2]

Patrick Bllingsley,Convergence of Probability Measures, Wiley, 1999

work page 1999
[3]

J. R. Blum, H. Chernoff, M. Rosenblatt, and H. Teicher,Central Limit Theorems For Inter- changeable Processes, Canadian Journal of Mathematics10(1958), 222–229

work page 1958
[4]

Louis-Pierre Chaintron and Antoine Diez,Propagation of chaos: A review of models, methods and applications. I. Models and methods, Kinetic and Related Models15(2022), 895–1015

work page 2022
[5]

,Propagation of chaos: A review of models, methods and applications. II. Applications, Kinetic and Related Models15(2022), 1017–1173

work page 2022
[6]

Christiansen and Boualem Djehiche,Non-linear reserving and multiple contract modifications in life insurance, Insurance: Mathematics and Economics93(2020), 187–195

Marcus C. Christiansen and Boualem Djehiche,Non-linear reserving and multiple contract modifications in life insurance, Insurance: Mathematics and Economics93(2020), 187–195

work page 2020
[7]

,As-if-Markov reserves for reserve-dependent payments, Insurance: Mathematics and Economics124(2025), 103–129

work page 2025
[8]

Boualem Djehiche and Ingemar Kaj,The rate function for some measure-valued jump pro- cesses, The Annals of Probability23(1995), 1414–1438

work page 1995
[9]

Boualem Djehiche and Bj¨ orn L¨ ofdahl,Nonlinear reserving in life insurance: Aggregation and mean-field approximation, Insurance: Mathematics and Economics69(2016), 1–13

work page 2016
[10]

Fahrenwaldt, Stefan Weber, and Kerstin Weske,Pricing of cyber insurance con- tracts in a network model, ASTIN Bulletin48(2018), 1175–1218

Matthias A. Fahrenwaldt, Stefan Weber, and Kerstin Weske,Pricing of cyber insurance con- tracts in a network model, ASTIN Bulletin48(2018), 1175–1218

work page 2018
[11]

Feinberg, Manasa Mandava, and Albert N

Eugene A. Feinberg, Manasa Mandava, and Albert N. Shiryaev,On solutions of Kolmogorov’s equations for nonhomogeneous jump Markov processes, Journal of Mathematical Analysis and Applications411(2014), 261–270

work page 2014
[12]

Willy Feller,On the Integro-Differential Equations of Purely Discontinuous Markoff Pro- cesses, Transactions of the American Mathematical Society48(1940), 488–515

work page 1940
[13]

Nicolas Fournier and Arnaud Guillin,On the rate of convergence in Wasserstein distance of the empirical measure, Probability Theory and Related Fields162(2015), 707–738

work page 2015
[14]

Laura Francis and Mogens Steffensen,Individual life insurance during epidemics, Annals of Actuarial Science18(2024), 152–175

work page 2024
[15]

Hornung,Disability insurance with collective health claims: A mean-field approach, 2025, In preparation

Christian Furrer and Philipp C. Hornung,Disability insurance with collective health claims: A mean-field approach, 2025, In preparation

work page 2025
[16]

Alexander David Gottlieb,Markov Transitions and the Propagation of Chaos, 1998, Ph.D. Thesis

work page 1998
[17]

Carl Graham,McKean-Vlasov Ito-Skorohod equations and nonlinear diffusions with discrete jump sets, Stochastic Processes and their Applications40(1992), 69–82

work page 1992
[18]

,Nonlinear diffusion with jumps, Ann. Inst. Henri Poincar´ e Probab. Stat.29(1992), 393–402

work page 1992
[19]

Maxime Hauray and St´ ephane Mischler,On Kac’s chaos and related problems, Journal of Functional Analysis266(2014), 6055–6157

work page 2014
[20]

Shiryaev,Limit Theorems for Stochastic Processes, Springer, 2003

Jean Jacod and Albert N. Shiryaev,Limit Theorems for Stochastic Processes, Springer, 2003

work page 2003
[21]

Kac,Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability3(1956), 171–197

M. Kac,Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability3(1956), 171–197

work page 1956
[22]

Kurtz,Equivalence of Stochastic Equations and Martingale Problems, Stochastic Anal- ysis 2010 (D

T.G. Kurtz,Equivalence of Stochastic Equations and Martingale Problems, Stochastic Anal- ysis 2010 (D. Crisan, ed.), Springer, 2010, pp. 113–130

work page 2010
[23]

G¨ unther Last and Andreas Brandt,Marked Point Processes on the Real Line, Springer, 1999

work page 1999
[24]

H. P. McKean,A class of Markov processes associated with nonlinear parabolic equations, Proceedings of the National Academy of Sciences of the United States of America56(1966), 1907–1911

work page 1966
[25]

McKean,Propagation of chaos for a class of non-linear parabolic equations, Lecture Series in Differential Equations, Volume 2 (A

Henry P. McKean,Propagation of chaos for a class of non-linear parabolic equations, Lecture Series in Differential Equations, Volume 2 (A. K. Aziz, ed.), Van Nostrand Reinhold Company, 1969, pp. 177–194

work page 1969
[26]

Marco Rehmeier and Michael R¨ ockner,On Nonlinear Markov Processes in the Sense of McK- ean, Journal of Theoretical Probability38(2025)

work page 2025
[27]

36 PHILIPP C

Tokuzo Shiga and Hiroshi Tanaka,Central Limit Theorem for a System of Markovian Par- ticles with Mean Field Interactions, Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und verwandte Gebiete69(1985), 439–459. 36 PHILIPP C. HORNUNG

work page 1985
[28]

Dario Trevisan,Well-posedness of multidimensional diffusion processes with weakly differen- tiable coefficients, Electronic Journal of Probability21(2016), 1–41

work page 2016
[29]

MEAN-FIELD APPROXIMATIONS IN INSURANCE 37 AppendixA.Proof of Theorems 2.2 and 2.7 Before starting with the proofs, we will introduce some notation

C´ edric Villani,Optimal Transport, Old and New, Springer, 2009. MEAN-FIELD APPROXIMATIONS IN INSURANCE 37 AppendixA.Proof of Theorems 2.2 and 2.7 Before starting with the proofs, we will introduce some notation. Letπ t :Dprτ, Ts, Eq ÑEbe the projectionπ tpωq “ω t. In the following we will work on the canoncial spacepDprτ, Ts, Eq,BpDprτ, Ts, Eqq,Fq, where...

work page 2009

[1] [1]

Luisa Andreis, Paolo Dai Pra, and Markus Fischer,McKean-Vlasov limit for interacting systems with simultaneous jumps, Stochastic Analysis and Applications36(2018), 960–995

work page 2018

[2] [2]

Patrick Bllingsley,Convergence of Probability Measures, Wiley, 1999

work page 1999

[3] [3]

J. R. Blum, H. Chernoff, M. Rosenblatt, and H. Teicher,Central Limit Theorems For Inter- changeable Processes, Canadian Journal of Mathematics10(1958), 222–229

work page 1958

[4] [4]

Louis-Pierre Chaintron and Antoine Diez,Propagation of chaos: A review of models, methods and applications. I. Models and methods, Kinetic and Related Models15(2022), 895–1015

work page 2022

[5] [5]

,Propagation of chaos: A review of models, methods and applications. II. Applications, Kinetic and Related Models15(2022), 1017–1173

work page 2022

[6] [6]

Christiansen and Boualem Djehiche,Non-linear reserving and multiple contract modifications in life insurance, Insurance: Mathematics and Economics93(2020), 187–195

Marcus C. Christiansen and Boualem Djehiche,Non-linear reserving and multiple contract modifications in life insurance, Insurance: Mathematics and Economics93(2020), 187–195

work page 2020

[7] [7]

,As-if-Markov reserves for reserve-dependent payments, Insurance: Mathematics and Economics124(2025), 103–129

work page 2025

[8] [8]

Boualem Djehiche and Ingemar Kaj,The rate function for some measure-valued jump pro- cesses, The Annals of Probability23(1995), 1414–1438

work page 1995

[9] [9]

Boualem Djehiche and Bj¨ orn L¨ ofdahl,Nonlinear reserving in life insurance: Aggregation and mean-field approximation, Insurance: Mathematics and Economics69(2016), 1–13

work page 2016

[10] [10]

Fahrenwaldt, Stefan Weber, and Kerstin Weske,Pricing of cyber insurance con- tracts in a network model, ASTIN Bulletin48(2018), 1175–1218

Matthias A. Fahrenwaldt, Stefan Weber, and Kerstin Weske,Pricing of cyber insurance con- tracts in a network model, ASTIN Bulletin48(2018), 1175–1218

work page 2018

[11] [11]

Feinberg, Manasa Mandava, and Albert N

Eugene A. Feinberg, Manasa Mandava, and Albert N. Shiryaev,On solutions of Kolmogorov’s equations for nonhomogeneous jump Markov processes, Journal of Mathematical Analysis and Applications411(2014), 261–270

work page 2014

[12] [12]

Willy Feller,On the Integro-Differential Equations of Purely Discontinuous Markoff Pro- cesses, Transactions of the American Mathematical Society48(1940), 488–515

work page 1940

[13] [13]

Nicolas Fournier and Arnaud Guillin,On the rate of convergence in Wasserstein distance of the empirical measure, Probability Theory and Related Fields162(2015), 707–738

work page 2015

[14] [14]

Laura Francis and Mogens Steffensen,Individual life insurance during epidemics, Annals of Actuarial Science18(2024), 152–175

work page 2024

[15] [15]

Hornung,Disability insurance with collective health claims: A mean-field approach, 2025, In preparation

Christian Furrer and Philipp C. Hornung,Disability insurance with collective health claims: A mean-field approach, 2025, In preparation

work page 2025

[16] [16]

Alexander David Gottlieb,Markov Transitions and the Propagation of Chaos, 1998, Ph.D. Thesis

work page 1998

[17] [17]

Carl Graham,McKean-Vlasov Ito-Skorohod equations and nonlinear diffusions with discrete jump sets, Stochastic Processes and their Applications40(1992), 69–82

work page 1992

[18] [18]

,Nonlinear diffusion with jumps, Ann. Inst. Henri Poincar´ e Probab. Stat.29(1992), 393–402

work page 1992

[19] [19]

Maxime Hauray and St´ ephane Mischler,On Kac’s chaos and related problems, Journal of Functional Analysis266(2014), 6055–6157

work page 2014

[20] [20]

Shiryaev,Limit Theorems for Stochastic Processes, Springer, 2003

Jean Jacod and Albert N. Shiryaev,Limit Theorems for Stochastic Processes, Springer, 2003

work page 2003

[21] [21]

Kac,Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability3(1956), 171–197

M. Kac,Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability3(1956), 171–197

work page 1956

[22] [22]

Kurtz,Equivalence of Stochastic Equations and Martingale Problems, Stochastic Anal- ysis 2010 (D

T.G. Kurtz,Equivalence of Stochastic Equations and Martingale Problems, Stochastic Anal- ysis 2010 (D. Crisan, ed.), Springer, 2010, pp. 113–130

work page 2010

[23] [23]

G¨ unther Last and Andreas Brandt,Marked Point Processes on the Real Line, Springer, 1999

work page 1999

[24] [24]

H. P. McKean,A class of Markov processes associated with nonlinear parabolic equations, Proceedings of the National Academy of Sciences of the United States of America56(1966), 1907–1911

work page 1966

[25] [25]

McKean,Propagation of chaos for a class of non-linear parabolic equations, Lecture Series in Differential Equations, Volume 2 (A

Henry P. McKean,Propagation of chaos for a class of non-linear parabolic equations, Lecture Series in Differential Equations, Volume 2 (A. K. Aziz, ed.), Van Nostrand Reinhold Company, 1969, pp. 177–194

work page 1969

[26] [26]

Marco Rehmeier and Michael R¨ ockner,On Nonlinear Markov Processes in the Sense of McK- ean, Journal of Theoretical Probability38(2025)

work page 2025

[27] [27]

36 PHILIPP C

Tokuzo Shiga and Hiroshi Tanaka,Central Limit Theorem for a System of Markovian Par- ticles with Mean Field Interactions, Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und verwandte Gebiete69(1985), 439–459. 36 PHILIPP C. HORNUNG

work page 1985

[28] [28]

Dario Trevisan,Well-posedness of multidimensional diffusion processes with weakly differen- tiable coefficients, Electronic Journal of Probability21(2016), 1–41

work page 2016

[29] [29]

MEAN-FIELD APPROXIMATIONS IN INSURANCE 37 AppendixA.Proof of Theorems 2.2 and 2.7 Before starting with the proofs, we will introduce some notation

C´ edric Villani,Optimal Transport, Old and New, Springer, 2009. MEAN-FIELD APPROXIMATIONS IN INSURANCE 37 AppendixA.Proof of Theorems 2.2 and 2.7 Before starting with the proofs, we will introduce some notation. Letπ t :Dprτ, Ts, Eq ÑEbe the projectionπ tpωq “ω t. In the following we will work on the canoncial spacepDprτ, Ts, Eq,BpDprτ, Ts, Eqq,Fq, where...

work page 2009