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## Links with the existing literature

The indifference price p of a reinsurance layer is solution of an equation indicating that the utility of an insurance company, when it buys the contract and pays the price p, is equal to its utility when it does not enter the transaction. When the contract contains reinstatements, the total premium paid is random (see equation (3.2.3)). This can complicate the calculation of the indifference price. In the next sections we will bound this price by two easily computable values. Despite the huge literature on reinsurance problems, there are little references concerning the pricing of reinsurance contracts containing reinstatements. Sundt [114] obtained a formula for the pure premium of contracts with reinstatements ; this formula coincide with ours in this particular case. Sundt [114] and Wahlin [119] also give formulas corresponding to other premium principles, such that the standard deviation principle or the PH-transform. Mata [86] studies the properties of different pricing principles. Now that we better understand the properties and axioms of convex monetary risk measures, we can compare it with the properties stated by Mata, who defines indifference prices with respect to various comonotonic risk measures, but using the approximation (X) (X). If.

is convex and normalized, we can only say that (X) (X). That is why the values computed in [86] correspond, in the particular case where there is no cost of capital, to our upper bound (p2 appearing in equation (3.2.6)).

### Quadratic g-martingales with jumps

The theory of g-expectations was introduced by Peng in [99] as an example of non-linear expectations. Since then, numerous authors have generalized his results, extending them notably to the case of quadratic coefficients (see Ma and Yao [84]). An extension to discontinuous filtrations was obtained by Royer [106] and Lin [83]. In particular, Royer [106] gave domination conditions under which we can write a non-linear expectation as a g-expectation. We refer the interested reader to these papers for more details about these filtration-consistent operators, and we recall for simplicity some of their general properties below.

**Non-linear Doob Meyer decomposition**

We start by proving that the non-linear Doob Meyer decomposition first proved by Peng in [100] still holds in our context. We have two different sets of assumptions under which this result holds, and they are both related to the assumptions under which our comparison theorem 4.2.7 holds. From a technical point of view, our proof consists in approximating our generator by a sequence of Lipschitz generators. However, the novelty here is that because of the dependence of the generator in u, we cannot use the classical exponential transformation and then use some truncation arguments, as in [74] and [84]. Indeed, since u lives in an infinite dimensional space, those truncation type arguments no longer work a priori. Instead, inspired by [8], we will only use regularizations by inf-convolution, which are known to work in any Banach space.

#### Dual Representation and Inf-Convolution

We generalize in this section some results of Barrieu and El Karoui [6] to the case of quadratic BSDEs with jumps. We give a dual representation of the related g-expectations, viewed as convex dynamic risk measures and then we compute in an explicit manner the inf-convolution of two convex g-expectations.

**Table of contents :**

**1 Introduction générale **

1.1 Problèmes récents de mesure et de gestion du risque

1.1.1 Des mesures de risque aux intégrales de Choquet

1.1.2 Une problématique de tarification en Réassurance

1.2 Une étude des EDSRs quadratiques à sauts

1.2.1 Préliminaires sur les EDSRs

1.2.2 Une approche par point fixe

1.2.3 Applications aux mesures de risque dynamiques

1.3 Un saut dans les EDSRs du second ordre

1.3.1 EDSRs et incertitude de modèle

1.3.2 Résultats nouveaux

**I Risk Measures, BSDEs and Applications **

**2 Choquet Integrals and Applications **

2.1 Introduction

2.1.1 Choquet integrals

2.1.2 Reinsurance and risk transfer

2.2 Preliminaries on Risk Measures

2.2.1 Introduction

2.2.2 VaR Representation of Convex Risk Measures

2.2.3 Optimal risk transfer and inf-convolution

2.3 The case of Comonotonic Risk Measures

2.3.1 Representation results on Lp

2.3.2 A quantile weighting result for general Choquet integrals

2.3.3 Inf-Convolution of Choquet Integrals

2.3.4 An extension to subadditive comonotonic functionals

2.4 Examples

2.4.1 AVaR Examples

2.4.2 Epsilon-contaminated capacities

2.4.3 Volatility uncertainty

2.4.4 Involving the Entropic Risk Measure

**3 A Reinsurance Pricing Problem **

3.1 Introduction

3.1.1 The collective risk theory

3.1.2 Links with the existing literature

3.2 Pricing of Layers with Reinstatements

3.2.1 The contract payoff

3.2.2 The pricing bounds

**4 Quadratic Backward SDEs with Jumps **

4.1 Introduction

4.2 An existence and uniqueness result

4.2.1 Notations

4.2.2 Standard spaces and norms

4.2.3 A word on càdlàg BMO martingales

4.2.4 The non-linear generator

4.2.5 A priori estimates

4.2.6 Existence and uniqueness for a small terminal condition

4.2.7 Existence for a bounded terminal condition

4.2.8 A uniqueness result

4.2.9 A priori estimates and stability

4.3 Quadratic g-martingales with jumps

4.3.1 Non-linear Doob Meyer decomposition

4.3.2 Upcrossing inequality

4.4 Dual Representation and Inf-Convolution

4.4.1 Dual Representation of the g-expectation

4.4.2 Inf-Convolution of g-expectations

4.4.3 Examples of inf-convolution

**II Model Uncertainty and 2BSDEs with Jumps **

**5 Second Order Backward SDEs with Jumps **

5.1 Introduction

5.2 Issues related to aggregation

5.2.1 The stochastic basis

5.2.2 The main problem

5.2.3 Characterization by martingale problems

5.2.4 Notations and definitions

5.3 Preliminaries on 2BSDEs

5.3.1 The non-linear Generator

5.3.2 The Spaces and Norms

5.3.3 Formulation

5.3.4 Connection with standard backward SDEs with jumps

5.3.5 Connection with G-expectations and G-Lévy processes

5.4 Uniqueness result

5.4.1 Representation of the solution

5.4.2 A priori estimates and uniqueness of the solution

**6 Existence of 2BSDEs with Jumps and Applications **

6.1 Introduction

6.2 A direct existence argument

6.2.1 Notations

6.2.2 Existence when is in UCb(

6.2.3 Main result

6.2.4 An extension of the representation formula

6.3 Application to a robust utility maximization problem

6.3.1 The Market

6.3.2 Solving the optimization problem with a Lipschitz 2BSDEJ

6.3.3 A link with a particular quadratic 2BSDEJ

**A Technical Results for Quadratic BSDEs with Jumps **

**B Technical Results for 2BSDEs with Jumps **

B.1 The measures P;

B.2 Other properties

B.2.1 Upward directed set

B.2.2 Lr-Integrability of exponential martingales

**Bibliography**