Second-order BSDEs with jumps: Formulation and uniqueness

In this paper, we define a notion of second-order backward stochastic differential equations with jumps (2BSDEJs for short), which generalizes the continuous case considered by Soner, Touzi and Zhang [Probab. Theory Related Fields 153 (2012) 149-190]. However, on the contrary to their formulation, where they can define pathwise the density of quadratic variation of the canonical process, in our setting, the compensator of the jump measure associated to the jumps of the canonical process, which is the counterpart of the density in the continuous case, depends on the underlying probability measures. Then in our formulation of 2BSDEJs, the generator of the 2BSDEJs depends also on the underlying probability measures through the compensator. But the solution to the 2BSDEJs can still be defined universally. Moreover, we obtain a representation of the $Y$ component of a solution of a 2BSDEJ as a supremum of solutions of standard backward SDEs with jumps, which ensures the uniqueness of the solution.Comment: Published at http://dx.doi.org/10.1214/14-AAP1063 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Second Order BSDEs with Jumps: Existence and probabilistic representation for fully-nonlinear PIDEs

In this paper, we pursue the study of second order BSDEs with jumps (2BSDEJs for short) started in our accompanying paper [15]. We prove existence of these equations by a direct method, thus providing complete wellposedness for 2BSDEJs. These equations are a natural candidate for the probabilistic interpretation of some fully non-linear partial integro-differential equations, which is the point of the second part of this work. We prove a non-linear Feynman-Kac formula and show that solutions to 2BSDEJs provide viscosity solutions of the associated PIDEs.Comment: 39 page

Quadratic BSDEs with jumps: a fixed-point approach

In this article, we prove the existence of bounded solutions of quadratic backward SDEs with jumps, that is to say for which the generator has quadratic growth in the variables (z,u). From a technical point of view, we use a direct fixed point approach as in Tevzadze [38], which allows us to obtain existence and uniqueness of a solution when the terminal condition is small enough. Then, thanks to a well-chosen splitting, we recover an existence result for general bounded solution. Under additional assumptions, we can obtain stability results and a comparison theorem, which as usual implies uniqueness.Comment: 29 page

Contribution à la modélisation probabiliste des propriétés géo-mécaniques des sols pour l’analyse de la fiabilité structurelle des fondations et conduites enterrées

This present study synthesizes a numerical approach to optimize foundation systems design and underground pipes based on the combination of elastoplastic soil behavior and probabilistic modeling of its geo-mechanical characteristics. Also, this work allows quantifying the effects of soil spatial variability on the mechanical and structural reliability of civil engineering infrastructures anchored at shallow depths in the ground. The illustrated simulations are based on simultaneous combination of the elastoplastic soil behavior and its spatial variability. The obtained results show that the probabilistic analysis of the spatial variability of soil properties into structure numerical modeling has a significant effect on the structural responses of civil engineering infrastructures, mainly if the structure length is relatively significant.Cette présente étude synthétise une approche numérique permettant d’optimiser la conception des systèmes de fondation et des conduites souterraines sur la base de la combinaison entre le comportement élastoplastique du sol et la modélisation probabiliste de ses caractéristiques géo-mécaniques. Ce travail permet aussi de quantifier l’influence de la variabilité spatiale de ces propriétés géo-mécaniques sur la fiabilité mécanique et structurelle des infrastructures ancrées à faible profondeur dans le sol. Les simulations présentées sont basées sur la prise en compte simultanée de l’aspect élastoplastique du sol et la variabilité spatiale de ses propriétés géo-mécaniques. Les résultats montrent que l’analyse probabiliste de variabilité spatiale des propriétés du sol dans la modélisation numérique, présente une influence considérable sur les réponses des infrastructures d’autant plus, si l’ouvrage en question possède une longueur relativement importante.Cette présente étude synthétise une approche numérique permettant d’optimiser la conception des systèmes de fondation et des conduites souterraines sur la base de la combinaison entre le comportement élastoplastique du sol et la modélisation probabiliste de ses caractéristiques géo-mécaniques. Ce travail permet aussi de quantifier l’influence de la variabilité spatiale de ces propriétés géo-mécaniques sur la fiabilité mécanique et structurelle des infrastructures ancrées à faible profondeur dans le sol. Les simulations présentées sont basées sur la prise en compte simultanée de l’aspect élastoplastique du sol et la variabilité spatiale de ses propriétés géo-mécaniques. Les résultats montrent que l’analyse probabiliste de variabilité spatiale des propriétés du sol dans la modélisation numérique, présente une influence considérable sur les réponses des infrastructures d’autant plus, si l’ouvrage en question possède une longueur relativement importante

Stopping with expectation constraints: 3 points suffice

We consider the problem of optimally stopping a one-dimensional continuous-time Markov process with a stopping time satisfying an expectation constraint. We show that it is sufficient to consider only stopping times such that the law of the process at the stopping time is a weighted sum of 3 Dirac measures. The proof uses recent results on Skorokhod embeddings in order to reduce the stopping problem to a linear optimization problem over a convex set of probability measures

Mechanical Strengths of Modified PET Mortar Composites in Aggressive MgSO4 Medium: ACI & B.S Predictions

Composites mortars based on plastic aggregates are often considered as an innovative materials of the future because of their potential and the advantages they present. In this paper, a comparative study was carried out on the effect of magnesium sulfate MgSO4 (5%) attack on the durability of composite mortars modified by recycled polyethylene terephthalate (PET). Laboratory tests were accomplished on limestone sand and cement mortars where the blended Portland cement was partially replaced by various volume fractions of PET plastic aggregates. Mechanical properties measured on specimens were used to assess the changes in the compressive strengths of PET-mortar composites exposed to MgSO4 attack at different ages, mainly the Young modulus of elasticity. Based on experimental compressive tests on selected specimens and there densities, the evolution of static Young modulus of elasticity has been discussed in accordance to predicted models proposed by (ACI-318) and (BS-8110) codes of practice. In addition, a comparative analysis has been carried out for corrosion resistance coefficients K of referenced mortar to those modified with plastic aggregates. It can be noted that, the corrosion resistance coefficients decrease as much as composite specimens are exposed to MgSO4 corrosive medium. For the case of modified composites, the values of K based on predicted Young modulus before and after immersion are better than the ones calculated for the unmodified mortar. Therefore, ACI 318 prediction model is recommended code for design and investigation works related to reparation mortars, screeds, pavements…etc. Also, it can be concluded that adding PET plastic aggregates by volume to blend Portland cement act to improve the corrosive resistance of this cement against MgSO4 aggressive medium

Contribution à la modélisation probabiliste des propriétés géo-mécaniques des sols pour l’analyse de la fiabilité structurelle des fondations et conduites enterrées

This present study synthesizes a numerical approach to optimize foundation systems design and underground pipes based on the combination of elastoplastic soil behavior and probabilistic modeling of its geo-mechanical characteristics. Also, this work allows quantifying the effects of soil spatial variability on the mechanical and structural reliability of civil engineering infrastructures anchored at shallow depths in the ground. The illustrated simulations are based on simultaneous combination of the elastoplastic soil behavior and its spatial variability. The obtained results show that the probabilistic analysis of the spatial variability of soil properties into structure numerical modeling has a significant effect on the structural responses of civil engineering infrastructures, mainly if the structure length is relatively significant.Cette présente étude synthétise une approche numérique permettant d’optimiser la conception des systèmes de fondation et des conduites souterraines sur la base de la combinaison entre le comportement élastoplastique du sol et la modélisation probabiliste de ses caractéristiques géo-mécaniques. Ce travail permet aussi de quantifier l’influence de la variabilité spatiale de ces propriétés géo-mécaniques sur la fiabilité mécanique et structurelle des infrastructures ancrées à faible profondeur dans le sol. Les simulations présentées sont basées sur la prise en compte simultanée de l’aspect élastoplastique du sol et la variabilité spatiale de ses propriétés géo-mécaniques. Les résultats montrent que l’analyse probabiliste de variabilité spatiale des propriétés du sol dans la modélisation numérique, présente une influence considérable sur les réponses des infrastructures d’autant plus, si l’ouvrage en question possède une longueur relativement importante.Cette présente étude synthétise une approche numérique permettant d’optimiser la conception des systèmes de fondation et des conduites souterraines sur la base de la combinaison entre le comportement élastoplastique du sol et la modélisation probabiliste de ses caractéristiques géo-mécaniques. Ce travail permet aussi de quantifier l’influence de la variabilité spatiale de ces propriétés géo-mécaniques sur la fiabilité mécanique et structurelle des infrastructures ancrées à faible profondeur dans le sol. Les simulations présentées sont basées sur la prise en compte simultanée de l’aspect élastoplastique du sol et la variabilité spatiale de ses propriétés géo-mécaniques. Les résultats montrent que l’analyse probabiliste de variabilité spatiale des propriétés du sol dans la modélisation numérique, présente une influence considérable sur les réponses des infrastructures d’autant plus, si l’ouvrage en question possède une longueur relativement importante

Structural Characteristics of Composite Mortars and their Evolution with PET Substitution Level for Several Specimens’ Ages

This present study aims to investigate the evolution of structural response of PET-Mortar composite test with a short-beam specimen in three-point bending tests, with composite mortar ages and volumetric polymer rate and this, based on compression strength tests. The ultimate PET-mortar composite structural responses are calculated at the mid span of the short-beam by the mean of mechanics-of-materials theory basis. According to this theory, the distribution of bending moments and shear forces at any point of the composite short-beam specimen doesn’t depend on material mechanical properties especially the young modulus of modified mortar composite; so, the structural response analysis has been limited to investigate the evolution of ultimate deflection with several volumetric PET rates and composite mortar ages. In the other hand, we present a comparative study between experimental test results of splitting tensile and compressive strengths with the ones predicted by codale previsions (ACI-363 and B.S) codes in terms of PET mortar ages and volumetric PET rates in order to recommend the most suitable design code for PET-mortar composite applications in construction industries

On the monotone stability approach to BSDEs with jumps: Extensions, concrete criteria and examples

We show a concise extension of the monotone stability approach to backward stochastic differential equations (BSDEs) that are jointly driven by a Brownian motion and a random measure for jumps, which could be of infinite activity with a non-deterministic and time inhomogeneous compensator. The BSDE generator function can be non convex and needs not to satisfy global Lipschitz conditions in the jump integrand. We contribute concrete criteria, that are easy to verify, for results on existence and uniqueness of bounded solutions to BSDEs with jumps, and on comparison and a-priori $L^{\infty}$-bounds. Several examples and counter examples are discussed to shed light on the scope and applicability of different assumptions, and we provide an overview of major applications in finance and optimal control.Comment: 28 pages. Added DOI https://link.springer.com/chapter/10.1007%2F978-3-030-22285-7_1 for final publication, corrected typo (missing gamma) in example 4.1

Indifference Pricing of Reinsurance with Reinstatements Using Coherent Monetary Criteria

We consider the problem of indifference pricing of reinsurance contracts that contain a reinstatement clause. We define the indifference price relative to both a monetary utility function and a risk measure, to take into account both the risk reduction and the relief of capital immobilization provided by reinsurance. We characterize the indifference price as the unique solution to a fixed point equation and we bound the price by two easily computable values, if one has access to losses simulations. We illustrate our results on a European catastrophe insurance portfolio, and we conduct a simulation study for comparison and reproducibility purposes, where we include the case of dependence between claim arrivals, using Hawkes processes