Insider trading consists in having an additional information, unknown from the common investor, and using it on the financial market. Mathematical modeling can study such behaviors, by modeling this additional information within the market, and comparing the investment strategies of an insider trader and a non-informed investor. Research on this subject has already been carried out by A. Grorud and M. Pontier since 1996, studying the problem in a wealth optimization point of view. This work focuses more on option hedging problems. We have chosen to study wealth equations as backward stochastic differential equations (BSDE), and we use Jeulin's method of enlargement of filtration to model the information of our insider trader. We will try to compare the strategies of an insider trader and a non-insider one. Different models are studied: at first prices are driven only by a Brownian motion and in a second part, we add jump processes (Poisson point processes) to the model. r

This paper is concerned with the determination of credit risk premia of defaultable contingent claims by means of indifference valuation principles. Assuming exponential utility preferences we derive representations of indifference premia of credit risk in terms of solutions of Backward Stochastic Differential Equations (BSDE). The class of BSDEs needed for that representation allows for quadratic growth generators and jumps at random times. Since the existence and uniqueness theory for this class of BSDEs has not yet been developed to the required generality, the first part of the paper is devoted to fill that gap. By using a simple constructive algorithm, and known results on continuous quadratic BSDEs, we provide sufficient conditions for the existence and uniqueness of quadratic BSDEs with discontinuities at random times.

This work extends the study of hedging problems in markets with asymmetrical information: an agent is supposed to possess an additional information on market prices, unknown to the common investor. The financial hedging problem for the influential and informed trader is modeled by a forward-backward stochastic differential equation, to be solved under an initial enlargement of the Brownian filtration. An existence and uniqueness theorem is proved under standard assumptions. The financial interpretation is derived, in terms of investment strategy for the informed and influential agent, as well as the conclusions concerning the general influenced market, in terms of completeness of the market. An example of such influenced and informed model is provided.

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BackgroundThe outbreak of SARS-CoV-2 virus has caused a major international health crisis with serious consequences in terms of public health and economy. In France, two lockdown periods were decided in 2020 to avoid the saturation of intensive care units (ICU) and an increase in mortality. The rapid dissemination of variant SARS-CoV-2 VOC 202012/01 has strongly influenced the course of the epidemic. Vaccines have been rapidly developed. Their efficacy against the severe forms of the disease has been established, and their efficacy against disease transmission is under evaluation. The aim of this paper is to compare the efficacy of several vaccination strategies in the presence of variants in controlling the COVID-19 epidemic through population immunity.MethodsAn agent-based model was designed to simulate with different scenarios the evolution of COVID-19 pandemic in France over 2021 and 2022. The simulations were carried out ignoring the occurrence of variants then taking into account their diffusion over time. The expected effects of three Non-Pharmaceutical Interventions (Relaxed-NPI, Intensive-NPI, and Extended-NPI) to limit the epidemic extension were compared. The expected efficacy of vaccines were the values recently estimated in preventing severe forms of the disease (75% and 94%) for the current used vaccines in France (Pfizer-BioNTech and Moderna since January 11, 2021, and AstraZeneca since February 2, 2021). All vaccination campaigns reproduced an advanced age-based priority advised by the Haute Autorité de Santé. Putative reductions of virus transmission were fixed at 0, 50, 75 and 90%. The effects of four vaccination campaign durations (6-month, 12-month, 18-month and 24-month) were compared.ResultsIn the absence of vaccination, the presence of variants led to reject the Relaxed-NPI because of a high expected number of deaths (170 to 210 thousands) and the significant overload of ICUs from which 35 thousand patients would be deprived. In comparison with the situation without vaccination, the number of deaths was divided by 7 without ICU saturation with a 6-month vaccination campaign. A 12-month campaign would divide the number of death by 3 with Intensive-NPI and by 6 with Extended-NPI (the latter being necessary to avoid ICU saturation). With 18-month and 24-month vaccination campaigns without Extended-NPI, the number of deaths and ICU admissions would explode.ConclusionAmong the four compared strategies the 6-month vaccination campaign seems to be the best response to changes in the dynamics of the epidemic due to the variants. The race against the COVID-19 epidemic is a race of vaccination strategy. Any further vaccination delay would increase the need of strengthened measures such as Extended-NPI to limit the number of deaths and avoid ICU saturation.