The scope of this thesis is to discuss a stochastic differential game in non-life insurance, in particular in the area of reinsurance. More in detail, in the first Chapter theoretical results are presented. Firstly, the theory regarding stochastic processes and stochastic dif- ferential equation is settled; then we introduce formally the notions of classic Cramér-Lundberg risk model and its diffusion approxima- tion. Since a significative role in the model is played by reinsurance, one part will be devoted to the discussion of this topic.

A consistent section regards stochastic control, especially in rein- surance. Then, the introduction of game theory will lead to the notion of differential games, at the intersection with controlled sys- tem.

In the second Chapter, we analyse exhaustively the paper by As- mussen et al. (2019), in which he treats a stochastic differential game and provides a Nash equilibrium for beta distributed market frictions.

Finally, in the last chapter we consider the setting proposed by Asmussen, basing the analysis on the diffusion approximation to a Cramér-Lundberg process extended to allow investment in a risk- free asset. Both companies are assumed to sign a reinsurance con- tract and hence a reinsurance component alterates the picture. Af- ter the presentation of the mathematical model underlying, we dis- cuss the client’s problem and rewrite the theoretical results seen in Chapter 2 for this case. Lastly, a numerical illustration is provided, in order to properly show the computation of the optimal premia and the Nash equilibrium.