The prospective reserve of a life insurance contract with modifications in a Non-Markovian setting

In this thesis we inspect the prospective reserve of a life insurance contract. The objective is to generalize the concepts from the Markovian framework into the non-Markovian setting. A Markov process has independent increments which is not assumed for pure jump processes. The changes of the state of the life insurance contract can therefore posses dependencies among themselves. The prospective reserve will have a backward stochastic differential equation representation even in the non-Markovian setting. Furthermore we will consider the case of non-linear reserving where the payment process is allowed to be depended on the prospective reserve. This occurs under contract modifications where the current premium reserve is utilized to cover the liabilities induced by the modification and the rest is viewed as the assets of the customer. In other words the charged premiums in the life insurance contract are allowed to be calculated utilizing the present expected premium reserve as a part of the payment process. This creates a iterative cycle which questions the validity of the definition of the prospective reserve. The main theorems in this thesis are analogous extensions of the Thiele equation and the Cantelli Theorem to the non-Markovian setting. The Thiele equation is utilized to prove the BSDE representation for the prospective reserve and the Cantelli Theorem yields means to sustain the actuarial equivalence at contract modifications. Lastly we construct a lot of theory around jump processes, their compensators and compensated martingales even providing an explicit formula for the stochastic intensities and an Itˆo type of isometry for the compensated jump processes. We also prove an explicit solution to the Martingale Representation Theorem for a specific type of a stochastic process, which is applied to the prospective reserve.