Thomas Bernhardt

The University of Manchester · School of Mathematics

The income stability of a closed pooled annuity fund is studied. The focus is on quantifying the impact of inhomogeneous initial savings amounts on idiosyncratic longevity risk. Besides wealth inhomogeneity, the members of the pool are independent and identical copies of each other. We ignore systematic investment risk or mortality risk and define...

We show that the barrier function in Root's solution to the Skorokhod embedding problem is continuous and finite at every point where the target measure has no atom and its absolutely continuous part is locally bounded away from zero.

The number of people who receive a stable income for life from a closed pooled annuity fund is studied. Income stability is defined as keeping the income within a specified tolerance of the initial income in a fixed proportion of future scenarios. The focus is on quantifying the effect of the number of members, which drives the level of idiosyncrat...

We consider the problem of optimally stopping a general one-dimensional stochastic differential equation (SDE) with generalised drift over an infinite time horizon. First, we derive a complete characterisation of the solution to this problem in terms of vari- ational inequalities. In particular, we prove that the problem’s value function is the dif...

We consider the problem of optimally stopping a general one-dimensional stochastic differential equation (SDE) with generalised drift over an infinite time horizon. First, we derive a complete characterisation of the solution to this problem in terms of variational inequalities. In particular, we prove that the problem's value function is the diffe...

We introduce a new pension product that offers retirees the opportunity for a lifelong income and a bequest for their estate. Based on a tontine mechanism, the product divides pension savings between a tontine account and a bequest account. The tontine account is given up to a tontine pool upon death while the bequest account value is paid to the r...

In the first part of the thesis, the solvability of stochastic differential equations with reflecting boundary conditions is studied. Such equations arise in singular stochastic control problems as a way for determining the optimal strategies. The stochastic differential equations represent homogeneous one-dimensional diffusions while the boundarie...