A note on applications of stochastic ordering to control problems in insurance and finance

Abstract

We consider a controlled diffusion process ðX t Þ t$0 where the controller is allowed to choose drift m t and volatility s t from a set KðxÞ , R £ ð0; 1Þ when X t ¼ x. By choosing the largest m=s 2 at every point in time, an extremal process is constructed which is under suitable time changes stochastically larger than any other admissible process. This observation immediately leads to a very simple solution of problems where ruin or hitting probabilities have to be minimized. Under further conditions this extremal process also minimizes 'drawdown' probabilities.

Recombining Tree Approximations for Optimal Stopping for Diffusions

“…is in fact is a lower bound to b for β < 0, which can be proved using the comparison arguments in [5]. .8)), from which we conclude that the unique root is in (60,70).…”

Optimal investment to minimize the probability of drawdown

“…Pestien and Sudderth show that to maximize the probability of reaching b before a, one maximizes the ratio of the drift of the diffusion divided by its volatility squared. Bäuerle and Bayraktar (2014) prove, via pathwise arguments, that maximizing the ratio of the drift of a diffusion divided by its volatility squared minimizes the probability of ruin, as in Pestien and Sudderth (1985). The additional pathwise arguments demonstrate, among other things, that the same optimizer also optimizes any decreasing measurable function of the running minimum of a given, but arbitrary, diffusion.…”

Minimizing the Probability of Lifetime Ruin Under Ambiguity Aversion

“…Same as the non-robust case, we also show that the optimally controlled wealth process never reaches the so-called "safe level". This is different from the zero-hazard rate case, goes back to the work of Pestien and Sudderth [33] (also see [3]). …”

“…See, for example, chapter 1 of [23]. 3 To simplify the discussion, we only work with constant consumption rate. But the main techniques can be applied to proportional consumption rate, and more generally, to the case when the consumption rate is a non-negative, Lipschitz continuous function of wealth.…”