risk Sensitive control of the lifetime ruin problem Asaf Cohen (joint work with Erhan Bayraktar) Department of Mathematics University of Michigan International Conference on Financial Risks and their Management Mar, 2016

Contents The Model The Differential Game The Approximation Summary

Contents The Model The Differential Game The Approximation Summary

The classical model The Model An investor trades continuously in a Black-Scholes market (with no transaction costs): Riskless asset entire wealth Risky asset (geometric Brownian motion) are constants. Wealth process consumption function The classical optimization problem

Literature – (classical) lifetime ruin problem The Model Literature – (classical) lifetime ruin problem [1] M.A. Milevsky and C. Robinson. Self-annuitization and ruin in retirement. North American Actuarial Journal, 2000. [2] V.R. Young. Optimal investment strategy to minimize the probability of lifetime ruin. North American Actuarial Journal, 2004. [3] E. Bayraktar and V.R. Young. Minimizing the probability of lifetime ruin under borrowing constraints. Insurance: Mathematics and Economics, 2007. [4] H. Yener. Minimizing the lifetime ruin under borrowing and short-selling constraints. Scandinavian Actuarial Journal, 2014. [5] E. Bayraktar and Y. Zhang. Minimizing the probability of lifetime ruin under ambiguity aversion. SIAM J. Control Optimization, 2015.  [6] A. C. and V. Young. Minimizing lifetime poverty with penalty for ruin, ??, 2016. 

Small noise, which means small ruin probability The Model Criticism and our model – why “small noise scaling”? While there are many papers in the classical framework, there are no papers with the small noise scaling. The latter’s solution is very robust (as will be seen). In the classical framework the probability of ruin is of order 1, while in the small noise scaling the order is very low. Moreover, we give high punishment for ruin. We consider the case . If then ruin is avoided by Now, and the probability of ruin is low. Define a sequence of models, indexed by that differ from each other only by the consumption function, such that is Lipschitz. Scaling the time: , and scaling the control: we get… Small noise, which means small ruin probability

Small noise, which means small ruin probability The Model Time scaling Consider the process with the time of death and an interval . By stretching the time, we get , and the equivalent time interval is now . The scaled time of death should therefore be . Define a sequence of models, indexed by that differ from each other only by the consumption function, such that is Lipschitz. Scaling the time: , and scaling the control: we get… Small noise, which means small ruin probability

Risk sensitive control The Model Risk sensitive control In the standard “risk-neutral” general control problem we minimize the expectation of the cost, . We minimize the expectation of a function of the cost . Why? We are interested in a minimization criteria that is sensitive to risk. The function measures the risk. If then this is the “risk-neutral” case. Large indicates a great sensitivity to risk. We consider , and so . Then taking .

MIN Risk sensitive control with small noise diffusion The Model Process: time of ruin poverty punishment is decreasing time of death Risk sensitive Cost: MIN This is a discounted version of the risk sensitive control with small noise diffusion. Difficulties: Discounted cost. The volatility depends on the control and can be degenerate. Smoothness: the indicator part is not continuous and depends on the past.

smoothness volatility The Model Literature – risk sensitive control with small noise diffusion PDE approach: [1] W.H. Fleming and W.M. McEneaney. Risk-sensitive control on an infinite time horizon. SIAM J. Control Optimization, 1995. [2] W.H. Fleming and H.M Soner. Controlled Markov processes and viscosity solutions. Stochastic modelling and applied probability, 2006. smoothness Large deviation approach: [3] P. Dupuis and H. Kushner. Minimizing escape probabilities: a large deviations approach. SIAM J. Control Optimization, 1989. [4] Huyên Pham. Some applications and methods of large deviations in finance and insurance. Lecture Notes in Mathematics, 2007. volatility Queueing: [1] R. Atar and A. Biswas. Control of the multiclass G/G/1 queue in the moderate- deviation regime. Ann. Appl. Probab., 2014. [2] R. Atar and A. C. An asymptotically optimal control for a multiclass queueing model in the moderate-deviation heavy-traffic regime. Math. Of O.R., 2015 Considers discounted

Contents The Model The Differential Game The Approximation Summary

Intuition The Differential Game Two approaches: 1. Large-deviations : Freidlin–Wentzell theorem and Varadhan’s lemma. 2. Change of measure: Girsanov’s theorem. Intuitive construction of the game:

Intuition The Differential Game So, Consider measures of the form: relative entropy By Jensen’s inequality

Intuition The Differential Game Now the problem becomes, Where, under , negative Therefore, the associated differential game is: Under optimality, or

maximizer stops immediately The Differential Game Solution of the game Theorem maximizer stops immediately Moreover, for , under and , the state process follows and is attained (independently of !).

Contents The Model The Differential Game The Approximation Summary

Main Result The Approximation Let and be the value functions in the stochastic model and the game, respectively. Then, Moreover, is an asymptotically optimal control in the stochastic model. The proof follows by the following two steps: 1) : this is done by choosing a measure driven by the maximizer’s path . : we provide an asymptotically optimal policy that follows by the minimizer’s optimal policy, .

Proof of The Values of the Stochastic Model and the Differential Game If , then . If , recall that, Moreover, for , under and , the state process follows and is attained (independently of !). Also, under , given by

Proof of The Approximation denote by With high probability, and would be close to and .

Proof of The Approximation discount factor Jensen’s inequality deterministic high probability

Proof of , and asymptotic optimality of . The Approximation Proof of , and asymptotic optimality of .

Proof of , and asymptotic optimality of . The Approximation Proof of , and asymptotic optimality of . denote by By Jensen’s inequality and equality holds for defined by So,

Proof of , and asymptotic optimality of . The Approximation Proof of , and asymptotic optimality of . Goal: replace and with and . Under , and the differential game’s dynamics: Main difficulty: Although and are close the each other with high probability, showing that and are close to each other is not trivial.

Proof of , and asymptotic optimality of . The Approximation Proof of , and asymptotic optimality of . The idea is to discretize the paths space: The set is compact and Therefore, we can find a finite cover with sufficiently small balls such that on each one, with high probability, and are close, and also , . denote by

Proof of , and asymptotic optimality of . The Approximation Proof of , and asymptotic optimality of . Eventually, in the limit

Contents The Model The Differential Game The Approximation Summary

Current and future work Summary Contribution We provide a risk sensitive control framework for the lifetime ruin probability framework. We found an asymptotically optimal control by using a differential game. Some technical difficulties that we managed to deal with: 1) Discounted cost. 2) The volatility depends on the control and can be degenerate. In fact, under optimality the volatility is degenerate. 3) Smoothness: the indicator part is not continuous and depends on the past. Current and future work We use PDE techniques to analyze a general discounted risk sensitive control with small noise diffusions. We use some of the mentioned techniques to deal with the general model of: [3] P. Dupuis and H. Kushner. Minimizing escape probabilities: a large deviations approach. SIAM J. Control Optimization, 1989. where we do not restrict the volatility to be independent of the control and nondegenerate.

Thank you!