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Asaf Cohen (joint work with Erhan Bayraktar) Department of Mathematics University of Michigan International Conference on Financial Risks and their Management Mar, 2016 1/24 Contents The Model The Differential Game The Approximation Summary 2/24 Contents The Model The Differential Game The Approximation Summary 2/24 The Model The classical 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 3/24 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. 4/24 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 . Now, and the probability of ruin is low. If then ruin is avoided by • Define a sequence of models, indexed by the consumption function, such that that differ from each other only by is Lipschitz. • Scaling the time: 5/24 , and scaling the control: Small noise, which means small ruin probability we get… The Model Time scaling • Consider the process • By stretching the time, we get • The scaled time of death should therefore be • Define a sequence of models, indexed by the consumption function, such that • Scaling the time: 5/24 with the time of death and an interval , and the equivalent time interval is now . that differ from each other only by is Lipschitz. , and scaling the control: Small noise, which means small ruin probability we get… . . 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 a great sensitivity to risk. • We consider 6/24 then this is the “risk-neutral” case. Large , and so . Then taking . indicates The Model Risk sensitive control with small noise diffusion Process: Risk sensitive Cost: time of ruin poverty punishment is decreasing time of death MIN This is a discounted version of the risk sensitive control with small noise diffusion. Difficulties: 1) Discounted cost. 2) The volatility depends on the control and can be degenerate. 3) Smoothness: the indicator part is not continuous and depends on the past. 7/24 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. 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. Queueing: [1] R. Atar and A. Biswas. Control of the multiclass G/G/1 queue in the moderatedeviation 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 8/24 Contents The Model The Differential Game The Approximation Summary 9/24 The Differential Game Intuition Two approaches: 1. Large-deviations : Freidlin–Wentzell theorem and Varadhan’s lemma. 2. Change of measure: Girsanov’s theorem. Intuitive construction of the game: 10/24 The Differential Game Intuition So, Consider measures of the form: By Jensen’s inequality 11/24 relative entropy The Differential Game Intuition Now the problem becomes, Where, under , Therefore, the associated differential game is: • Under optimality, 12/24 or negative The Differential Game Solution of the game Theorem maximizer stops immediately Moreover, for and 13/24 , under is attained (independently of and !). , the state process follows Contents The Model The Differential Game The Approximation Summary 14/24 The Approximation Main Result 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) 2) : this is done by choosing a measure the maximizer’s path . : we provide an asymptotically optimal policy that follows by the minimizer’s optimal policy, 15/24 driven by . The Values of the Stochastic Model and the Differential Game Proof of If , then . If , recall that, Moreover, for and Also, under 16/24 , under is attained (independently of , given by and !). , the state process follows The Approximation Proof of With high 17/24 probability, and would be close to and . denote by The Approximation Proof of 18/24 The Approximation Proof of 19/24 , and asymptotic optimality of . The Approximation Proof of , and asymptotic optimality of . denote by By Jensen’s inequality and equality holds for So, 19/24 defined by The Approximation Proof of Goal: replace Under , , and asymptotic optimality of . and with and . 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. 19/24 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 20/24 . The Approximation Proof of Eventually, in the limit 21/24 , and asymptotic optimality of . Contents The Model The Differential Game The Approximation Summary 22/24 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. 23 24/24