1. 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

2. Contents
The Model
The Differential Game
The Approximation
Summary

3. Contents
The Model
The Differential Game
The Approximation
Summary

4. 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

5. 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. 

6. 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

7. 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

8. 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

9. 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.

10. 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

11. Contents
The Model
The Differential Game
The Approximation
Summary

12. 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:

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

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

15. 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 !).

16. Contents
The Model
The Differential Game
The Approximation
Summary

17. 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,

18. 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

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

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

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

22. 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,

23. 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.

24. 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

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

26. Contents
The Model
The Differential Game
The Approximation
Summary

27. 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.

28. Thank you!