Download 3740_cohen - Mathematics

Asaf Cohen (joint work with Erhan Bayraktar)
Department of Mathematics
University of Michigan
International Conference on Financial Risks
and their Management
Mar, 2016
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Contents
The Model
The Differential Game
The Approximation
Summary
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Contents
The Model
The Differential Game
The Approximation
Summary
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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
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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.
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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:
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, 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:
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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
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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.
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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
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Contents
The Model
The Differential Game
The Approximation
Summary
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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:
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The Differential Game
Intuition
So,
Consider measures of the form:
By Jensen’s inequality
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relative entropy
The Differential Game
Intuition
Now the problem becomes,
Where, under
,
Therefore, the associated differential game is:
•
Under optimality,
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or
negative
The Differential Game
Solution of the game
Theorem
maximizer stops
immediately
Moreover, for
and
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, under
is attained (independently of
and
!).
, the state process
follows
Contents
The Model
The Differential Game
The Approximation
Summary
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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,
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driven by
.
The Values of the Stochastic Model and the Differential Game
Proof of
If
, then
.
If
, recall that,
Moreover, for
and
Also, under
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, under
is attained (independently of
, given by
and
!).
, the state process
follows
The Approximation
Proof of
With high
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probability,
and
would be close to
and
. denote by
The Approximation
Proof of
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The Approximation
Proof of
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, and asymptotic optimality of .
The Approximation
Proof of
, and asymptotic optimality of .
denote by
By Jensen’s inequality
and equality holds for
So,
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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.
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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
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.
The Approximation
Proof of
Eventually, in the limit
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, and asymptotic optimality of .
Contents
The Model
The Differential Game
The Approximation
Summary
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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.
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