Stochastic Perron's Method: Obstacle Problems and Dynkin … · Stochastic Perron’s Method:...

Stochastic Perron’s Method: Obstacle Problemsand Dynkin Games

Erhan Bayraktar

Department of Mathematics,

University of Michigan

Joint work with Mihai Sırbu, UT Austin

Outline

Overview of DPP

Objective

Main Idea of Stochastic Perron’s method

Obstacle problems and Dynkin games

Classical approaches to solving stochastic control problems

I State equation:

= b(t,Xt

)dt + �(t,Xt

I Reward:

J(s, x ,↵) = E [

R(s,Xt

)dt + g(XT

I Value function:

v(s, x) = sup↵

J(s, x ,↵).

Classical approaches to solving stochastic control problems

I State equation:

= b(t,Xt

)dt + �(t,Xt

I Reward:

J(s, x ,↵) = E [

R(s,Xt

)dt + g(XT

I Value function:

v(s, x) = sup↵

J(s, x ,↵).

Characterizing v as the solution of the HJB

Two approaches:

I Analytic (direct)

I Probabilistic (analyzing the properties of the value function)

Characterizing v as the solution of the HJB

Two approaches:

I Analytic (direct)

I Probabilistic (analyzing the properties of the value function)

Analytical Approach

I Prove that the HJB has a smooth solution.

I Verification.

Analytical Approach

I Prove that the HJB has a smooth solution.

I Verification.

Probabilistic Approach

I Prove DPP. (Requires a priori regularity).

I Show that DPP ! v is a viscosity solution.

I Then comparison implies that v is the unique viscositysolution.

References:

I By discrete-time approximations (Krylov)

I Working with the value function directly (El Karoui, Borkar,Haussmann, Bouchard-Touzi (weak DPP)).

References:

Probabilistic approach++

Some cases viscosity solutions are regular, then use

Perron ! viscosity solution ! smooth solution ! uniqueness

References: Shreve Soner (1994), Janecek and Sırbu (2012).

Perron’s method, Ishii (1987)

� = supw2U�,visc

+ = infw2U+,visc

are viscosity super- and sub-solutions.Here U�,visc and U+,visc are the collections of viscosity sub- andsuper-solutions respectively.

Question: why not inf/sup over classical super/sub-solutions?

Answer: Because one cannot prove (in general/directly) the resultis a viscosity solution. The classical solutions are not enough (theset of classical solutions is not stable under max or min).

+ = infw2U+,visc

are viscosity super- and sub-solutions.

Here U�,visc and U+,visc are the collections of viscosity sub- andsuper-solutions respectively.

+ = infw2U+,visc

Objective

Provide a method to replace the first two steps in

Perron ! visc .solution ! smoothsolution ! uniqueness + (DPP)

(in case one cannot prove that the viscosity solutions are smooth)

and construct a viscosity solutions to the HJB that one candirectly compare with the value function.

Remarks on Perron’s method

+ = supw2U+,visc

I Test function w does NOT compare to the value function vUNLESS one proves v is a viscosity solutions already AND theviscosity comparison

I If we ask w to be classical semi-solutions, we cannot provethat the inf/sup are viscosity solutions

Main Idea

Perform Perrons Method over a class of semi-solutions which are

I weak enough to conclude (in general/directly) that v� andv

+ are viscosity super/sub-solutions.

I strong enough to compare with the value function withoutstudying the properties of the value function

Stochastic solutions do the job! (Generalization of a solutionconcept introduced by Stroock and Varadhan for linear PDEs.)

Main Idea

Stochastic Perron

General statement:

I sup over stochastic sub-solutions is a viscosity

super-solution

v⇤ = supw2U�,stoch

I inf over stochastic super-solutions is a viscosity sub-solution

⇤ = infw2U+,stoch

Conclusion: v⇤ v v

If a viscosity comparison holds, then v is the unique viscositysolution. In other words:

SP+visc. comparison ! v is the unique disc sol (DPE)+DPP.

Stochastic Perron

General statement:

super-solution

⇤ = infw2U+,stoch

Stochastic Perron

General statement:

super-solution

⇤ = infw2U+,stoch

Stochastic Perron

General statement:

super-solution

⇤ = infw2U+,stoch

If a viscosity comparison holds, then v is the unique viscositysolution.

In other words:

Stochastic Perron

General statement:

super-solution

⇤ = infw2U+,stoch

Remarks

I Morally, substituting Ito’s lemma with comparison principle inverification.

I Set of stochastic sub and super-solutions have to be carefullydefined (depending on the control problem) as to obtainviscosity solutions as sup/inf (and to retain the comparisonbuild in).

Obstacle Problems and Dynkin Games

State process:

= b(t,Xt

)dt + �(t,Xt

Assumption: Continuous coe�cients with linear growth. [Thereexist (possibly non-unique) weak solutions of the SDE.] We willmake a selection (not necessarily Markovian).

Let g : Rd ! R l , u : [0,T ]⇥Rd ! R be bounded and measurablefunctions satisfying l u and l(T , ·) g u(T , ·).

Denote by T s,x the set of stopping times ⌧ (with respect to thefiltration (F s,x

) which satisfy s ⌧ T .

The first player (⇢) pays to the second player (⌧) the amount

J(s, x , ⌧, ⇢) :=

Es,x⇥I{⌧<⇢}l(⌧,X

s,x⌧ ) + I{⇢⌧,⇢<T})u(⇢,X

s,x⇢ ) + I{⌧=⇢=T}g(X

Obstacle Problems and Dynkin GamesState process:

= b(t,Xt

)dt + �(t,Xt

J(s, x , ⌧, ⇢) :=

Es,x⇥I{⌧<⇢}l(⌧,X

s,x⌧ ) + I{⇢⌧,⇢<T})u(⇢,X

s,x⇢ ) + I{⌧=⇢=T}g(X

= b(t,Xt

)dt + �(t,Xt

J(s, x , ⌧, ⇢) :=

Es,x⇥I{⌧<⇢}l(⌧,X

s,x⌧ ) + I{⇢⌧,⇢<T})u(⇢,X

s,x⇢ ) + I{⌧=⇢=T}g(X

= b(t,Xt

)dt + �(t,Xt

J(s, x , ⌧, ⇢) :=

Es,x⇥I{⌧<⇢}l(⌧,X

s,x⌧ ) + I{⇢⌧,⇢<T})u(⇢,X

s,x⇢ ) + I{⌧=⇢=T}g(X

= b(t,Xt

)dt + �(t,Xt

J(s, x , ⌧, ⇢) :=

Es,x⇥I{⌧<⇢}l(⌧,X

s,x⌧ ) + I{⇢⌧,⇢<T})u(⇢,X

s,x⇢ ) + I{⌧=⇢=T}g(X

Dynkin Games

The lower value of the Dynkin game

v⇤(s, x) := sup⌧2T s,x

inf⇢2T s,x

J(s, x , ⌧, ⇢),

and the upper value of the game

⇤(s, x) := inf⇢2T s,x

sup⌧2T s,x

J(s, x , ⌧, ⇢).

DPE for Dynkin games

⇢F (t, x , v , v

) = 0,u(T , x) = g(x),

F (t,x , v , vt

max{v � u,min{�v

v , v � l}}= min{v � l ,max{�v

, v � u}},

Stochastic super-solution

U+, is the set of functions v : [0,T ]⇥ Rd ! R which have thefollowing properties:

1. These functions are upper semicontinuous (USC) andbounded on [0,T ]⇥ Rd . In addition, they satisfy v � l andthe terminal condition v(T , ·) � g(·).

2. For each (s, x) 2 [0,T ]⇥ Rd , and any stopping times⇢1 ⇢2 2 T s,x , we have

v(⇢1,X s,x⇢1 ) � E

hv(⇢2 ^ ⇢+,X s,x

⇢2^⌧+)|Fs,x⇢1

where the stopping time ⌧+(v ; s, x , ⇢1) is defined as

⌧+(v , s, x , ⌧1) := inf{t 2 [⇢1,T ] : v(t,X s,x) � u(t,X s,x)}.

Question: Why the starting stopping time? No Markov property.

Stochastic super-solution

U+, is the set of functions v : [0,T ]⇥ Rd ! R which have thefollowing properties:

1. These functions are upper semicontinuous (USC) andbounded on [0,T ]⇥ Rd . In addition, they satisfy v � l andthe terminal condition v(T , ·) � g(·).

2. For each (s, x) 2 [0,T ]⇥ Rd , and any stopping times⇢1 ⇢2 2 T s,x , we have

v(⇢1,X s,x⇢1 ) � E

hv(⇢2 ^ ⇢+,X s,x

⇢2^⌧+)|Fs,x⇢1

where the stopping time ⌧+(v ; s, x , ⇢1) is defined as

⌧+(v , s, x , ⌧1) := inf{t 2 [⇢1,T ] : v(t,X s,x) � u(t,X s,x)}.

Question: Why the starting stopping time? No Markov property.

Stochastic Perron for obstacle problems

Define U�, stochastic sub-solutions, symmetrically. Let

� := supw2U�

w v⇤ v

+ := infw2U+

Cannot show v

� 2 U� or v+ 2 U+, but what really matters is theclosedness under max/min .(Thanks to an argument similar to theproof of Dini’s lemma).

This is the reason we can assume continuity.

Theorem.

� is a viscosity supersoln of the DPE.

+ is a viscosity sub solution of the DPE.

� := supw2U�

w v⇤ v

+ := infw2U+

Cannot show v

Theorem.

� := supw2U�

w v⇤ v

+ := infw2U+

Cannot show v

Theorem.

� := supw2U�

w v⇤ v

+ := infw2U+

Cannot show v

Theorem.

� := supw2U�

w v⇤ v

+ := infw2U+

Cannot show v

Theorem.

Verification by comparison

Theorem

I If comparison holds, then there exist a unique, continuousviscosity solution equal to v

� = v⇤ = v

⇤ = v

I The first hitting times are optimal.

Theorem

� = v⇤ = v

⇤ = v

Theorem

� = v⇤ = v

⇤ = v

Remarks about optimal stoppin, i.e. u = 1

References: El Karoui, Shiryaev, Fakeev: In the Markov case, thevalue function is the least excessive function, i.e.

+ = infw2U+

w = v .

The proof requires to show that v 2 U+. We avoid showing thatby using

� v v

and use comparison.

We prove the continuity of the value function for free. (See studyof the continuity in Basan Ceci.)

+ = infw2U+

w = v .

� v v

and use comparison.

+ = infw2U+

w = v .

� v v

and use comparison.

+ = infw2U+

w = v .

� v v

and use comparison.

References

1) Stochastic Perron’s method and verification without smoothnessusing viscosity comparison: the linear case, (EB and Mihai Sirbu),Proceedings of the American Mathematical Society, 140,3645-3654, 2012.

2) Stochastic Perron’s method and verification without smoothnessusing viscosity comparison: obstacle problems and Dynkin games,(EB and Mihai Sirbu), to appear in the Proceedings of the

American Mathematical Society.

Thank you for your attention!

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