Stochastic Perron for Stochastic Target Games...Stochastic Perron for Stochastic Target Games Jiaqi...

IntroductionThe Set-up

Sub-solution property for v+

Super-solution property for v−

Future Work

Stochastic Perron for Stochastic Target Games

Jiaqi Li(Joint work with Erhan Bayraktar)

Department of MathematicsUniversity of Michigan

September 3, 2014

E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games




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OUTLINE

1 Introduction

2 The Set-up

3 Sub-solution property for v+

4 Super-solution property for v−

5 Future Work





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The stochastic target game

Consider a stochastic target game:

Two players: controller and nature.A state process X can be manipulated by the naturethrough the selection of α. Another process Y is driven byboth players, where the controller reacts to the nature.The controller tries to find a strategy such that thecontrolled state process almost-surely reaches a giventarget.The nature may choose a parametrization of the model tobe totally adverse to the controller.





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Related Work

The formulation in our paper has been considered byBouchard and Nutz (2013).Related literature on the use of Perron’s method:

Linear problems, Bayraktar and Sîrbu (2012).Dynkin games involving free-boundary games, Bayraktarand Sîrbu (2014).Stochastic control problems, Bayraktar and Sîrbu (2013).Symmetric stochastic differential games, Sîrbu (2014).





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Our goal

Characterize the inf (sup) of the stochastic super-solutions(sub-solutions) v+ (v−) as the viscosity sub-solution(super-solution) of the HJB equation without going throughthe geometric dynamic programming principle first ( henceavoiding the abstract measurable selection theorem ).Give a more elementary proof to the result. As a result, weare able to avoid using Krylov’s method of shakencoefficients and hence avoid assuming the concavity of theHamiltonian, which was required in Bouchard and Nutz’spaper.





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Methodology

Define the set of stochastic super- and sub-solutionsappropriately, denoted by U+ and U− respectively.Show: v+ := infw∈U+ is a viscosity sub-solution.Show: v− = supw∈U− is a viscosity super-solution.Since v+, v and v− compare by definition, then

v+ = v = v−

holds under a comparison principle.





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OUTLINE

1 Introduction

2 The Set-up



5 Future Work





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Notations

Time horizon: T > 0;D := [0,T ]× Rd , D<T := [0,T )× Rd , DT := T × Rd ;Ω := C([0,T ]; Rd );

Wt (ω) = ωt : the canonical process;F = (Fs)0≤s≤T : the augmented filtration generated by W ;P: the Wiener measure on Ω;Ft = (F t

s)t≤s≤T : the augmented filtration generated by(Ws −Wt )s≥t , for t ≤ s ≤ T ;





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Notations

U: Borel subset of Rd ;A: compact subset of Rd ;St : the set of F t -stopping times valued in [t ,T ];U t (At ) : the collection of all Ft -predictable processes inLp(P⊗ dt) with values in U(A);





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Assumptions on µX , σX , µY and σY

For u ∈ U t , α ∈ At , let (Xαt ,x ,Y

u,αt ,x ,y ) denote processes satisfying

dXs = µX (s,Xs, αs)ds + σX (s,Xs, αs)dWs,

dYs = µY (s,Xs,Ys,us, αs)ds + σY (s,Xs,Ys),us, αs)dWs.

(2.1)with initial data (Xt ,Yt ) = (x , y).

Assume: µX , µY , σY and σX are continuous in all variables.Moreover there exists K > 0 such that, for all(x , y), (x ′, y ′) ∈ Rd × R and u ∈ U,





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Assumptions on µX , σX , µY and σY

|µX (·, x , ·)− µX (·, x ′, ·)|+ |σX (·, x , ·)− σX (·, x ′, ·)| ≤ K |x − x ′|,|µX (·, x , ·)|+ |σX (·, x , ·)| ≤ K ,

|µY (·, y , ·)− µY (·, y ′, ·)|+ |σY (·, y , ·)− σY (·, y ′, ·)| ≤ K |y − y ′|,|µY (·, y ,u, ·)|+ |σY (·, y ,u, ·)| ≤ K (1 + |u|+ |y |).

This implies that for ∀(t , x , y) ∈ D<T × R and anyu ∈ U t , α ∈ At , (2.1) admits a unique strong solution.

Also assume g: Rd → R is a bounded and measurable function.





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Definition 1 (Admissible strategies)

A map u : At → U t , α 7→ u[α] is a t-admissible strategy if

ω ∈ Ω : α(ω)|[t ,s] = α′(ω)|[t ,s] ⊂ω ∈ Ω : u[α](ω)|[t ,s] = u[α′](ω)|[t ,s] -a.s.

for all s ∈ [t ,T ] and α, α′ ∈ At . Denote u ∈ U(t).

Definition 2 (Value function)

v(t , x) := infy ∈ R : ∃ u ∈ U(t) s.t. Y u,αt ,x ,y (T ) ≥ g(Xα

t ,x (T )) -a.s.

∀ α ∈ At.





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Non-anticipating family of stopping times

Definition 3

Let ταα∈At ⊂ St be a family of stopping times. This family ist-non-anticipating if

ω ∈ Ω : α(ω)|[t ,s] = α′(ω)|[t ,s] ⊂

ω ∈ Ω : t ≤ τα(ω) = τα′(ω) ≤ s ∪ ω ∈ Ω : s < τα, s < τα

′-a.s.

Denote the set of t-non-anticipating families of stopping timesby St .





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Strategies starting at a non-anticipating family ofstopping times

Definition 4

Fix t and let τα ∈ St . We say that a map u : At → U t ,α 7→ u[α] is a (t , τα)-admissible strategy if it isnon-anticipating in the sense that

ω ∈ Ω : α(ω)|[t ,s] = α′(ω)|[t ,s] ⊂ ω ∈ Ω : s < τα, s < τα′ ∪

ω ∈ Ω : t ≤ τα = τα′ ≤ s, u[α](ω)|[τα(ω),s] = u[α′](ω)|[τα′ (ω),s],

for all s ∈ [t ,T ] and α, α′ ∈ At , denoted by u ∈ U(t , τα).





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ConcatenationDefinition 5 (Concatenation)

Let α1, α2 ∈ At , τ ∈ St is a stopping time. The concatenation ofα1, α2 is defined as follows:

α1 ⊗τ α2 := α11[t ,τ) + α21[τ,T ].

The concatenation of elements in U t is defined in the similarfashion.

Lemma 2.1

Fix t and let τα ∈ St . For u ∈ U(t) and u ∈ U(t , τα), defineu∗[α] := u[α]⊗τα u[α]. Then u∗ ∈ U(t). Use u⊗τα u[α] torepresent u[α]⊗τα u[α].





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Stochastic super-solutions

Definition 6 (Stochastic super-solutions)

A function w : [0,T ]× Rd → R is called a stochasticsuper-solution of (2.2) if

1 it is bounded, continuous and w(T , ·) ≥ g(·),2 for fixed (t , x , y) ∈ D × R and τα ∈ St , there exists a

strategy u ∈ U(t , τα) such that, for any u ∈ U(t), α ∈ At

and each stopping time ρ ∈ St , τα ≤ ρ ≤ T with thesimplifying notation X := Xα

t ,x ,Y := Y u⊗τα u[α],αt ,x ,y we have

Y (ρ) ≥ w(ρ,X (ρ)) P−a.s. on Y (τα) > w(τα,X (τα)).





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The set of stochastic super-solutions is denoted by U+.Assume it is nonempty and v+ := infw∈U+ w . For any stochasticsuper-solution w , choose τα = t for all α and ρ = T , then thereexists u ∈ U(t) such that, for any α ∈ At and

Y u,αt ,x ,y (T ) ≥ w

(T ,Xα

t ,x (T ))≥ g

(Xα

t ,x (T ))

-a.s. on y > w(t , x).

Hence, y > w(t , x) implies y ≥ v(t , x). This gives w ≥ v andv+ ≥ v .





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Stochastic sub-solutionsDefinition 7 (Stochastic sub-solutions)

A function w : [0,T ]× Rd → R is called a stochasticsub-solution of (2.2) if

1 it is bounded, continuous and w(T , ·) ≤ g(·),2 for fixed (t , x , y) ∈ D × R and τα ∈ St , for any u ∈ U(t),α ∈ At , there exists α ∈ At (may depend on u, α and τα)such that for each stopping time ρ ∈ St , τα ≤ ρ ≤ T withthe simplifying notation X := Xα

t ,x ,Y := Y u,α⊗τα αt ,x ,y , we have

P (Y (ρ) < w (ρ,X (ρ)) | B) > 0,

for any B ⊂ Y (τα) < w(τα,X (τα), B ∈ F tτα and

P(B) > 0.





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The set of stochastic sub-solutions is denoted by U−. Assumeit is nonempty and let v− := supw∈U− w . For any stochasticsub-solution w , choose τα = t for all α and ρ = T . Hence forany u ∈ U(t), there exists α ∈ At , such that

P(

Y u,αt ,x ,y (T ) < w

(T ,X α

t ,x (T ))≤ g(Xα

t ,x (T )) | y < w(t , x))> 0.

Hence, y < w(t , x) implies y < v(t , x). This gives w ≤ v andv− ≤ v . Clearly,

v− , supw∈U−

w ≤ v ≤ infw∈U+

w , v+.





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Assumptions

Assumption 2.1

supu∈U

|µY (·,u, ·)|1 + |σY (·,u, ·)|

is locally bounded.

Assumption 2.2

(1) There exists u ∈ U such that µY (t , x , y ,u,a) = 0,σY (t , x , y ,u,a) = 0 for all (t , x , y ,a) ∈ D<T × R× A.(2) There exists a ∈ A, µY (t , x , y ,u,a) = 0, σY (t , x , y ,u,a) = 0for all (t , x , y ,u) ∈ D<T × R× U.





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AssumptionsAssumption 2.3

Let N(t , x , y , z,a) = u ∈ U : σY (t .x , y ,u,a). There exists ameasurable map u : D × R× Rd × A→ U such that N = u.Moreover, the map u is continuous.

Assumption 2.4

The map (y , z) 7→ µuY := µY (t , x , y , u(t , x , y , z,a),a) is Lipschitz

continuous and has linear growth, uniformly in (t , x ,a).

Assumption 2.5 (Comparison principle)

Let v (resp. v) be a LSC (resp. USC) bounded viscositysuper-solution (resp. sub-solution) of (2.2). Then, v ≥ v on D.





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HJB equation

Define (t , x , y ,p,M) ∈ D × R× Rd ×Md ,

H(t , x , y ,p,M) := supa∈A

− µu

Y (t , x , y , σX (t , x ,a)p,a) + µX (t , x ,a)>p

+12

Tr[σXσ>X (t , x ,a)M]

,

where µuY (t , x , y , z,a) := µY (t , x , y , u(t , x , y , z,a),a), z ∈ Rd .

The HJB equation is

−φt − H(t , x , φ,Dφ,D2φ) = 0 on D<Tφ− g = 0 on DT

(2.2)





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Preparatory lemmasLemma 2.2Under Assumption 2.2 the sets U+ and U− are nonempty.

Proof.We will only prove U+ is nonempty under Assumption 2.2 (1).Choose u[α] = u. For any given τα ∈ St , we haveu ∈ U(t , τα) and ∀α ∈ At and ρ ∈ St such that τα ≤ ρ ≤ T ,

Y u⊗τα u[α],αt ,x ,y (ρ) = Y u⊗τα u[α],α

t ,x ,y (τα).

From the boundedness of g, there exists an C, such thatg(x) < C. Now take w(t , x) ≡ C. On the setY (τα) > w(τα,X (τα)), we clearly have thatY (ρ) > w(ρ,X (ρ)).





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Lemma 2.31 if w1,w2 ∈ U+, then w1 ∧ w2 ∈ U+;2 if w1,w2 ∈ U−, then w1 ∨ w2 ∈ U−.

Proof of (1): for w1,w2 ∈ U+, let w = w1 ∧ w2. For fixed(t , x , y) ∈ D<T × R and τα ∈ St ,

u[α] = u1[α]1A + u2[α]1Ac ,

where u1 and u2 are the strategies starting at τα for w1 andw2, respectively and A = w1(τα,X (τα)) < w2(τα,X (τα)). uworks for w in the definition of stochastic super-solutions.





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Lemma 2.4There exists a non-increasing sequence U+ 3 wn v+ and annon-decreasing sequence U− 3 vn v−.

Lemma 2.5

f (x ,a) is defined on X × A ∈ Rm × Rn and f (x ,a) is uniformlycontinuous. Assume F (x) := supa∈A f (x ,a) <∞, then F (x) iscontinuous.





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OUTLINE

1 Introduction

2 The Set-up



5 Future Work





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The viscosity sub-solutionProposition 3.1

Under all standing assumptions, if g is USC, the function v+ isa bounded upper semi-continuous (USC) viscosity sub-solutionof (2.2);

Proof of the proposition:1.1 The interior sub-solution property: Assume for some

(t0, x0) ∈ D<T ,

ϕt + H(t , x , ϕ,Dϕ,D2ϕ) < 0 at (t0, x0).

where the test function ϕ strictly touches v+ from above at(t0, x0). Then the map (t , x , y)→ H(t , x , y ,Dϕ(t , x),D2ϕ(t , x))is continuous near (t0, x0, ϕ(t0, x0)) from Lemma 2.5.





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Proof

There exists a ε > 0 and δ > 0 such that

ϕt + H(t , x , y ,Dϕ,D2ϕ) < 0,∀ (t , x) ∈ B(t0, x0, ε) and |y − ϕ(t , x)| ≤ δ, (3.1)

On T = B(t0, x0, ε)− B(t0, x0, ε/2), ϕ > v+ + η on T for someη > 0.

wn v+ and Dini type argument⇒ there exists a n such thatϕ > wn + η/2 on T and ϕ > wn − δ on B(t0, x0, ε/2) . Letw = wn. Define, for small κ < η

2 ∧ δ,

wκ ,

(ϕ− κ) ∧ w on B(t0, x0, ε),

w outside B(t0, x0, ε).





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Proof

We will obtain a contradiction if we show wκ ∈ U+ andwκ(t0, x0) < v+(t0, x0). Fix t and τα ∈ St .

(i) if wκ(τα,X (τα)) = w(τα,X (τα)): set u = u1, which is the"optimal" strategy in the definition of stochastic super-solutionsfor w starting at τα.

(ii) if wκ(τα,X (τα)) < w(τα,X (τα)): let Y be the solution to

Y (t) =Y u,αt ,x ,y (τα) +

∫ τα∨t

ταµu

Y(s,Xα

t ,x ,Y (s), σX (s,Xαt ,x , αs)Dϕ, αs

)ds

+

∫ τα∨t

τασX (s,Xα

t ,x (s), αs)Dϕ(s,Xαt ,x (s))dWs, t ≥ τα,





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Proof

Let θα1 = infs ≥ τα, (s,X (s)) /∈ B(t0, x0; ε/2),θα2 = infs ≥ τα, |Y (s)− ϕ(s,X (s))| ≥ δ and θα = θα1 ∧ θα2 .θα ∈ St (Example 1, Bayraktar and Huang). We will set u tobe

u0[α](s) := u(s,Xαt ,x (s),Y (s), σX (s,Xα

t ,x (s), αs)Dϕ(s,Xαt ,x (s)), αs)

until θα. Starting at θα, we will then follow the strategy uθ whichis "optimal" for w . In other words, define u by

u[α] := 1Au1[α] + 1Ac (u0[α]1[t ,θα) + uθ[α]1[θα,T ]) ∈ U(t , τα),

where A = wκ(τα,X (τα)) = w(τα,X (τα)).





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Proof

We need to show that

Y (ρ) ≥ wκ(ρ,X (ρ)) on Y (τα) > wκ(τα,X (τα)),

where X := Xαt ,x ,Y := Y u⊗τα u[α],α

t ,x ,y . Note that Y = Y u⊗τα u0[α],αt ,x ,y

and

Y = 1AY u⊗τα u1[α],αt ,x ,y + 1Ac Y u⊗τα u0[α],α

t ,x ,y for τα ≤ s ≤ θα. (3.2)

(i) On the set A ∩ Y (τα) > wκ(τα,X (τα)), from (3.2) and the"optimality" of u1 (for w),

Y (ρ) = Y u⊗τα u1[α],αt ,x ,y (ρ) ≥ w(ρ,X (ρ)) ≥ wκ(ρ,X (ρ)) P-a.s.





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Proof

(ii) On the set Ac ∩ Y (τα) > wκ(τα,X (τα), by the definition ofu0, (3.1) and (3.2), using Itô’s Formula,Y (· ∧ θα)− ϕ(· ∧ θα,X (· ∧ θα)) is increasing on [τα, θα] and

Y (θα)−ϕ(θα,X (θα))+κ ≥ Y (τα)−ϕ(τα,X (τα))+κ > 0. (3.3)

From (3.3), we can prove

Y (θα)− w(θα,X (θα)) > 0 on Ac ∩ Y (τα) > wκ(τα,Xα).

It follows from this conclusion and the fact the "optimality" of uθ

starting at θα that on Ac ∩ Y (τα) > wκ(τα,Xα)

Y (ρ∨θα)−wκ(ρ∨θα,X (ρ∨θα)) ≥ Y (ρ∨θα)−w(ρ∨θα,X (ρ∨θα)) ≥ 0.(3.4)





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Proof

Also, since Y (· ∧ θα)− ϕ(· ∧ θα,X (· ∧ θα)) is increasing on[τα, θα], then

(Y (ρ ∧ θα)− ϕ(ρ ∧ θα,X (ρ ∧ θα)) + κ

)> 0, which

further gives(Y (ρ∧θα)−wκ(ρ∧θα,X (ρ∧θα))

)> 0, on Ac∩Y (τα) > wκ(τα,Xα).

(3.5)From (3.4) and (3.5) we have

Y (ρ)− wκ(ρ,X (ρ)) ≥ 0, on Ac ∩ Y (τα) > wκ(τα,Xα).





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Proof

1.2 The boundary condition:Step A: Let µu

Y be non-decreasing in its y-variable. Assumethat for some xo ∈ Rd , we have v+(T , xo) > g(xo). There existsε > 0 such that v+(T , xo) > g(x) + ε for |x − xo| ≤ ε. Chooseβ, such that v+(T , xo) + ε2

4β > ε+ supT v+(t , x). There exists aw ∈ U+, such that

v+(T , xo) +ε2

4β> ε+ sup

Tw(t , x).

Define for C > 0,

ϕβ,ε,C = v+(T , xo) +|x − xo|2

β+ C(T − t).





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Proof

There exists for large enough C > 0, such that

− ϕβ,ε,Ct − H(·, y ,Dϕβ,ε,C ,D2ϕβ,ε,C)(t , x) > 0,

∀ (t , x , y) ∈ B(T , xo; ε)× R s.t. y ≥ ϕβ,ε,C(t , x)− ε.

Making sure that C ≥ ε/2β, we obtain that

ϕβ,ε,C ≥ ε+ w on T.

Also,

ϕβ,ε,C(T , x) ≥ v+(T , x0) > g(x) + ε for |x − x0| ≤ ε. (3.6)





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Proof

Now we can choose κ < ε and define

wβ,ε,C,κ ,

(ϕβ,ε,C − κ) ∧ w on B(T , x0, ε),

w outside B(T , x0, ε).(3.7)

Note that wβ,ε,C,κ(T , x) ≥ g(x) and wβ,ε,C,κ(T , xo) < v+(T , xo).Similar to Step 1.1, wβ,ε,C,κ is a stochastic super-solution.





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Proof

Step B: Fix c > 0 and define Y u,αt ,x ,y as the strong solution of

dY (s) = µY (s,Xαt ,x (s), Y (s), u[α]s, αs)ds +

σY (s,Xαt ,x (s), Y (s), u[α]s, αs)dWs with Y (t) = y , where

µY (t , x , y ,u,a) := cy + ectµY (t , x ,e−cty ,u,a),

σY (t , x , y ,u,a) := ectσY (t , x ,e−cty ,u,a).

Then, Y u,αt ,x ,y (s)e−cs = Y u,α

t ,x ,ye−ct (s) for any s ∈ [t ,T ]. Setg(x) := ecT g(x) and define v(t , x) := infy ∈ R : ∃ u ∈Ut s.t. Y u,α

t ,x ,y (T ) ≥ g(Xαt ,x (T )) -a.s. ∀ α ∈ At.





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Proof

Therefore, v(t , x) = ectv(t , x) and for large c > 0 the map

µuY : (t , x , y , z,a) 7→ cy + ectµu

Y (t , x ,e−cty ,e−ctz,a)

is non-decreasing in its y -variable. v+ is a sub-solution of thenew PDE with

H(t , x , y ,p,M) := −cy − supa∈A

ectµu

Y (t , x ,e−cty ,e−ctσX (t , x ,a)p,a)

+µX (t , x ,a)>p + 12

[σXσ

>X (t , x ,a)M

] ,

w(t , x) is a stochastic super-solution of (2.2) if and only ifw(t , x) = ectw(t , x) is a stochastic super-solution of the newHJB equation with H ⇒ v+(t , x) = ectv+(t , x). This concludesthe proof.





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OUTLINE

1 Introduction

2 The Set-up



5 Future Work





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The viscosity super-solutionProposition 4.1

Under all standing assumptions, if g is LSC, the function v− is abounded lower semi-continuous (LSC) viscosity super-solutionof (2.2).

Proof of the proposition:2.1 The interior super-solution property: Assume for some

(t0, x0) ∈ D<T , ϕt + H(t , x , ϕ,Dϕ,D2ϕ) > 0 at (t0, x0), wherethe test function ϕ strictly touches v− from below at (t0, x0).Similar to Step 1.1, we get

ϕt + Hu,a0(·, y ,Dϕ,D2ϕ) > 0 ∀ (t , x) ∈ Bε and (y ,u) ∈ R × U s.t.|y − ϕ(t , x)| ≤ δ and |σY (t , x , y ,u,a0)− σX (t , x ,a0)Dϕ(t , x)| ≤ δ,





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Proof

where u0 = u(t0, x0, ϕ(t0, x0), σX (·,a0)Dϕ(t0, x0),D2ϕ(t0, x0))and Hu,a(t , x , y ,p,M) := −µY (t , x , y ,u,a) + µX (t , x ,a)>p+1

2Tr[σXσ

>X (t , x ,a)M

].

Similarly, there exits a w ∈ U−, such that ϕ+ η/2 < w for someη > 0 on T and ϕ < w + δ on B(t0, x0, ε/2) . Define, for smallκ << η

2 ∧ δ,

wκ ,

(ϕ+ κ) ∨ w on B(t0, x0, ε),

w outside B(t0, x0, ε).

and we want to show wκ ∈ U− with wκ(t0, x0) > v−(t0, x0). Fora given u ∈ Ut and α ∈ At , we need to construct a processα ∈ At in the definition of stochastic sub-solutions for wκ.





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Proof

(i) if w(τα,X (τα)) = wκ(τα,X (τα)): set α = α1, which isoptimal" for the nature given u, α and τα.

(ii) if w(τα,X (τα)) < wκ(τα,X (τα)): defineθα1 = infs ≥ τα, (s,Xα⊗ταa0

t ,x ) /∈ B(t0, x0; ε/2),θα2 = infs ≥ τα, |Y u,α⊗ταa0

t ,x ,y − ϕ(s,Xα⊗ταa0t ,x ))| ≥ δ and

θα = θα1 ∧ θα2 . The nature choose a0 until θα. Starting from θα,choose α∗ which is "optimal" for the nature for w given α, u. Inother words, let A = w(τα,X (τα)) = wκ(τα,X (τα)),

α = 1Aα1 + 1Ac (a01[t ,θα) + α∗1[θα,T ]).





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Proof

we abbreviate (X ,Y ) = (Xα⊗τα αt ,x ,Y u,α⊗τα α

t ,x ,y ). Note that

X = 1AXα⊗τα α1t ,x + 1Ac Xα⊗ταa0

t ,x for τα ≤ s ≤ θα,Y = 1AY u,α⊗τα α1

t ,x ,y + 1Ac Y u,α⊗ταa0t ,x ,y for τα ≤ s ≤ θα.

(4.1)

For simplicity, let

E = Y (τα) < wκ(τα,X (τα)), E0 = Y (τα) < w(τα,X (τα)),E1 = w(τα,X (τα)) ≤ Y (τα) < wκ(τα,X (τα)),

G = Y (ρ) < wκ(ρ,X (ρ), G0 = Y (ρ) < w(ρ,X (ρ).

Note that E = E0 ∪ E1, E0 ∩ E1 = ∅ and G0 ⊂ G. The goal isto show P(G|B) > 0 given that B ⊂ E and P(B) > 0. It sufficesto show P(G ∩ B) = P(G ∩ B ∩ E0) + P(G ∩ B ∩ E1) > 0.





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Proof

(i) P(B ∩ E0) > 0: Directly from the way α1 is defined and thedefinition of the stochastic sub-solutions, we get

P(G0|B∩E0) = P(Y u,α⊗τα α1t ,x ,y (ρ) < w(ρ,Xα⊗τα α1

t ,x (ρ))|B∩E0) > 0.

This further implies that P(G ∩ B ∩ E0) ≥ P(G0 ∩ B ∩ E0) > 0.

(ii) P(B ∩ E1) > 0: from (4.1), we have thatP(Y (θα) < wκ(θα,X (θα))|B ∩ E1) = P(Y u,α⊗ταa0

t ,x ,y (θα) <

wκ(θα,Xα⊗ταa0t ,x (θα))|B ∩ E1). Let

∆(s) = Y (s ∧ θα)− (ϕ(s ∧ θα,X (s ∧ θα)) + κ) .





Future Work

Proof

We still study ∆. However, the rest of the proof relies on thesuper-martingale property of a process involving ∆ (in contrastto the increasing property of ∆). In fact, we prove M∆ is asuper-martingale on [τα, θα], where M is given by followingformula

λ = σY (·,X ,Y , u[a0]·,a0)− σX (·,X ,a0)Dϕ(·,X ),

β =(ϕt (·,X ) + Hu[a0]·,a0(·,Y , ϕ,Dϕ,D2ϕ)(·,X )

)‖λ‖−2λ1|λ|>δ,

M(·) = 1 +

∫ ·∧θατα

M(s)β>s dWs.





Future Work

Proof

From the definition of E1 and wκ, ∆(τα) < 0 on B ∩ E1. Thesuper-martingale property of M∆ implies that there exists anon-null H ⊂ B ∩ E1, H ∈ F t

τα such that ∆(θα ∧ ρ) < 0 on H.Therefore, we see that

Y (θα1 )− (ϕ(θα1 ,X (θα1 )) + κ) < 0 on H ∩ θα1 < θα2 ∧ ρ, (4.2)

Y (θα2 )− (ϕ(θα2 ,X (θα2 )) + κ) < 0 on H ∩ θα2 ≤ θα1 ∧ ρ, (4.3)

and that

Y (ρ)− (ϕ(ρ,X (ρ)) + κ) < 0 on H ∩ ρ < θα. (4.4)

From (4.2) and (4.3), we can show that





Future Work

Proof

Y (θα) < w(θα,X (θα)) on H ∩ θα ≤ ρ.

Now from the definition of stochastic sub-solutions and of α∗,

P(G0|H ∩ θα ≤ ρ) > 0 if P(H ∩ θα ≤ ρ) > 0. (4.5)

On the other hand, (4.4) implies that

P(G|H ∩ θα > ρ) > 0 if P(H ∩ θα > ρ) > 0. (4.6)

Since P(H) > 0,G0 ⊂ G, and H ⊂ E1 ∩ B, (4.5) and (4.6) implyP(G ∩ E1 ∩ B) > 0.

2.2 The boundary condition: Similar to that in Step 1.2.





Future Work

Restate the theorem,

Theorem 5.1 (Stochastic Perron for stochastic target games)

Under all standing assumptions1 If g is USC, the function v+ is a bounded upper

semi-continuous (USC) viscosity sub-solution of (2.2);2 If g is LSC, the function v− is a bounded lower

semi-continuous (LSC) viscosity super-solution of (2.2).

Corollary 5.1Under all standing assumptions, if g is continuous, then v iscontinuous and is the unique bounded viscosity solution of(2.2).





Future Work

OUTLINE

1 Introduction

2 The Set-up



5 Future Work





Future Work

Future work

(1) Add a stopping time in the definition of v :

v(t , x) := infy ∈ R : ∃ u ∈ U(t) s.t. Y u,αt ,x ,y (τ) ≥ g(Xα

t ,x (τ)) -a.s.

∀ α ∈ At , τ ∈ St.

In the super-hedging context in mathematical finance, it issuper-hedging a American financial contract (with modeluncertainty).

Realistic model. The holder of the contract may playagainst the controller since she has the right to exercise atany time τ .Controller and stopper game version of stochastic targetproblem with controlled loss, Bayraktar and Huang.





Future Work

Future Work

(2) Find a weaker sufficient condition which guarantees that U−is not empty.

Under some suitable conditions, Girsanov’s Theorem⇒ Yis a martingale under a measure Q which is equivalent toP.Assume |uY |

‖σY ‖ is bounded on N = (t , x , y ,u,a) : σY 6= 0and for fixed (t , x , y) ∈ D<T × R, τα ∈ St , assume thatany given u, α, there exits a α, such that

EP

∫ T

0‖σY‖2(s,X (s),Y (s), u[α⊗τα α]s, [α⊗τα α]s)ds,

where X = Xα⊗τα αt ,x ,Y = Y u,α⊗τα α

t ,x ,y .





Future Work

Future Work

Sketch of the proof: Take w(t , x) = c, where c is a lowerbound of g. For any given u, α, choose the α in the assumption.Let B ⊂ Y (τ) < w(τ,X (τ)) and P(B > 0). Let

θs ,

uYσY‖σY ‖2 (s,X (s),Y (s),u[α⊗τα α]s, α⊗τα α), σY 6= 0C, otherwise,

for any constant C. Therefore, θs satisfies the Novikov’scondition and W (s) = W (s)−

∫ sτα θudu is a Brownian motion

under some probability measure Q which is equivalent to P .Therefore, we have Q(B) > 0 .

Under Q, dY = σY dWs. Under square integrability condition, Yis a martingale.





Future Work

Future Work

From the martingale property of Y under Q , we know thatY (ρ) ≤ Y (τ) on some set H ⊂ B with Q(H) > 0 (P(H) > 0). Inaddition,

Y (ρ) ≤ Y (τ) < m = w(t , x) on H.

This implies Q(Y (ρ) < m |B) > 0 and P(Y (ρ) < m |B) > 0.Therefore, w(t , x) = m is a stochastic sub-solution.

Remark: the relative growth condition is similar to Assumption2.1.

(3) Relax the continuity assumption for u.





Future Work

Thank you!





Future Work

E. BAYRAKTAR AND Y.-J. HUANG, On the multidimensionalcontroller-and-stopper games, SIAM J. Control Optim., 51(2013), pp. 1263–1297.

E. BAYRAKTAR AND M. SÎRBU, Stochastic Perron’s methodand verification without smoothness using viscositycomparison: the linear case, Proc. Amer. Math. Soc., 140(2012), pp. 3645–3654.

, Stochastic Perron’s method forHamilton-Jacobi-Bellman equations, SIAM J. ControlOptim., 51 (2013), pp. 4274–4294.

, Stochastic Perron’s method and verification withoutsmoothness using viscosity comparison: obstacle problemsand Dynkin games, Proc. Amer. Math. Soc., 142 (2014),pp. 1399–1412.





Future Work

B. BOUCHARD AND M. NUTZ, Stochastic Target Gamesand Dynamic Programming via Regularized ViscositySolutions, ArXiv e-prints, (2013).

M. SÎRBU, Stochastic Perron’s method and elementarystrategies for zero-sum differential games, SIAM J. ControlOptim., 52 (2014), pp. 1693–1711.


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