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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
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
OUTLINE
1 Introduction
2 The Set-up
3 Sub-solution property for v+
4 Super-solution property for v−
5 Future Work
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
OUTLINE
1 Introduction
2 The Set-up
3 Sub-solution property for v+
4 Super-solution property for v−
5 Future Work
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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).
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
OUTLINE
1 Introduction
2 The Set-up
3 Sub-solution property for v+
4 Super-solution property for v−
5 Future Work
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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 ;
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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);
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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,
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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 .
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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).
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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].
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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)).
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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 .
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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+.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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)
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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)).
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
OUTLINE
1 Introduction
2 The Set-up
3 Sub-solution property for v+
4 Super-solution property for v−
5 Future Work
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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).
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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,
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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)).
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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)
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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).
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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).
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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)
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
OUTLINE
1 Introduction
2 The Set-up
3 Sub-solution property for v+
4 Super-solution property for v−
5 Future Work
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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,
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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 ]).
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
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) .
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
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
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
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).
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
OUTLINE
1 Introduction
2 The Set-up
3 Sub-solution property for v+
4 Super-solution property for v−
5 Future Work
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
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 .
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
Thank you!
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
Future Work
E. BAYRAKTAR AND Y.-J. HUANG, On the multidimensionalcontroller-and-stopper games, SIAM J. Control Optim., 51(2013), pp. 1263–1297.
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, 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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games
IntroductionThe Set-up
Sub-solution property for v+
Super-solution property for v−
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.
E. Bayraktar, J. Li Stochastic Perron for Stochastic Target Games