A two player zerosum game where one player
observes a Brownian motion.
common work with Fabien Gensbittel (TSE Toulouse)
Erice, May 2017
Introduction : Idea of the game
• 0 ≤ t ≤ T,
R
m ∈ P2 = {m prob. on Rd , s.t. |x|2 m(dx) < ∞},
• (Bst,m )s∈[t,T ] Rd -valued Brownian motion, such that
L(Btt,m ) = m,
• Controls: (us , vs )s∈[t,T ] → U × V compact, metric,
hR
i
T
• Payo: J(t, m, u· , v· ) = E t f (s, Bst,m , us , vs )ds ,
for f suciently regular.
Introduction : Idea of the game
• 0 ≤ t ≤ T,
R
m ∈ P2 = {m prob. on Rd , s.t. |x|2 m(dx) < ∞},
• (Bst,m )s∈[t,T ] Rd -valued Brownian motion, such that
L(Btt,m ) = m,
• Controls: (us , vs )s∈[t,T ] → U × V compact, metric,
hR
i
T
• Payo: J(t, m, u· , v· ) = E t f (s, Bst,m , us , vs )ds ,
for f suciently regular.
Zero-sum game with asymmetric information:
•
•
Player 1 plays (us ), tries to minimize the payo
J(t, m, u· , v· ),
player 2 plays (vs ), tries to maximize this payo,
Introduction : Idea of the game
• 0 ≤ t ≤ T,
R
m ∈ P2 = {m prob. on Rd , s.t. |x|2 m(dx) < ∞},
• (Bst,m )s∈[t,T ] Rd -valued Brownian motion, such that
L(Btt,m ) = m,
• Controls: (us , vs )s∈[t,T ] → U × V compact, metric,
hR
i
T
• Payo: J(t, m, u· , v· ) = E t f (s, Bst,m , us , vs )ds ,
for f suciently regular.
Zero-sum game with asymmetric information:
•
•
•
•
Player 1 plays (us ), tries to minimize the payo
J(t, m, u· , v· ),
player 2 plays (vs ), tries to maximize this payo,
both players observe the action of their opponent, not the
payo,
player 1 observes the Brownian motion,
player 2 not.
(Bst,m ) Brownian motion, such that L(Btt,m ) = m,
i
hR
Payo: J(t, m, u· , v· ) = E tT f (s, Bst,m , us , vs )ds
Player 1 : (us ), minimize the payo, player 2: (vs ), maximize the payo,
player 1 observes the Brownian motion, Player 2 not.
Questions:
1) Is it possible to formalize the game in order to have a value ?
−→ nd adapted sets of strategies such that
inf
sup J(t, m, α, β) = sup inf
α(B t,m ) β
β α(B t,m )
J(t, m, α, β) ?
(Bst,m ) Brownian motion, such that L(Btt,m ) = m,
i
hR
Payo: J(t, m, u· , v· ) = E tT f (s, Bst,m , us , vs )ds
Player 1 : (us ), minimize the payo, player 2: (vs ), maximize the payo,
player 1 observes the Brownian motion, Player 2 not.
Questions:
1) Is it possible to formalize the game in order to have a value ?
−→ nd adapted sets of strategies such that
inf
sup J(t, m, α, β) = sup inf
α(B t,m ) β
β α(B t,m )
J(t, m, α, β) ?
2) If yes, do we have a characterization of the value in terms
of a PDE ?
(Bst,m ) Brownian motion, such that L(Btt,m ) = m,
i
hR
Payo: J(t, m, u· , v· ) = E tT f (s, Bst,m , us , vs )ds
Player 1 : (us ), minimize the payo, player 2: (vs ), maximize the payo,
player 1 observes the Brownian motion, Player 2 not.
Questions:
1) Is it possible to formalize the game in order to have a value ?
−→ nd adapted sets of strategies such that
inf
sup J(t, m, α, β) = sup inf
α(B t,m ) β
Answer to 1): ???
β α(B t,m )
J(t, m, α, β) ?
(Bst,m ) Brownian motion, such that L(Btt,m ) = m,
i
hR
Payo: J(t, m, u· , v· ) = E tT f (s, Bst,m , us , vs )ds
Player 1 : (us ), minimize the payo, player 2: (vs ), maximize the payo,
player 1 observes the Brownian motion, Player 2 not.
Questions:
1) Is it possible to formalize the game in order to have a value ?
−→ nd adapted sets of strategies such that
inf
sup J(t, m, α, β) = sup inf
α(B t,m ) β
β α(B t,m )
J(t, m, α, β) ?
Answer to 1): ???
→ discretePtime games,
−1
payo: E[ Nk=1
(tk+1 − tk )f (tk , Btk , uk , vk )],
then let supk |tk − tk−1 | → 0.
for t = t0 < . . . < tN = T ,
(Bst,m ) Brownian motion, such that L(Btt,m ) = m,
i
hR
Payo: J(t, m, u· , v· ) = E tT f (s, Bst,m , us , vs )ds
Player 1 : (us ), minimize the payo, player 2: (vs ), maximize the payo,
player 1 observes the Brownian motion, Player 2 not.
Questions:
1) Is it possible to formalize the game in order to have a value ?
−→ nd adapted sets of strategies such that
inf
sup J(t, m, α, β) = sup inf
α(B t,m ) β
β α(B t,m )
J(t, m, α, β) ?
Answer to 1): ???
→ discretePtime games,
−1
payo: E[ Nk=1
(tk+1 − tk )f (tk , Btk , uk , vk )],
then let supk |tk − tk−1 | → 0.
for t = t0 < . . . < tN = T ,
Remark: Approach used in Cardaliaguet- R.-Rosenberg-Vieille
(2016), Gensbittel (2016), see also Sorin (2017).
A discrete time game with vanishing stage duration
Fix (t, m) ∈ [0, T ] × P2 .
For any partition π = {t = t1 < . . . < tN = T }, we dene a
N -stage game Γπ (t, m):
• at each stage k ∈ {1, . . . , N − 1},
•
•
•
•
•
player 1 observe Btt,m
k
both players choose simultaneously a pair of controls
(uk , vk ) ∈ U × V ,
the actions are observed after each stage
stage payo : f (tk , Btk , uk , vk ) (not observed)
total expected payo :
"N −1
#
X
E
(tk+1 − tk )f (tk , Btk , uk , vk ) .
k=1
A discrete time game with vanishing stage duration
Fix (t, m) ∈ [0, T ] × P2 .
For any partition π = {t = t1 < . . . < tN = T }, we dene a
N -stage game Γπ (t, m):
• at each stage k ∈ {1, . . . , N − 1},
•
•
•
•
•
player 1 observe Btt,m
k
both players choose simultaneously a pair of controls
(uk , vk ) ∈ U × V ,
the actions are observed after each stage
stage payo : f (tk , Btk , uk , vk ) (not observed)
total expected payo :
"N −1
#
X
E
(tk+1 − tk )f (tk , Btk , uk , vk ) .
k=1
−→ For each π , the game Γπ (t, m) has a value Vπ (t, m).
Theorem: Vπ (t, m) converges (up to a subsequence) when
|π| & 0.
An alternative formulation
Let M(t, m) be the set of processes (Ms )t≤s≤T with values in
P2 , such that,
for all s ∈ [t, T ], Ms = L(Bst,m |Fs ),
for (Fs )s∈[t,T ] ltration such that (Bst,m )
is still a (σ(Brt,m , t ≤ r ≤ s) ∨ Fs )-Brownian motion.
Interpretation: if (Fs ) is generated by a control of player 1 in
continuous time, Ms is the belief of player 2 on Bst,m .
An alternative formulation
Let M(t, m) be the set of processes (Ms )t≤s≤T with values in
P2 , such that,
for all s ∈ [t, T ], Ms = L(Bst,m |Fs ),
for (Fs )s∈[t,T ] ltration such that (Bst,m )
is still a (σ(Brt,m , t ≤ r ≤ s) ∨ Fs )-Brownian motion.
Interpretation: if (Fs ) is generated by a control of player 1 in
continuous time, Ms is the belief of player 2 on Bst,m .
Examples:
is trivial : L(Bst,m |Fs ) = L(Bst,m ) ,
(Bst,m ) is adapted to (Fs ): L(Bst,m |Fs ) = δBst,m .
• (Fs )
•
An alternative formulation
M(t, m) = {(Ms )t≤s≤T → P2 , Ms = L(Bst,m |Fs ), (Fs ) convinient}.
Suppose that Isaac's assumption holds :
for all (t, m) ∈ [0, T ] × P2 ,
Z
inf sup
u∈U v∈V
Z
f (t, x, u, v)m(dx) = sup inf
Rd
v∈V u∈U
Set V (t, m) := inf M ∈M(t,m) E[
RT
t
f (t, x, u, v)m(dx) := H(t, m).
Rd
H(s, Ms )ds].
An alternative formulation
M(t, m) = {(Ms )t≤s≤T → P2 , Ms = L(Bst,m |Fs ), (Fs ) convinient}.
Suppose that Isaac's assumption holds :
for all (t, m) ∈ [0, T ] × P2 ,
Z
inf sup
u∈U v∈V
Z
f (t, x, u, v)m(dx) = sup inf
v∈V u∈U
Rd
Set V (t, m) := inf M ∈M(t,m) E[
RT
t
f (t, x, u, v)m(dx) := H(t, m).
Rd
H(s, Ms )ds].
Theorem: For all (t, m),
V (t, m) = lim Vπ (t, m).
|π|&0
Characterization of the value : Derivative in the space of
probability measures
Proposition [Lyons 201?], [Cardaliaguet-Delarue-Lasry-Lyons
2015]:
Let L2d = {X random variables → Rd , E[|X|2 ] < ∞}.
For U : P2 → R, dene U : L2d → R, X 7→ U(X) := U (L(X)).
Characterization of the value : Derivative in the space of
probability measures
Proposition [Lyons 201?], [Cardaliaguet-Delarue-Lasry-Lyons
2015]:
Let L2d = {X random variables → Rd , E[|X|2 ] < ∞}.
For U : P2 → R, dene U : L2d → R, X 7→ U(X) := U (L(X)).
Then (under convenient assumptions), there exists
Dm U : P2 × Rd → Rd such that, for all X, Y ∈ L2d ,
lim
h→0+
U(X + hY ) − U(X)
= E[Dm U (L(X), Y )Y ].
h
Characterization : the equation
, us , vs )]ds.
payo J(t, m, u, v) = tT E[f (s, Bst,m
R
Hamiltonian H(t, m) := inf u supv Rd f (t, x, u, v)m(dx).
R
For U : [0, T ] × P2 → R, consider the equation
R
∂t U (t, m) + 12 Rd divx [Dm U ](t, m, x)m(dx) + H(t, m) = 0,
U (T, m) = 0, m ∈ P2 ,
(1)
Characterization : the equation
, us , vs )]ds.
payo J(t, m, u, v) = tT E[f (s, Bst,m
R
Hamiltonian H(t, m) := inf u supv Rd f (t, x, u, v)m(dx).
R
For U : [0, T ] × P2 → R, consider the equation
R
∂t U (t, m) + 12 Rd divx [Dm U ](t, m, x)m(dx) + H(t, m) = 0,
U (T, m) = 0, m ∈ P2 ,
(1)
Proposition: Set pt,m
:= L(Bst,m ). Suppose that H is
s
suciently smooth. Then (1) has a unique regular solution:
Z
U0 (t, m) :=
t
T
H(r, pt,m
r )dr.
Characterization : the equation
, us , vs )]ds.
payo J(t, m, u, v) = tT E[f (s, Bst,m
R
Hamiltonian H(t, m) := inf u supv Rd f (t, x, u, v)m(dx).
R
For U : [0, T ] × P2 → R, consider the equation
R
∂t U (t, m) + 12 Rd divx [Dm U ](t, m, x)m(dx) + H(t, m) = 0,
U (T, m) = 0, m ∈ P2 ,
(1)
Proposition: Set pt,m
:= L(Bst,m ). Suppose that H is
s
suciently smooth. Then (1) has a unique regular solution:
Z
U0 (t, m) :=
T
H(r, pt,m
r )dr.
t
Remark: U0 is the value of a continuous time game where the
Brownian motion is observed by nobody.
Characterization of the value
R
∂t U (t, m) + 21 Rd div[Dm U ](t, m, x)m(dx) + H(t, m) = 0,
U (T, m) = 0, m ∈ P2 .
(1)
Denition: We call a subsolution of (1) a map
U : [0, T ] × P2 → R such that, for all (t, m) ∈ [0, T ) × P2 , and
for all ϕ suciently smooth, such that ϕ − U has a minimum at
(t, m),
Z
1
div[Dm ϕ](t, m, x)m(dx) + H(t, m) ≥ 0.
∂t ϕ(t, m) +
2 Rd
Theorem: The value function V is the largest bounded,
continuous subsolution of (1), which is convex in m and satises
the terminal condition V (T, ·) = 0.
Open questions and perspectives
•
Characterization of V as the solution of a Hamilton-Jacobi
equation
•
Observation by only the rst Player of a controlled diusion
•
Does the continuous time game have a value ?
Thank you for your attention!