The value of a zero-sum stochastic differential game
involving impulse control
SIAM Conference on Financial Mathematics and Engineering (FM16)
November 20, 2016
Parsiad Azimzadeh
Agenda
The value of a
zero-sum stochastic
differential game
involving impulse
control
Parsiad Azimzadeh
Introduction
Framework and results
Introduction
Framework and results
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
Introduction
Zero-sum game
The value of a
zero-sum stochastic
differential game
involving impulse
control
Parsiad Azimzadeh
2
A zero-sum game is a competition between player I and player
II whose rewards add up to zero
Introduction
Framework and results
A stochastic differential game is a game whose state evolves
according to a stochastic differential equation (SDE)
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
Dynamics
The value of a
zero-sum stochastic
differential game
involving impulse
control
The state (of our game) evolves according to
Parsiad Azimzadeh
3
dXs = µ(Xs , bs )ds + σ(Xs , bs )dWs
Introduction
Framework and results
where b ≡ (bs )s is chosen by player II
Player I chooses an impulse control a ≡ (τj , zj )j≥1 that
influences the state “instantaneously:”
Xτj = Xτj − + Γ(τj , zj )
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
Cost functional
The value of a
zero-sum stochastic
differential game
involving impulse
control
Parsiad Azimzadeh
At the end of the game, player II pays player I
4
Introduction
Framework and results
J(t, x; a ≡ (τj , zj )j≥1 , b ≡ (bs )s )


Z T
X
f (s, Xs , bs )ds +
K (τj , zj ) + g(XT )
= E(t,x) 
t
j
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
Why impulses?
The value of a
zero-sum stochastic
differential game
involving impulse
control
Parsiad Azimzadeh
Impulses enable more realistic financial models
5
Introduction
Framework and results
e.g., fixed transaction costs: K (t, z) = − const.
e.g., proportional transaction costs: K (t, z) = − const. |z|
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
Value of the game
The value of a
zero-sum stochastic
differential game
involving impulse
control
Define upper and lower values in the Elliot-Kalton sense:
Parsiad Azimzadeh
6
inf sup J(t, x; a, β(a)) and sup inf J(t, x; α(b), b)
β
a
α
Introduction
Framework and results
b
The game is said to admit a value if the lower and upper values
coincide
i.e., no player has an “informational advantage”
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
Goal
The value of a
zero-sum stochastic
differential game
involving impulse
control
Parsiad Azimzadeh
7
Introduction
Framework and results
Establish that the game admits a value by viscosity theory
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
Framework and results
Controls
The value of a
zero-sum stochastic
differential game
involving impulse
control
Parsiad Azimzadeh
A [t, T ] impulse control is a tuple a := (τj , zj )j where
τ1 ≤ τ2 ≤ . . . are (Ft,s )s∈[t,T ] -stopping times, each zj is an
Ft,τj ∧T -measurable r.v. taking values in Z ⊂ RdZ , and
I E[#a] < ∞ where #a := sup{j ≥ 1 : τj ≤ T };
I
Introduction
8
Framework and results
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
each τj takes values in ([t, T ] ∩ Q) ∪ {T , +∞}.
The set of all such controls is denoted A(t).
A [t, T ] stochastic control is an (Ft,s )s∈[t,T ] -progressively
measurable process b := (bs )s∈[t,T ] taking values in B ⊂ RdB .
The set of all such controls is denoted B(t).
Admissible control
The value of a
zero-sum stochastic
differential game
involving impulse
control
The integral form of the SDE is
Parsiad Azimzadeh
Introduction
Z
Xs = x +
s
Z
µ(Xu , bu )du +
t
t
s
9
Framework and results
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
σ(Xu , bu )dWu
X
+
Γ(τj , zj ) for s ∈ [t, T ].
τj ≤s
A [t, T ] impulse control a ∈ A(t) is admissible at x ∈ Rd if for all
b ∈ B(t), a solution of the SDE exists and is unique
The set of all such controls is denoted A(t, x).
Capped control
The value of a
zero-sum stochastic
differential game
involving impulse
control
Parsiad Azimzadeh
Introduction
dZ
q,Q
For each integer q ≥ 1 and Borel set Q ⊂ R , A
set of all controls with
I
less than q impulses (i.e. τq = +∞);
I
zj ∈ Q for all j.
10
Framework and results
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
(t) is the
Strategies
The value of a
zero-sum stochastic
differential game
involving impulse
control
α : B(t) → A(t) is a player I strategy if it is
I
I
nonanticipative: for any b, b0 ∈ B(t) and s ∈ [t, T ], if b ≡ b0
on [t, s], then α(b) ≡ α(b0 ) on [t, s]
delayed: there is a partition t = t0 < t1 < · · · < tm = T such
that for all b, b0 ∈ B(t) and i < m, if b ≡ b0 on [t, ti ],
α(b) ≡ α(b0 ) on [t, ti+1 ]
Parsiad Azimzadeh
Introduction
11
Framework and results
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
β : A(t) → B(t) is defined similarly, with the additional
requirement that β is an r-strategy
The set of all player I (resp. player II) strategies is A (t) (resp.
B(t)).
A (t, x) and A q,Q (t) defined in the obvious ways.
Assumptions
The value of a
zero-sum stochastic
differential game
involving impulse
control
Most assumptions are fairly standard; we refer the reader to
[Azimzadeh, 2016]. One (more exotic) assumption is worth
mentioning...
Parsiad Azimzadeh
Introduction
12
Framework and results
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
sup K < 0 and z 7→ K (t, z) ∈ ω(1) as |z| → ∞ (uniformly in t)
ω is the Bachmann–Landau symbol. Precisely, we mean that
for each c > 0, there exists an r > 0 such that for all
(t, z) ∈ [0, T ] × Z with |z| > r , |k (t, z)| ≥ c.
Upper and lower values
The value of a
zero-sum stochastic
differential game
involving impulse
control
Parsiad Azimzadeh
Introduction
v + (t, x) := inf
sup J(t, x; a, β(a))
13
Framework and results
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
β∈B(t) a∈A(t,x)
v − (t, x) :=
sup
inf J(t, x; α(b), b)
α∈A (t,x) b∈B(t)
The simple inequality
The value of a
zero-sum stochastic
differential game
involving impulse
control
Lemma
For each (α, β) ∈ A (t) × B(t), there exists (a, b) ∈ A(t) × B(t)
such that α(b) = a and β(a) = b.
Parsiad Azimzadeh
Introduction
14
Framework and results
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
Proof.
Induction on delay partition.
For each (t, x),
v − (t, x) =
sup
inf
J(t, x; α, β)
α∈A (t,x) β∈B(t)
≤ inf
sup
J(t, x; α, β) = v + (t, x)
β∈B(t) α∈A (t,x)
Nonanticipative stopping times
The value of a
zero-sum stochastic
differential game
involving impulse
control
The concept of a nonanticipative family of stopping times
formalizes the intuitive notion that the decision to stop should
not depend on future information from the controls
Parsiad Azimzadeh
Introduction
15
Framework and results
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
Let t ∈ [0, T ] and C(t) be a subset of A(t) × B(t). {θa,b }(a,b)∈C(t)
is a nonanticipative family of (Ft,s )s∈[t,T ] -stopping times if
I
θa,b is an (Ft,s )s∈[t,T ] -stopping time for each (a, b) ∈ C(t);
I
for each s ∈ [t, T ] and (a, b), (a0 , b0 ) ∈ C(t), if a ≡ a0 and
0 0
b ≡ b0 on [t, s], then P(θa,b = θa ,b on [t, s]) = 1.
Dynamic programming principle (DPP)
The value of a
zero-sum stochastic
differential game
involving impulse
control
Theorem
Parsiad Azimzadeh
There exists a q ≥ 1 and a compact Q such that for each
(t, x) ∈ [0, T ) × Rd and each nonanticipative family of
[t, T ] ∩ Q-valued (Ft,s )s∈[t,T ] -stopping times
{θa,b }(a,b)∈Aq,Q (t)×B(t) ,
v + (t, x) ≤ inf
Introduction
16
Framework and results
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
sup
β∈B(t) a∈Aq,Q (t)

Z
E
t

θ
f (s, Xs , bs )ds +
X
K (τj , zj ) + (v + )∗ (θ, Xθ )
τj ≤θ
where it is understood that X := X t,x;a,β(a) and θ := θa,β(a) .
There is a symmetric result for v −
Dynamic programming equation (DPE)
The value of a
zero-sum stochastic
differential game
involving impulse
control
Consider the HJBQVI
0 = F (·, u, Du(·), D 2 u(·))
(
min{− infb∈B {(∂t + Łb )u + f b }, u − Mu}
:=
min{u − g, u − Mu}
Parsiad Azimzadeh
Introduction
on [0, T ) × Rd
on {T } × Rd
17
Framework and results
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
where Łb is the generator of the SDE (w/o impulses) and
Mu(t, x) := sup {u(t, x + Γ(t, z)) + K (t, z)}
z∈Z
Theorem
(v + )∗ (resp. (v − )∗ ) is a viscosity subsolution (resp.
supersolution) of the HJBQVI.
Hurdles in DPE proof
The value of a
zero-sum stochastic
differential game
involving impulse
control
Recall that our DPP only allows for stopping times θ that take
rational values
Parsiad Azimzadeh
i.e., we cannot use début-type stopping time
Introduction
18
Framework and results
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
θ := inf {s > t : (s, Xs ) ∈
/ Nh }
in the proof!
• (θ, Xθ )
(t, x) •
Nh
Approximating θ
The value of a
zero-sum stochastic
differential game
involving impulse
control
This problem can be avoided by imposing a priori uniform
continuity on v ± (but we do not want to impose so much
regularity on K !)
Idea: Approximate θ by θm ↓ θ where each θm takes rational
values
Parsiad Azimzadeh
Introduction
19
Framework and results
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
N2h
•
(θ, Xθ )
• (θm , Xθm )
(t, x) •
Nh
Approximating θ (cont’d)
The value of a
zero-sum stochastic
differential game
involving impulse
control
Parsiad Azimzadeh
Let Am := {(θm , Xθm ) ∈ N2h } so that 1Am → 1 almost surely by
continuity of sample paths (between impulses)
Introduction
20
Framework and results
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
Consider the problem on Am and its complement
Use an argument involving compact test functions to show that
the contributions from the complement are as close to zero as
we desire
Comparison principle
The value of a
zero-sum stochastic
differential game
involving impulse
control
Parsiad Azimzadeh
Introduction
Theorem
21
Framework and results
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
If u is a bounded subsolution and w is a bounded
supersolution of the HJBQVI, u ∗ ≤ w∗ pointwise.
This gives us v + ≤ (v + )∗ ≤ (v − )∗ ≤ v − pointwise!
Thank you!
Bibliography
The value of a
zero-sum stochastic
differential game
involving impulse
control
Parsiad Azimzadeh
Introduction
22
Framework and results
22
David Cheriton School of
Computer Science
University of Waterloo
Waterloo, Ontario
[Azimzadeh, 2016] Azimzadeh, P. (2016).
The value of a zero-sum stochastic differential game
involving impulse control.
arXiv preprint arXiv:1609.09092.