Homogenization and other multi-scale problems
in Bellman-Isaacs equations
Martino Bardi
Dipartimento di Matemetica Pura ed Applicata
Università di Padova
New Trends in Analysis and Control of Nonlinear PDEs
Roma, June 13–15, 2011
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
1 / 29
Plan
1
2
Homogenization of non-coercive Hamilton–Jacobi equations
(joint work with G. Terrone)
I
Some previous results
I
Examples of non-homogenization
I
Homogenization on subspaces
I
Convex-concave eikonal equations and differential games
Optimal control with stochastic volatility
(joint work with A. Cesaroni and L. Manca)
I
Motivation: financial models
I
Singular perturbations of H-J-B equations
I
Merton portfolio optimization with stochastic volatility
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
2 / 29
Plan
1
2
Homogenization of non-coercive Hamilton–Jacobi equations
(joint work with G. Terrone)
I
Some previous results
I
Examples of non-homogenization
I
Homogenization on subspaces
I
Convex-concave eikonal equations and differential games
Optimal control with stochastic volatility
(joint work with A. Cesaroni and L. Manca)
I
Motivation: financial models
I
Singular perturbations of H-J-B equations
I
Merton portfolio optimization with stochastic volatility
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
2 / 29
1. Homogenization of Hamilton–Jacobi Equations
Problem: let ε → 0+ in
utε + H x, xε , Du ε = 0
in (0, T ) × RN
u ε (0, x) = h x, xε .
ZN − periodic case:
H(x, ξ + k , p) = H(x, ξ, p),
h(x, ξ + k ) = h(x, ξ)
∀k ∈ ZN .
Goal: find continuous effective Hamiltonian and initial data H, h s.t.
uε → u
locally uniformly, and u solves
ut + H (x, Du) = 0
in (0, T ) × RN
u(0, x) = h (x) .
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
3 / 29
Classical result for coercive H
Coercivity in p is
lim H(x, ξ, p) = +∞
|p|→∞
uniformly in x, ξ.
P.-L. Lions - Papanicolaou - Varadhan ’86, L.C. Evans ’92:
The effective Hamiltonian H = H(x, p) is the unique constant such
that the cell problem, with x, p frozen parameters,
H(x, ξ, Dξ χ + p) = H,
in RN ,
has a ZN −periodic (viscosity) solution χ(ξ), called the corrector;
if h = h(x) is independent of xε , u ε → u locally uniformly.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
4 / 29
Classical result for coercive H
Coercivity in p is
lim H(x, ξ, p) = +∞
|p|→∞
uniformly in x, ξ.
P.-L. Lions - Papanicolaou - Varadhan ’86, L.C. Evans ’92:
The effective Hamiltonian H = H(x, p) is the unique constant such
that the cell problem, with x, p frozen parameters,
H(x, ξ, Dξ χ + p) = H,
in RN ,
has a ZN −periodic (viscosity) solution χ(ξ), called the corrector;
if h = h(x) is independent of xε , u ε → u locally uniformly.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
4 / 29
Classical result for coercive H
Coercivity in p is
lim H(x, ξ, p) = +∞
|p|→∞
uniformly in x, ξ.
P.-L. Lions - Papanicolaou - Varadhan ’86, L.C. Evans ’92:
The effective Hamiltonian H = H(x, p) is the unique constant such
that the cell problem, with x, p frozen parameters,
H(x, ξ, Dξ χ + p) = H,
in RN ,
has a ZN −periodic (viscosity) solution χ(ξ), called the corrector;
if h = h(x) is independent of xε , u ε → u locally uniformly.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
4 / 29
The effective initial data
If the initial data h = h(x, xε ) must find the initial condition h(x).
O. Alvarez - M. B. (ARMA ’03):
Assume there exists the recession function of H
H(x, ξ, λp)
.
λ→+∞
λ
H 0 (x, ξ, p) := lim
Consider, for frozen x,
If
then
wt + H 0 (x, ξ, Dξ w) = 0,
w(0, ξ) = h(x, ξ).
limt→+∞ w(t, ξ) = constant =: h(x),
limt→0 limε→0 u ε = h(x).
In the coercive case
Martino Bardi (Università di Padova)
h (x) = minξ h(x, ξ).
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
5 / 29
The effective initial data
If the initial data h = h(x, xε ) must find the initial condition h(x).
O. Alvarez - M. B. (ARMA ’03):
Assume there exists the recession function of H
H(x, ξ, λp)
.
λ→+∞
λ
H 0 (x, ξ, p) := lim
Consider, for frozen x,
If
then
wt + H 0 (x, ξ, Dξ w) = 0,
w(0, ξ) = h(x, ξ).
limt→+∞ w(t, ξ) = constant =: h(x),
limt→0 limε→0 u ε = h(x).
In the coercive case
Martino Bardi (Università di Padova)
h (x) = minξ h(x, ξ).
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
5 / 29
The effective initial data
If the initial data h = h(x, xε ) must find the initial condition h(x).
O. Alvarez - M. B. (ARMA ’03):
Assume there exists the recession function of H
H(x, ξ, λp)
.
λ→+∞
λ
H 0 (x, ξ, p) := lim
Consider, for frozen x,
If
then
wt + H 0 (x, ξ, Dξ w) = 0,
w(0, ξ) = h(x, ξ).
limt→+∞ w(t, ξ) = constant =: h(x),
limt→0 limε→0 u ε = h(x).
In the coercive case
Martino Bardi (Università di Padova)
h (x) = minξ h(x, ξ).
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
5 / 29
Bellman-Isaacs equations
In control theory the Hamiltonian is, for A compact,
H(x, ξ, p) = max {−f (x, ξ, a) · p − l (x, ξ, a)} ,
a∈A
ξ=
x
ε
and in 0-sum differential games it is, for B compact,
H(x, ξ, p) = min max {−f (x, ξ, a, b) · p − l (x, ξ, a, b)} .
b∈B a∈A
The associated control system in an oscillating periodic medium is
x
ẋ = f x, , a, b
ε
and cost - payoff
Z t x(t)
x(s)
, a(s), b(s) ds + h x(t),
J :=
l x(s),
ε
ε
0
that player a wants to minimize and player b to maximize.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
6 / 29
Bellman-Isaacs equations
In control theory the Hamiltonian is, for A compact,
H(x, ξ, p) = max {−f (x, ξ, a) · p − l (x, ξ, a)} ,
a∈A
ξ=
x
ε
and in 0-sum differential games it is, for B compact,
H(x, ξ, p) = min max {−f (x, ξ, a, b) · p − l (x, ξ, a, b)} .
b∈B a∈A
The associated control system in an oscillating periodic medium is
x
ẋ = f x, , a, b
ε
and cost - payoff
Z t x(t)
x(s)
, a(s), b(s) ds + h x(t),
J :=
l x(s),
ε
ε
0
that player a wants to minimize and player b to maximize.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
6 / 29
Alternative conditions to coercivity
coercivity of H is equivalent to Small Time Local Controllability in
a time proportional to the distance (by one of the players for any
control of the other),
it is a STRONG assumption, can be replaced by
Controllability in a uniformly Bounded Time (by one of the players
for any control of the other) (M.B. - Alvarez Mem. AMS 2010)
Example: hypoelliptic eikonal equation
utε +
k
X
i=1
|fi
x x
· Du ε | = l x,
ε
ε
in (0, T ) × RN
with f1 , ..., fk ∈ C ∞ satisfying Hörmander bracket-generating
condition.
Non-resonance conditions (Arisawa - Lions ’99, M.B. - Alvarez ’10
for control problems, Cardaliaguet ’10 for games).
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
7 / 29
Alternative conditions to coercivity
coercivity of H is equivalent to Small Time Local Controllability in
a time proportional to the distance (by one of the players for any
control of the other),
it is a STRONG assumption, can be replaced by
Controllability in a uniformly Bounded Time (by one of the players
for any control of the other) (M.B. - Alvarez Mem. AMS 2010)
Example: hypoelliptic eikonal equation
utε +
k
X
i=1
|fi
x x
· Du ε | = l x,
ε
ε
in (0, T ) × RN
with f1 , ..., fk ∈ C ∞ satisfying Hörmander bracket-generating
condition.
Non-resonance conditions (Arisawa - Lions ’99, M.B. - Alvarez ’10
for control problems, Cardaliaguet ’10 for games).
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
7 / 29
Alternative conditions to coercivity
coercivity of H is equivalent to Small Time Local Controllability in
a time proportional to the distance (by one of the players for any
control of the other),
it is a STRONG assumption, can be replaced by
Controllability in a uniformly Bounded Time (by one of the players
for any control of the other) (M.B. - Alvarez Mem. AMS 2010)
Example: hypoelliptic eikonal equation
utε +
k
X
i=1
|fi
x x
· Du ε | = l x,
ε
ε
in (0, T ) × RN
with f1 , ..., fk ∈ C ∞ satisfying Hörmander bracket-generating
condition.
Non-resonance conditions (Arisawa - Lions ’99, M.B. - Alvarez ’10
for control problems, Cardaliaguet ’10 for games).
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
7 / 29
Examples of non-homogenization
Example 1
utε + |uxε + uyε |β = cos 2π x−y
ε
u ε (0, x, y ) = 0
is solved by
x, y ∈ R (β ≥ 1)
u ε (t, x, y ) = t cos 2π x−y
.
ε
Example 2
utε + |uxε + uyε |β = 0
x, y ∈ R
u ε (0, x, y ) = cos 2π x−y
ε
is solved by
u ε (t, x, y ) = cos 2π x−y
.
ε
In both cases
u ε has no limit as ε → 0 if x 6= y .
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
8 / 29
Example 3
utε + |uxε | − |uyε | = cos 2π x−y
ε
u ε (0, x, y ) = 0
is solved by
x, y ∈ R
u ε (t, x, y ) = t cos 2π x−y
.
ε
Example 4
utε + |uxε | − |uyε | = 0 x, y ∈ R
u ε (0, x, y ) = 2π x−y
ε
is solved by
Again
u ε (t, x, y ) = cos 2π x−y
.
ε
u ε has no limit as ε → 0 if x 6= y .
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
9 / 29
H vanishing in a direction
The previous examples are special cases of
Proposition
Assume ∃ p0 ∈ ZN \ {0} such that
z ,
utε + H(Dz u ε ) = l
ε
H(λp0 ) = 0 ∀ λ ∈ R in
z u ε (0, z) = h
ε
If h = 0 ∃ smooth ZN −periodic l such that limε→0 u ε @
if l = 0 ∃ smooth ZN −periodic h such that limε→0 u ε @
The 1st statement is SHARP if H is 1-homogeneous and convex:
∀ p0 ∈ ZN \ {0} H(p0 ) 6= 0 is nonresonance condition
⇒ homogenization!
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
10 / 29
H vanishing in a direction
The previous examples are special cases of
Proposition
Assume ∃ p0 ∈ ZN \ {0} such that
z ,
utε + H(Dz u ε ) = l
ε
H(λp0 ) = 0 ∀ λ ∈ R in
z u ε (0, z) = h
ε
If h = 0 ∃ smooth ZN −periodic l such that limε→0 u ε @
if l = 0 ∃ smooth ZN −periodic h such that limε→0 u ε @
The 1st statement is SHARP if H is 1-homogeneous and convex:
∀ p0 ∈ ZN \ {0} H(p0 ) 6= 0 is nonresonance condition
⇒ homogenization!
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
10 / 29
Homogenization on subspaces
If H satisfies the assumption of the negative Proposition, must restrict
the data l, h to oscillate only in directions where H does not vanish.
Let V M–dimensional subspace of RN , zV := ΠV (z) projection
z
z z V
V
V
utε + H
, Dz u ε = l z,
, u ε (0, z) = h z,
ε
ε
ε
N
Assume coercivity in V : ∀ p̄ ∈ R
lim
|p|→+∞, p∈V
H(θ, p + p̄) = +∞,
uniformly in θ ∈ V .
Theorem
∃ H̄(z, p) continuous such that, ∀h(z, θ) periodic in θ, u ε converges
uniformly on compacta of (0, +∞) × RN to u, unique solution of
ut + H̄(z, Du) = 0,
Martino Bardi (Università di Padova)
u(0, z) = min h(z, θ)
Multi-scale Bellman-Isaacs
θ∈V
Roma, June 15th, 2011
11 / 29
Homogenization on subspaces
If H satisfies the assumption of the negative Proposition, must restrict
the data l, h to oscillate only in directions where H does not vanish.
Let V M–dimensional subspace of RN , zV := ΠV (z) projection
z
z z V
V
V
utε + H
, Dz u ε = l z,
, u ε (0, z) = h z,
ε
ε
ε
N
Assume coercivity in V : ∀ p̄ ∈ R
lim
|p|→+∞, p∈V
H(θ, p + p̄) = +∞,
uniformly in θ ∈ V .
Theorem
∃ H̄(z, p) continuous such that, ∀h(z, θ) periodic in θ, u ε converges
uniformly on compacta of (0, +∞) × RN to u, unique solution of
ut + H̄(z, Du) = 0,
Martino Bardi (Università di Padova)
u(0, z) = min h(z, θ)
Multi-scale Bellman-Isaacs
θ∈V
Roma, June 15th, 2011
11 / 29
Related results: Barles 2007, Viterbo 2008 (by symplectic methods).
Example 1
utε + |uxε + γuyε | = l x, y , x−y
ε
u ε (0, x, y ) = h x, y , x−y
ε
x, y ∈ RN/2
there is homogenization ∀ l, h periodic in θ =
x−y
ε
⇐⇒
γ 6= 1.
γ = 1 by the Examples 1 and 2 of non-homogenization;
γ 6= 1 fits in the Thm. with V = {(q, −q) : q ∈ RN/2 }.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
12 / 29
Related results: Barles 2007, Viterbo 2008 (by symplectic methods).
Example 1
utε + |uxε + γuyε | = l x, y , x−y
ε
u ε (0, x, y ) = h x, y , x−y
ε
x, y ∈ RN/2
there is homogenization ∀ l, h periodic in θ =
x−y
ε
⇐⇒
γ 6= 1.
γ = 1 by the Examples 1 and 2 of non-homogenization;
γ 6= 1 fits in the Thm. with V = {(q, −q) : q ∈ RN/2 }.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
12 / 29
Related results: Barles 2007, Viterbo 2008 (by symplectic methods).
Example 1
utε + |uxε + γuyε | = l x, y , x−y
ε
u ε (0, x, y ) = h x, y , x−y
ε
x, y ∈ RN/2
there is homogenization ∀ l, h periodic in θ =
x−y
ε
⇐⇒
γ 6= 1.
γ = 1 by the Examples 1 and 2 of non-homogenization;
γ 6= 1 fits in the Thm. with V = {(q, −q) : q ∈ RN/2 }.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
12 / 29
Convex-concave eikonal equation
|Dx u ε | − g2 x, y , x−y
|Dy u ε | = l(x, y , x−y
utε + g1 x, y , x−y
ε
ε
ε )
u ε (0, x, y ) = h(x, y , x−y
ε )
there is homogenization if
(g1 − g2 )(x, y , θ) ≥ γo > 0 ∀ x, y , θ,
or, symmetrically, g2 − g1 ≥ γo .
Proof: by the Thm. with V = {(q, −q) : q ∈ RN/2 }.
Motivation: Pursuit-Evasion type differential games, with pursuer
controlling the x variables, evader the y variables, and cost depending
on the distance between them.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
13 / 29
The result is sharp in the next
Example 2
utε + |uxε | − γ|uyε | = l x, y , x−y
ε
there is homogenization ∀ l, h periodic in θ =
x, y ∈ RN/2
x−y
ε
⇐⇒
γ 6= 1.
γ = 1 by the Examples 3 and 4 of non-homogenization;
γ 6= 1 fits in the preceding slide.
Other results
by control-game methods we can treat problems with different
assumptions: instead of strongly interacting fast variables,
decoupling of xε and yε , e.g., h or l has a saddle point in this
entries;
many problems on homogenization of convex-concave H-J
equations remain open!
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
14 / 29
The result is sharp in the next
Example 2
utε + |uxε | − γ|uyε | = l x, y , x−y
ε
there is homogenization ∀ l, h periodic in θ =
x, y ∈ RN/2
x−y
ε
⇐⇒
γ 6= 1.
γ = 1 by the Examples 3 and 4 of non-homogenization;
γ 6= 1 fits in the preceding slide.
Other results
by control-game methods we can treat problems with different
assumptions: instead of strongly interacting fast variables,
decoupling of xε and yε , e.g., h or l has a saddle point in this
entries;
many problems on homogenization of convex-concave H-J
equations remain open!
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
14 / 29
The result is sharp in the next
Example 2
utε + |uxε | − γ|uyε | = l x, y , x−y
ε
there is homogenization ∀ l, h periodic in θ =
x, y ∈ RN/2
x−y
ε
⇐⇒
γ 6= 1.
γ = 1 by the Examples 3 and 4 of non-homogenization;
γ 6= 1 fits in the preceding slide.
Other results
by control-game methods we can treat problems with different
assumptions: instead of strongly interacting fast variables,
decoupling of xε and yε , e.g., h or l has a saddle point in this
entries;
many problems on homogenization of convex-concave H-J
equations remain open!
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
14 / 29
2. Optimal control with stochastic volatility
In Financial Mathematics the evolution of a stock S is described by
d log Ss = γ ds + σ dWs
and the classical Black-Scholes formula for the option pricing problem
is derived assuming the parameters are constants.
However the volatility σ is not really a constant, it rather looks like an
ergodic mean-reverting stochastic process, so it is often modeled as
σ(ys ) with ys an Ornstein-Uhlenbeck diffusion process.
It is argued in the book
Fouque, Papanicolaou, Sircar: Derivatives in financial markets with
stochastic volatility, 2000,
that the process ys also evolves on a faster time scale than the stock
prices. Next picture shows the typical bursty behavior.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
15 / 29
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
16 / 29
The model with fast stochastic volatility σ proposed in [FPS] is
d log Ss = γ ds + σ(ys ) dWs
dys = 1ε (m − ys ) +
√ν d W̃s
ε
For the option pricing problem the limit is given by the Black-Scholes
formula of the model with (constant) mean historical volatility
d log Ss = γ ds + σ dWs ,
2
2
e−(y −m) /2ν
σ =
σ (y ) √
dy ,
2πν 2
R
2
Z
2
the mean being w.r.t. the (Gaussian) invariant measure of the
Ornstein-Uhlenbeck process driving ys .
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
17 / 29
Merton portfolio optimization problem
Invest βs in the stock Ss , 1 − βs in a bond with interest rate r .
Then the wealth xs evolves as
d xs = (r + (γ − r )βs )xs ds + xs βs σ dWs
and want to maximize the expected utility at time t,
for some h increasing and concave.
E[h(xt )],
If h(x) = ax δ /δ with a > 0, 0 < δ < 1, a HARA function, and the
parameters are constants the problem has an explicit solution.
The version with fast stochastic volatility is
d xs = (r + (γ − r )βs )xs ds + xs βs σ(ys ) dWs
dys = 1ε (m − ys ) +
√ν d W̃s
ε
QUESTIONS:
1. Is the limit as ε → 0 a Merton problem with constant volatility σ?
2. If so, is σ an average of σ(·) ?
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
18 / 29
Merton portfolio optimization problem
Invest βs in the stock Ss , 1 − βs in a bond with interest rate r .
Then the wealth xs evolves as
d xs = (r + (γ − r )βs )xs ds + xs βs σ dWs
and want to maximize the expected utility at time t,
for some h increasing and concave.
E[h(xt )],
If h(x) = ax δ /δ with a > 0, 0 < δ < 1, a HARA function, and the
parameters are constants the problem has an explicit solution.
The version with fast stochastic volatility is
d xs = (r + (γ − r )βs )xs ds + xs βs σ(ys ) dWs
dys = 1ε (m − ys ) +
√ν d W̃s
ε
QUESTIONS:
1. Is the limit as ε → 0 a Merton problem with constant volatility σ?
2. If so, is σ an average of σ(·) ?
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
18 / 29
By formal asymptotic expansions [FPS] argue that the answers are
1. Yes, the limit is a Merton problem with constant volatility σ
2. σ is a harmonic average of σ
Z
−1/2
1
,
σ :=
dµ(y )
σ 2 (y )
NOT the linear average as in Black-Scholes!
Practical consequence:
if I have N empirical data σ1 , ..., σN of the volatility, in the Black-Scholes
formula for option pricing I estimate the volatility by the arithmetic mean
σa2 =
N
1X 2
σi ,
N
i=1
whereas in the Merton problem I use the harmonic mean
!−1
N
1X 1
2
σh =
≤ σa2 .
2
N
σ
i=1 i
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
19 / 29
By formal asymptotic expansions [FPS] argue that the answers are
1. Yes, the limit is a Merton problem with constant volatility σ
2. σ is a harmonic average of σ
Z
−1/2
1
,
σ :=
dµ(y )
σ 2 (y )
NOT the linear average as in Black-Scholes!
Practical consequence:
if I have N empirical data σ1 , ..., σN of the volatility, in the Black-Scholes
formula for option pricing I estimate the volatility by the arithmetic mean
σa2 =
N
1X 2
σi ,
N
i=1
whereas in the Merton problem I use the harmonic mean
!−1
N
1X 1
2
σh =
≤ σa2 .
2
N
σ
i=1 i
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
19 / 29
By formal asymptotic expansions [FPS] argue that the answers are
1. Yes, the limit is a Merton problem with constant volatility σ
2. σ is a harmonic average of σ
Z
−1/2
1
,
σ :=
dµ(y )
σ 2 (y )
NOT the linear average as in Black-Scholes!
Practical consequence:
if I have N empirical data σ1 , ..., σN of the volatility, in the Black-Scholes
formula for option pricing I estimate the volatility by the arithmetic mean
σa2 =
N
1X 2
σi ,
N
i=1
whereas in the Merton problem I use the harmonic mean
!−1
N
1X 1
2
σh =
≤ σa2 .
2
N
σ
i=1 i
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
19 / 29
Conclusion: the correct average seems to depend on the problem!
QUESTIONS:
Why?
Can the formula for the Merton problem be rigorously justified and
what is the convergence as ε → 0?
Is there a unified "formula" for the two problems and for other
similar problems?
Note: the Black-Scholes model has no control, the associated PDE is
a heat-type linear equation;
Merton is a stochastic control problem whose Hamilton-Jacobi-Bellman
equation is fully nonlinear and degenerate parabolic.
I’ll present a result on two-scale Hamilton-Jacobi-Bellman equations
that will answer these (and other) questions.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
20 / 29
The H-J approach to Singular Perturbations
Control system
dxs = f (xs , ys , αs ) ds + σ(xs , ys , αs ) dWs ,
dys = 1ε g(xs , ys ) ds +
x0 = x,
√1 ν(xs , ys ) dWs ,
ε
xs ∈ Rn , αs ∈ A,
ys ∈ Rm ,
y0 = y
Cost functional
J ε (t, x, y , α) :=
Rt
u ε (t, x, y ) :=
inf E [J ε (t, x, y , α)]
0
l(xs , ys , αs ) ds + h(xt , yt )
Value function
α∈A(t)
A(t) admisible controls on [0, t].
GOAL: let ε → 0 and find a simplified model.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
21 / 29
The H-J approach to Singular Perturbations
Control system
dxs = f (xs , ys , αs ) ds + σ(xs , ys , αs ) dWs ,
dys = 1ε g(xs , ys ) ds +
x0 = x,
√1 ν(xs , ys ) dWs ,
ε
xs ∈ Rn , αs ∈ A,
ys ∈ Rm ,
y0 = y
Cost functional
J ε (t, x, y , α) :=
Rt
u ε (t, x, y ) :=
inf E [J ε (t, x, y , α)]
0
l(xs , ys , αs ) ds + h(xt , yt )
Value function
α∈A(t)
A(t) admisible controls on [0, t].
GOAL: let ε → 0 and find a simplified model.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
21 / 29
Assumptions on data:
h continuous, supy h(x, y ) ≤ K (1 + |x|2 )
f , σ, g, τ, l Lipschitz in (x, y ) (unif. in α) with linear growth
Then u ε solves the HJB equation
∂u ε
1
ε
2 ε 1
2 ε
+ H x, y , Dx u , Dxx u , √ Dxy u − Lu ε = 0 in R+ × Rn × Rm ,
∂t
ε
ε
n
o
H (x, y , p, M, Z ) := max −tr(σσ T M) − f · p − l − tr(σνZ T )
a∈A
2
L := tr(νν T Dyy
) + g · Dy .
Theorem (Da Lio - Ley). The value function u ε is the unique viscosity
solution with growth u ε (t, x, y ) ≤ C(1 + |x|2 + |y |2 ) of the Cauchy
problem with initial condition
u ε (0, x, y ) = h(x, y ).
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
22 / 29
Next assume on the linear operator L
Ellipticity:
Lyapunov:
∃ Λ(y ) > 0 s.t. ∀ x
ν(x, y )ν T (x, y ) ≥ Λ(y )I
∃w ∈ C(Rm ), k > 0, R0 > 0 s.t.

 −Lw ≥ k for |y | > R0 , ∀x,

w(y ) → +∞
as |y | → +∞.
Then,
the fast subsystem rescaled by τ = s/ε and with x ∈ Rn frozen
(FS)
dyτ = g(x, yτ ) dτ + ν(x, yτ ) dWτ ,
is uniquely ergodic, i.e., has a unique invariant measure µx .
Liouville property:
any bounded subsolution of −Lv = 0 is constant.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
23 / 29
Next assume on the linear operator L
Ellipticity:
Lyapunov:
∃ Λ(y ) > 0 s.t. ∀ x
ν(x, y )ν T (x, y ) ≥ Λ(y )I
∃w ∈ C(Rm ), k > 0, R0 > 0 s.t.

 −Lw ≥ k for |y | > R0 , ∀x,

w(y ) → +∞
as |y | → +∞.
Then,
the fast subsystem rescaled by τ = s/ε and with x ∈ Rn frozen
(FS)
dyτ = g(x, yτ ) dτ + ν(x, yτ ) dWτ ,
is uniquely ergodic, i.e., has a unique invariant measure µx .
Liouville property:
any bounded subsolution of −Lv = 0 is constant.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
23 / 29
Next assume on the linear operator L
Ellipticity:
Lyapunov:
∃ Λ(y ) > 0 s.t. ∀ x
ν(x, y )ν T (x, y ) ≥ Λ(y )I
∃w ∈ C(Rm ), k > 0, R0 > 0 s.t.

 −Lw ≥ k for |y | > R0 , ∀x,

w(y ) → +∞
as |y | → +∞.
Then,
the fast subsystem rescaled by τ = s/ε and with x ∈ Rn frozen
(FS)
dyτ = g(x, yτ ) dτ + ν(x, yτ ) dWτ ,
is uniquely ergodic, i.e., has a unique invariant measure µx .
Liouville property:
any bounded subsolution of −Lv = 0 is constant.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
23 / 29
A typical sufficient condition: ellipticity + recurrence condition
i
h
lim sup sup g(x, y ) · y + tr(νν T (x, y )) < 0
|y |→+∞
x
Proof by choosing as Lyapunov function w(y ) = |y |2 .
Example: Ornstein-Uhlenbeck process
dyτ = (m(x) − yτ )dτ + ν(x)dWτ ,
where m, ν are bounded.
GOAL: let ε → 0 in the Cauchy problem for the HJB equation.
Similarity with homogenization problems: the "fast" variable y plays the
role of xε .
Main difference with periodic homogenization: the state variables ys
are unbounded and uncontrolled.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
24 / 29
A typical sufficient condition: ellipticity + recurrence condition
i
h
lim sup sup g(x, y ) · y + tr(νν T (x, y )) < 0
|y |→+∞
x
Proof by choosing as Lyapunov function w(y ) = |y |2 .
Example: Ornstein-Uhlenbeck process
dyτ = (m(x) − yτ )dτ + ν(x)dWτ ,
where m, ν are bounded.
GOAL: let ε → 0 in the Cauchy problem for the HJB equation.
Similarity with homogenization problems: the "fast" variable y plays the
role of xε .
Main difference with periodic homogenization: the state variables ys
are unbounded and uncontrolled.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
24 / 29
Convergence Theorem
Under the previous assumptions, the (weak) limit u(t, x) as ε → 0 of
the value function u ε (t, x, y ) solves

R
∂u
2
n

 ∂t + H x, y , Dx u, Dxx u, 0 dµx (y ) = 0 in R+ × R
(CP)

R

u(0, x) = h(x, y ) dµx (y )
If, moreover, either g, ν do not depend on x or
g(x, ·), ν(x, ·) ∈ C 1 and g(·, y ), ν(·, y ) ∈ Cb1
Dx g, Dy g, Dx ν, Dy ν are Hölder in y uniformly w.r.t. x,
then u is the unique viscosity solution (with quadratic growth) of (CP)
and u ε (t, x, y ) → u(t, x) locally uniformly on (0, +∞) × Rn .
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
25 / 29
Examples
Denote hφi :=
R
φ(y )dµx (y ). Got simple formulas for effective H and h
H(x, p, M) = hH(x, ·, p, M, 0)i,
h(x) = hh(x, ·)i.
Corollary [see also Kushner, book 1990]
For split systems, i.e.,
σ = σ(x, y ),
f = f0 (x, y ) + f1 (x, a),
l = l0 (x, y ) + l1 (x, a),
the linear averaging of the data is the correct limit, i.e.,
Z t
ε
lim u (t, x, y ) = u(t, x) := inf E
hli(xs , αs ) ds + hhi(xt ) ,
α.
ε→0
0
dxs = hf i(xs , αs ) ds + hσσ T i1/2 (xs ) dWs
Proof: H(x, p, M) = −trace(Mhσσ T i)/2 + maxA {−hf i · p − hli} .
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
26 / 29
Examples
Denote hφi :=
R
φ(y )dµx (y ). Got simple formulas for effective H and h
H(x, p, M) = hH(x, ·, p, M, 0)i,
h(x) = hh(x, ·)i.
Corollary [see also Kushner, book 1990]
For split systems, i.e.,
σ = σ(x, y ),
f = f0 (x, y ) + f1 (x, a),
l = l0 (x, y ) + l1 (x, a),
the linear averaging of the data is the correct limit, i.e.,
Z t
ε
lim u (t, x, y ) = u(t, x) := inf E
hli(xs , αs ) ds + hhi(xt ) ,
α.
ε→0
0
dxs = hf i(xs , αs ) ds + hσσ T i1/2 (xs ) dWs
Proof: H(x, p, M) = −trace(Mhσσ T i)/2 + maxA {−hf i · p − hli} .
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
26 / 29
Example: we recover the Black-Scholes formula with stochastic
volatility.
In general, for system or cost NOT split,
H(x, p, M) = h max{...}i > max h{...}i
A
A
and the limit control problem is not obvious.
We try to write H as a Bellman Hamiltonian in some other way in order
to find an explicit effective control problem approximating the singularly
perturbed one as ε → 0. We did it for
Merton problem with stochastic volatility (see next slides),
Ramsey model of optimal economic growth with (fast) random
parameters,
Vidale - Wolfe advertising model with random parameters,
advertising game in a duopoly with Lanchester dynamics and
random parameters.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
27 / 29
Merton portfolio optimization with stochastic volatility
Given a control βs the wealth xs evolves as
dxs = (r + (γ − r )βs )xs ds + xs βs σ(ys ) dWs
dys = 1ε (m − ys ) ds +
and the value functions is
√ν d W̃s
ε
x0 = x
y0 = y
V ε (t, x, y ) := supβ. E[h(xt )].
Let ρ = correlation of Ws and W̃s . The HJB equation is
∂V ε
b2 x 2 σ2 ε
bxρσν ε
− rxVxε − max (γ − r )bxVxε +
Vxx + √ Vxy
∂t
2
ε
b
ε
(m − y )Vyε + ν 2 Vyy
=
ε
Assume the utility h has h0 > 0 and h00 < 0. Then expect a value
ε < 0.
function strictly increasing and concave in x, i.e., Vxε > 0, Vxx
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
28 / 29
The HJB equation becomes
ε
√ V ε ]2
[(γ − r )Vxε + xρσν
(m − y )Vyε + ν 2 Vyy
xy
∂V ε
ε
− rxVxε +
=
ε
∂t
ε
σ 2 (y )2Vxx
By the Theorem, V ε (t, x, y ) → V (t, x) as ε → 0 and V solves
(γ − r )2 Vx2
∂V
− rxVx +
∂t
2Vxx
Z
1
σ 2 (y )
dµ(y ) = 0
in R+ × R+
This is the HJB equation of a Merton problem with the harmonic
average of σ as constant volatility
Z
σ :=
−1/2
1
dµ(y )
.
σ 2 (y )
Therefore this is the limit control problem.
The limit of the optimal control βsε,∗ as ε → 0 can also be studied.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
29 / 29
The HJB equation becomes
ε
√ V ε ]2
[(γ − r )Vxε + xρσν
(m − y )Vyε + ν 2 Vyy
xy
∂V ε
ε
− rxVxε +
=
ε
∂t
ε
σ 2 (y )2Vxx
By the Theorem, V ε (t, x, y ) → V (t, x) as ε → 0 and V solves
(γ − r )2 Vx2
∂V
− rxVx +
∂t
2Vxx
Z
1
σ 2 (y )
dµ(y ) = 0
in R+ × R+
This is the HJB equation of a Merton problem with the harmonic
average of σ as constant volatility
Z
σ :=
−1/2
1
dµ(y )
.
σ 2 (y )
Therefore this is the limit control problem.
The limit of the optimal control βsε,∗ as ε → 0 can also be studied.
Martino Bardi (Università di Padova)
Multi-scale Bellman-Isaacs
Roma, June 15th, 2011
29 / 29