New results on (F)BSDE of quadratic growth
U. Horst, Y. Hu, P. Imkeller, A. Réveillac, J. Zhang
HU Berlin, U Rennes
http://wws.mathematik.hu-berlin.de/∼imkeller
Warwick, December 1, 2011
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RESULTS ON
(F)BSDE
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OF QUADRATIC GROWTH
1 Cross hedging, optimal investment, exponential utility
for convex constraints: (N. El Karoui, R. Rouge ’00; J. Sekine ’02; J. Cvitanic,
J. Karatzas ’92, Kramkov, Schachermayer ’99, Mania, Schweizer ’05, Pham
’07, Zariphopoulou ’01,...)
maximal expected exponential utility from terminal wealth
V (x) = sup EU (x + XTπ + H) = sup E(− exp(−α(x +
π∈A
Z
π∈A
T
πs[dWs + θsds] + H)))
0
wealth on [0, T ] by investment strategy π :
Z
0
T
dSu
i=
hπu,
Su
Z
T
πu[dWu + θudu] = XTπ ,
0
H liability or derivative, correlated to financial market S
π ∈ A subject to π taking values in C closed
aim: use BSDE to represent optimal strategy π ∗
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2 Martingale optimality
Idea: Construct family of processes Q(π) such that
(π)
(form 1)
Q0
= constant,
(π)
QT
= − exp(−α(x + XTπ + H)),
Q(π)
supermartingale, π ∈ A,
∗
Q(π )
martingale, for (exactly) one π ∗ ∈ A.
Then
(π)
E(− exp(−α[x + XTπ + H])) = E(QT )
≤ E(Qπ0 )
=
(π ∗ )
E(Q0 )
= E(− exp(−α[x +
Hence π ∗ optimal strategy.
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(π ∗ )
XT
+ H])).
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3 Solution method based on BSDE
Introduction of BSDE into problem
Find generator f of BSDE
Z
T
Z
ZsdWs −
Yt = H −
t
T
f (s, Zs)ds,
YT = H,
t
such that with
(π)
Qt
= − exp(−α[x + Xtπ + Yt]),
t ∈ [0, T ],
we have
(π)
(form 2)
Q0
= − exp(−α(x + Y0)) = constant,
(fulfilled)
(π)
QT
= − exp(−α(x + XTπ + H))
(fulfilled)
Q(π)
supermartingale, π ∈ A,
∗
Q(π )
martingale, for (exactly) one π ∗ ∈ A.
This gives solution of valuation problem.
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4 Construction of generator of BSDE
How to determine f :
Suppose f generator of BSDE. Then by Ito’s formula
(π)
Qt
= − exp(−α[x + Xtπ + Yt])
Z t
α
(π)
(π)
2
(π
−
Z
)
]ds,
αQ(π)
[−π
θ
−
f
(s,
Z
)
+
= Q0 + Mt +
s
s
s
s
s
s
2
0
with a local martingale M (π).
Q(π) satisfies (form 2) iff for
α
q(·, π, z) = −f (·, z)−πθ + (π − z)2,
2
π ∈ A, z ∈ R,
we have
(form 3)
q(·, π, z) ≥ 0,
q(·, π ∗, z) = 0,
π ∈ A (supermartingale)
for (exactly) one π ∗ ∈ A (martingale).
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4 Construction of generator of BSDE
Now
α
q(·, π, z) = −f (·, z)−πθ + (π − z)2
2
α
1
1
= −f (·, z)+ (π − z)2 − (π − z) · θ + θ2 −zθ− θ2
2
2α
2α
α
1
1
= −f (·, z)+ [π − (z + θ)]2 −zθ − θ2.
2
α
2α
Under non-convex constraint p ∈ C:
1 2
1
2
[π − (z + θ)] ≥ dist (C, z + θ).
α
α
with equality for at least one possible choice of π ∗ due to closedness of C.
Hence (form 3) is solved by the choice (predictable selection)
(form 4)
1 2
f (·, z) = α2 dist2(C, z + α1 θ)−z · θ − 2α
θ (supermartingale)
π∗
: dist(C, z + α1 θ) = dist(π ∗, z + α1 θ) (martingale).
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5 Summary of results, exponential utility
Solve utility optimization problem
sup EU (x + XTπ + H)
π∈A
by considering FBSDE
dXtπ
= πt[dWt + θtdt],
X0π = x,
dYt = ZtdWt + f (t, Zt)dt,
YT = H
with generator as described before; determine π ∗ by previsible selection;
coupling through requirement of martingale optimality
sup EU (x +
π∈A
XTπ
+ H) = EU (x +
∗
U 0(x + Xtπ + Yt)
π∗
XT
+ H),
martingale.
for general U : forward part depends on π ∗, get fully coupled FBSDE
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6 Cross hedging, optimal investment, utility on R
Lit: Mania, Tevzadze (2003)
U : R → R strictly increasing and concave; maximal expected utility from
terminal wealth
(1) V (x) = sup EU (x + XTπ + H)
π∈A
wealth on [0, T ] by investment strategy π :
Z
0
T
dSu
i=
hπu,
Su
Z
T
πu[dWu + θudu] = XTπ ,
0
H liability or derivative, correlated to financial market S, W d−dimensional
Wiener process, W 1 first d1 components of W
π ∈ A subject to convex constraint π = (π 1, 0), π 1 d1−dimensional, hence
incomplete market
aim: use FBSDE system to describe optimal strategy π ∗
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7 Verification theorems
Thm 1
Assume U is three times differentiable, U 0 regular enough. If there exists π ∗
∗
solving (1), and Y is the predictable process for which U 0(X π + Y ) is square
d
integrable martingale, then with Z = dt
hY, W i
∗ 1
(π ) =
0
∗
1U
−θ 00 (X π
U
+ Y ) − Z 1.
Pf:
α = E(U
0
π∗
(XT
0 −1
+ H)|F·), Y = (U )
π∗
(α) − X .
Use Itô’s formula and martingale property. Find
Z
Y =H−
T
Z
ZsdWs −
·
with
T
π∗
f (s, Xs , Ys, Zs)ds,
·
1 U (3) π∗
∗
2
∗
(X
+
Y
)|π
+
Z
|
−
π
=−
s
s
s θs .
00
2U
Use variational maximum principle to derive formula for π ∗.
∗
f (s, Xsπ , Ys, Zs)
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7 Verification theorems
From preceding theorem derive the FBSDE system
Thm 2
π∗
Assumptions of Thm 1; then optimal wealth process X given as component
X of solution (X, Y, Z) of fully coupled FBSDE system
·
Z
X
= x−
0
Z
Y
= H−
0
U
(θs1 00 (Xs
U
+ Ys) +
Zs1)dWs1
Z
−
0
·
0
1U
(θs 00 (Xs
U
+ Ys) + Zs1)θs1ds,
T
ZsdWs
·
Z
−
·
T
0
(3) 02
1
U
U
U
1
1
·
θ
)
[|θs1|2((−
+
)(X
+
Y
)
+
Z
s
s
s
s
2 (U 00)3
U 00
1 2 2 U (3)
− |Zs |
(Xs + Ys)]ds.
00
2
U
(2)
Pf:
Use expression for f and formula for π ∗ from Thm 1.
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8 Representation of optimal strategy
Invert conclusion of Thm 2 to give representation of optimal strategy
Thm 3
Let (X, Y, Z) be solution of (2), U (XT + H) integrable, U 0(XT + H) square
integrable. Then
0
U
(π ∗)1 = −θ1 00 (X + Y ) + Z 1
U
is optimal solution of (1).
Pf:
By concavity for any admissible π
U (X π + Y ) − U (X + Y ) ≤ U 0(X + Y )(X π − X).
Now prove that
0
π
0
U (X + Y )(X − X) = U (X
π∗
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π
π∗
+ Y )(X − X ) is a martingale!
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9 Utility function on R+
0
π∗
π∗
0
π∗
Replace U (X + Y ) with X U (X ) exp(Ỹ ). Then (X, Y, Z) satisfies (3) if
d
and only if (X, Ỹ , Z̃) satisfies (4) (Z̃ = dt
hW, Ỹ i):
Thm 4
Let (X, Ỹ , Z̃) be solution of the fully coupled FBSDE
·
U
U0
1
1
1
= x − ( 00 (Xs)(θs + Z̃s )dWs − ( 00 (Xs)(θs1 + Z̃s1)θs1ds,
0 U
0 U
Z T
U 0(XT + H)
= ln(
Z̃sdWs
)−
0
U (XT )
·
Z T
(3) 0
1
U
U
1
1 2
1
2
−
[|Z̃s + θs | ((1 −
)(X
)
−
|
Z̃
|
]ds. (4)
s
s
00 )2
2
(U
2
·
Z
X
Ỹ
·
0
π∗
U (XT
Z
π∗
0
π∗
such that
+ H) is integrable and X U (X ) exp(Ỹ ) is a true
martingale. Then
0
U
(π ∗)1 = − 00 (X)(Z 1 + θ1)
U
solves (1).
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10 The incomplete case
U (x) =
1 p
px
π∗
for x > 0, p < 1; (X , Ỹ , Z̃) solution of (4)
∗
∗
∗
then we saw that G := X π U 0(X π ) exp(Ỹ ) is a martingale; X π
π∗
supermartingale, hence U (X ) supermartingale; aim: use these two facts to
construct solution by iteration
Notation: Z̃tdWt := Z̃t1dWt1, dNt := Z̃t2dWt2; wlog: θ = 0 (otherwise measure
change, drift θ)
Initialization: Set Z̃ 0 := 0 and
1
0
X := xE
Z̃ ∗ W = x,
1−p
1
G1T
XT1
1 p
p
:=
≤
C|X
|
=
Cx
.
T
1
1−p
(XT + H)
Obtain (Z 1, N 1) ∈ H2 via the following consequence of martingale
representation
G1t = E[G1T |Ft] = G1T −
Z
t
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T
G1s
1
Z̃s1dWs + dNs1
1−p
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10 The incomplete case
n+1-st iteration
Z n and N n already obtained, define
X
n+1
1
n
Z̃ ∗ W ,
:= xE
1−p
Gn+1
T
XTn+1
n+1 p
:=
≤
C|X
|
T
(XTn+1 + H)1−p
p
with EGn+1
≤
Cx
< ∞.
T
Obtain (Z n+1, N n+1) via the following consequence of martingale
representation
n+1
Gn+1
= E[Gn+1
|F
]
=
G
−
t
t
T
T
• we obtain sequence
R
Z
T
Gn+1
s
t
1
Z̃sn+1dWs + dNsn+1
1−p
Z̃sndWs, N n n∈N
• convergence (possibly along subsequence)?
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10 The incomplete case
Thm 5
For n ∈ N we have
" Z
#
T
n
1
1
1
−
p
2
n
n
p
E
Z̃ ds + hN iT ≤
2 log C + log x
2 0 1−p s
1−p
" Z
#
T
1
1
pn−1
2
E
Z̃s1 ds + hN 1iT
+
2
2 0 1−p
< ∞.
=⇒
R
Z̃sndWs, N n n∈N
bounded in H2
=⇒ Delbaen & Schachermayer (A compactness principle for bounded
sequences of martingales with applications, 1996)
∃(Z̃, N ) such that Z̃ n → Z̃ and N n → N
Finally, obtain FBSDE solution (X, Ỹ , Z̃, N ) by using (Z̃, N ) for solving for X
and Ỹ
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