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Forward-backward systems for expected utility maximization
Ulrich Horst, Ying Hu, Peter Imkeller, Anthony Réveillac, Jianing Zhang
To cite this version:
Ulrich Horst, Ying Hu, Peter Imkeller, Anthony Réveillac, Jianing Zhang. Forward-backward systems
for expected utility maximization. Stochastic Processes and their Applications, Elsevier, 2014, 124
(5), pp.1813-1848. �10.1016/j.spa.2014.01.004�. �hal-00631727�
Forward-backward systems for expected utility maximization
Ulrich Horst
∗, Ying Hu
†, Peter Imkeller
‡, Anthony R´ eveillac
§and Jianing Zhang
¶October 12, 2011
Abstract
In this paper we deal with the utility maximization problem with a general utility function. We derive a new approach in which we reduce the utility maximization prob- lem with general utility to the study of a fully-coupled Forward-Backward Stochastic Differential Equation (FBSDE).
AMS Subject Classification: Primary 60H10, 93E20 JEL Classification: C61, D52, D53
1 Introduction
One of the most commonly studied topic in mathematical finance (and applied probably) is the problem of maximizing expected terminal utility from trading in a financial market. In such a situation, the stochastic control problem is of the form
V (0, x) := sup
π∈A
E [U (X
Tπ+ H)] (1.1)
for a real-valued function U , where A denotes the set of admissible trading strategies, T < ∞ is the terminal time, X
Tπis the wealth of the agent when he follows the strategy π ∈ A and his initial capital at the initial time zero is x > 0, and H is a liability that the agent must deliver at the terminal time. One is typically interested in establishing existence and uniqueness of optimal solutions and in characterizing optimal strategies and the value function V (t, x) which is defined as
V (t, x) := sup
π∈A
E[U (X
t,Tπ+ H)|F
t].
∗Institut f¨ur Mathematik, Humboldt-Universit¨at zu Berlin, Unter den Linden 6, 10099 Berlin, Germany, horst@mathematik.hu-berlin.de
†Universit´e de Rennes 1, campus Beaulieu, 35042 Rennes cedex, France,ying.hu@univ-rennes1.fr
‡Institut f¨ur Mathematik, Humboldt-Universit¨at zu Berlin, Unter den Linden 6, 10099 Berlin, Germany, imkeller@mathematik.hu-berlin.de
§CEREMADE UMR CNRS 7534, Universit´e Paris Dauphine, Place du Mar´echal De Lattre De Tassigny, 75775 PARIS CEDEX 16, France,anthony.reveillac@cremade.dauphine.fr
¶Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany, jianing.zhang@wias-berlin.de
Here X
t,Tdenotes the wealth of the agent when the investment period is [t, T ] and where the filtration (F
t)
t∈[0,T]defines the flow of information.
The question of existence of an optimal strategy π
∗can essentially be addressed using convex duality. The convex duality approach is originally due to Bismut [2] with its modern form dating back to Kramkov and Schachermayer [13]. For instance, given some growth condition on U or related quantities (such as the asymptotic elasticity condition for utilities defined on the half line) existence of an optimal strategy is guaranteed under mild regularity conditions on the liability and convexity assumptions on the set of admissible trading strategies (see e.g. [1] for details). However, the duality method is not constructive and does not allow for a characterization of optimal strategies and value functions.
One approach to simultaneously characterize optimal trading strategies and utilities uses the theory of forward-backward stochastic differential equations (FBSDE). When the filtration is generated by a standard Wiener process W and if either U (x) := − exp(−αx) for some α > 0 and H ∈ L
2, or U (x) :=
xγγfor γ ∈ (0, 1) or U (x) = ln x and H = 0, it has been shown by Hu, Imkeller and M¨ uller [9] that the control problem (1.1) can essentially be reduced to solving a BSDE of the form
Y
t= H − Z
Tt
Z
sdW
s− Z
Tt
f(s, Z
s)ds, t ∈ [0, T ], (1.2) where the driver f (t, z) is a predictable process of quadratic growth in the z-variable. Their results have since been extended beyond the Brownian framework and to more general utility optimization problems with complete and incomplete information in, e.g., [8], [19], [20], [21]
and [17]. The method used in [9] and essentially all other papers relies on the martingale optimality principle and can essentially only be applied to the standard cases mentioned above (exponential with general endowment and power, respectively logarithmic, with zero endowment). This is due to a particular “separation of variables” property enjoyed by the classical utility functions: their value function can be decomposed as V (t, x) = g(x)V
twhere g is a deterministic function and V is an adapted process. As a result, optimal future trading strategies are independent of current wealth levels.
More generally, there has recently been an increasing interest in dynamic translation in- variant utility functions. A utility function is called translation invariant if a cash amount added to a financial position increases the utility by that amount and hence optimal trad- ing strategies are wealth-independent
1. Although the property of translation invariance renders the utility optimization problem mathematically tractable, independence of the trading strategies on wealth is rather unsatisfactory from an economic point of view. In [18] the authors derive a verification theorem for optimal trading strategies for more gen- eral utility functions when H = 0. More precisely, given a general utility function U and assuming that there exists an optimal strategy regular enough such that the value function enjoys some regularity properties in (t, x), it is shown that there exists a predictable random field (ϕ(t, x))
(t,x)∈[0,T]×(0,∞)such that the pair (V, ϕ) is solution to the following backward
1It has been shown by [6] that essentially all such utility functions can be represented in terms of a BSDE of the form 1.2.
stochastic partial differential equation (BSPDE) of the form:
V (t, x) = U (x) − Z
Tt
ϕ(s, x)dW
s− Z
Tt
|ϕ
x(s, x)|
2V
xx(s, x) ds, t ∈ [0, T ] (1.3) where ϕ
xdenotes the partial derivative of ϕ with respect to x and V
xxthe second partial derivative of V with respect to the same variable. The optimal strategy π
∗can then be obtained from (V, ϕ). Unfortunately, the BSPDE-theory is still in its infancy and to the best of our knowledge the non-linearities arising in (1.3) cannot be handled except in the classical cases mentioned above where once again one benefits of the “separation of variables”
(see [11]). Moreover, the utility function U only appears in the terminal condition which is not very handy. In that sense this is exactly the same situation as the Hamilton-Jacobi- Bellman equation where U only appears as a terminal condition but not in the equation itself.
In this paper we propose a new approach to solving the optimization problem (1.1) for a larger class of utility function and characterize the optimal strategy π
∗in terms of a fully-coupled FBSDE-system. The optimal strategy is then a function of the current wealth and of the solution to the backward component of the system. In addition, the driver of the backward part is given in terms of the utility function and its derivatives. This adds enough structure to the optimization problem to deal with fairly general utilities functions, at least when the market is complete. We also derive the FBSDE system for the power case with general (non-hedgeable) liabilities; to the best of our knowledge we are the first to characterize optimal strategies for power utilities with general liabilities. Finally, we link our approach to the well established approaches using convex dual theory and stochastic maximum principles.
The remainder of this paper is organized as follows. In Section 2 we introduce our financial market model. In Section 3 we first derive a verification theorem in terms of a FBSDE for utilities defined on the real line along with a converse result, that is, we show that a solution to the FBSDE allows to construct the optimal strategy. Section 4 is devoted to the same question but for utilities defined on the positive half line. In Section 5 we relate our approach to the stochastic maximum principle obtained by Peng [22] and the standard duality approach. We use the duality-BSDE link to show that the FBSDE associated with the problem of maximizing power utility with general positive endowment has a solution.
2 Preliminaries
We consider a financial market which consists of one bond S
0with interest rate zero and of d ≥ 1 stocks given by
d S ˜
ti:= ˜ S
tidW
ti+ ˜ S
itθ
tidt, i ∈ {1, . . . , d}
where W is a standard Brownian motion on R
ddefined on a filtered probability space
(Ω, F, (F
t)
t∈[0,T], P ), (F
t)
t∈[0,T]is the filtration generated by W , and θ := (θ
1, . . . , θ
d) is
a predictable bounded process with values in R
d. Since we assume the process θ to be
bounded, Girsanov’s theorem implies that the set of equivalent local martingale measures
(i.e. probability measures under which ˜ S is a local martingale) is not empty, and thus according to the classical literature (see e.g. [7]), arbitrage opportunities are excluded in our model. For simplicity throughout we write
dS
ti:= d S ˜
tiS ˜
ti.
We denote by α ·β the inner product in R
dof vectors α and β and by | · | the usual associated L
2-norm on R
d. In all the paper C will denote a generic constant which can differ from line to line. We also define the following spaces:
S
2( R
d) :=
(
β : Ω × [0, T ] → R
d, predictable, E [ sup
t∈[0,T]
|β
t|
2] < ∞ )
,
H
2( R
d) :=
β : Ω × [0, T ] → R
d, predictable, E Z
T0
|β
t|
2dt
< ∞
.
Since the market price of risk θ is assumed to be bounded, the stochastic process E(−θ · W )
t:= exp
− Z
t0
θ
sdW
s− 1 2
Z
t 0|θ
s|
2ds
has finite moments of order p for any p > 0. We assume d
1+ d
2= d and that the agent can invest in the assets ˜ S
1, . . . , S ˜
d1while the stocks ˜ S
d1+1, . . . , S ˜
d2cannot be in- vested into. Denote S
H:= (S
1, . . . , S
d1, 0 . . . , 0), W
H:= (W
1, . . . , W
d1, 0 . . . , 0), W
O:=
(0, . . . , 0, W
d1+1, . . . , W
d2), and θ
H:= (θ
1, . . . , θ
d1, 0 . . . , 0) (the notation H refers to “hedge- able” and O to “orthogonal”). We define the set Π
xof admissible strategies with initial capital x > 0 as
Π
x:=
π : Ω × [0, T ] → R
d1, E Z
T0
|π
t|
2dt
< ∞, π is self-financing
(2.1) where for π in Π
xthe associated wealth process X
πis defined as
X
tπ:= x + Z
t0
π
rdS
rH= x +
d1
X
i=1
Z
t 0π
ridS
ri, t ∈ [0, T ].
Every π in Π
xis extended to an R
d-valued process by
˜
π := (π
1, . . . , π
d1, 0, . . . , 0).
In the following, we will always write π in place of ˜ π, i.e. π is an R
d-valued process where the last d
2components are zero. Moreover, we consider a utility function U : I → R where I is an interval of R such that U is strictly increasing and strictly concave. We seek for a strategy π
∗in Π
xsatisfying E [U (X
Tπ∗+ H)] < ∞ such that
π
∗= argmax
π∈Πx,E[|U(XTπ+H)|]<∞{ E [U (X
Tπ+ H)]} (2.2) where H is a random variable in L
2(Ω, F
T, P ) such that the expression above makes sense.
We concretize on sufficient conditions in the subsequent sections.
3 Utilities defined on the real line
In this section we consider a utility function U : R → R defined on the whole real line. We assume that U is strictly increasing and strictly concave and that the agent is endowed with a claim H ∈ L
2(Ω, F
T, P ). We introduce the following conditions.
(H1) U : R → R is three times differentiable
(H2) We say that condition (H2) holds for an element π
∗in Π
x, if E [|U
0(X
Tπ∗+ H)|
2] < ∞ and if for every bounded predictable process h : [0, T ] → R , the family of random variables
Z
T 0h
rdS
rHZ
10
U
0X
Tπ∗+ H + εr Z
T0
h
rdS
rHdr
ε∈(0,1)
is uniformly integrable.
Before presenting the first main result of this section, we prove that condition (H2) is satisfied for every strategy π
∗such that E [|U
0(X
Tπ∗+ H)|] < ∞ when one has an exponential growth condition on the marginal utility of the form:
U
0(x + y) ≤ C 1 + U
0(x)
(1 + exp(αy)) for some α ∈ R . Indeed, let G := R
T0
h
rdS
rHand d > 0. We will show that the quantity q(d) := sup
ε∈(0,1)
E
G Z
10
U
0(X
Tπ∗+ H + εrG)dr
1 |
GR10 U0(XTπ∗+H+εrG)dr
|
>dvanishes when d goes to infinity. For simplicity we write δ
ε,d:= 1 |
GR01U0(XTπ∗+H+εrG)dr|
>d. By the Cauchy-Schwarz inequality
q(d) ≤ sup
ε∈(0,1)
E
(1 + U
0(X
Tπ∗+ H))
G(1 + Z
10
exp(αεrG))dr
δ
ε,d≤ C E
h |U
0(X
Tπ∗+ H)|
2i
1/2sup
ε∈(0,1)
E
"
G Z
10
exp(αεrG)dr
2
δ
ε,d#
1/2. Since E
|U
0(X
Tπ∗+ H)|
2is assumed to be finite we deduce from the inequality exp(αζx) ≤ 1 + exp(αx) for all x ∈ R , 0 < ζ < 1
that
q(d) ≤ C sup
ε∈(0,1)
E h
|G(2 + exp(αG))|
2δ
ε,di
1/2.
Applying successively the Cauchy-Schwarz inequality and the Markov inequality, it holds that
q(d) ≤ C E
h |G(2 + exp(αG))|
4i
1/4sup
ε∈(0,1)
E [δ
ε,d]
1/4≤ CE h
|G(2 + exp(αG))|
4i
1/4d
−1/4sup
ε∈(0,1)
E
|G|
Z
1 0U
0(X
Tπ∗+ H + εrG)dr
1/4≤ C E
h |G(2 + exp(αG))|
4i
1/4d
−1/4E
|G(2 + exp(αG))|
21/8. Let p ≥ 2. Since h and θ are bounded it is clear that E
|G|
2p< ∞ and E [|G(2 + exp(αG))|
p]
≤ E
|G|
2p1/2E h
|2 + exp(αG)|
2pi
1/2≤ C 2 + E
h |exp(αG)|
2pi
1/2= C
2 + E
exp Z
T0
2pαh
rdW
rH− 1 2
Z
T 0|2pαh
r|
2dr
exp 1
2 Z
T0
|2pαh
r|
2+ 2pαh
r· θ
rdr
1/2≤ C.
Hence lim
d→∞q(d) = 0 which proves the assertion.
3.1 Characterization and verification: incomplete markets
We are now ready to state and prove the first main result of this paper: a verification theorem for optimal trading strategies.
Theorem 3.1. Assume that (H1) holds. Let π
∗∈ Π
xbe an optimal solution to the problem (2.2) which satisfies assumption (H2). Then there exists a predictable process Y with Y
T= H such that U
0(X
π∗+ Y ) is a martingale in L
2(Ω, F
T, P ) and
π
∗ti= −θ
tiU
0(X
tπ∗+ Y
t)
U
00(X
tπ∗+ Y
t) − Z
ti, t ∈ [0, T ], i = 1, . . . , d
1where Z
t:=
dhY,Widt t:=
dhY,Wiit
dt
, . . . ,
dhY,Wdtdit.
Proof. We first prove the existence of Y . Since E [|U
0(X
Tπ∗+H)|
2] < ∞, the stochastic process α defined as α
t:= E [U
0(X
Tπ∗+ H)|F
t], for t in [0, T ] is a square integrable martingale. Define Y
t:= (U
0)
−1(α
t) − X
tπ∗. Then Y is (F
t)
t∈[0,T]-predictable. Now Itˆ o’s formula yields
Y
t+ X
tπ∗= Y
T+ X
Tπ∗− Z
Tt
1
U
00(U
0−1(α
s)) dα
s+ 1 2
Z
T tU
(3)(U
0−1(α
s))
(U
00(U
0−1(α
s)))
3dhα, αi
s. (3.1) By definition, α is the unique solution of the zero driver BSDE
α
t= U
0(X
Tπ∗+ Y
T) − Z
Tt
β
sdW
s, t ∈ [0, T ], (3.2)
where β is a square integrable predictable process with valued in R
d. Plugging (3.2) into (3.1) yields
Y
t+ X
tπ∗=X
Tπ∗+ H − Z
Tt
1
U
00(X
sπ∗+ Y
s)) β
sdW
s+ 1 2
Z
T tU
(3)(X
sπ∗+ Y
s)
(U
00(X
sπ∗+ Y
s))
3|β
s|
2ds.
Setting ˜ Z :=
1U00(Xπ∗+Y))
β , we have Y
t+ X
tπ∗=X
Tπ∗+ H −
Z
Tt
Z ˜
sdW
s+ 1 2
Z
Tt
U
(3)U
00(X
sπ∗+ Y
s)| Z ˜
s|
2ds.
Now by putting Z
i:= ˜ Z
i− π
∗i, i = 1, . . . , d, we have shown that Y is a solution to the BSDE
Y
t= H − Z
Tt
Z
sdW
s− Z
Tt
f (s, X
sπ∗, Y
s, Z
s)ds, t ∈ [0, T ], (3.3) where f is given by
f (s, X
sπ∗, Y
s, Z
s) := − 1 2
U
(3)U
00(X
sπ∗+ Y
s)|π
s∗+ Z
s|
2− π
∗s· θ
s. (3.4) Finally, by construction we have U
0(X
tπ∗+ Y
t) = α
t, thus it is a martingale.
Now we deal with the characterization of the optimal strategy. To this end, let h : [0, T ] → R
d1be a bounded predictable process. We extend h into R
dby setting ˜ h :=
(h
1, . . . , h
d1, 0, . . . , 0) and use the convention that ˜ h is again denoted by h. Thus for every ε in (0, 1) the perturbed strategy π
∗+ εh belongs to Π
x. Since π
∗is optimal it is clear that for every such h it holds that
l(h) := lim
ε→0
1 ε E
U (x +
Z
T 0(π
r∗+ εh
r)dS
rH+ Y
T) − U (x + Z
T0
π
∗rdS
rH+ Y
T)
≤ 0. (3.5) Moreover we have
1 ε
U (x +
Z
T 0(π
r∗+ εh
r)dS
rH+ Y
T) − U (x + Z
T0
π
∗rdS
rH+ Y
T)
= Z
T0
h
rdS
rHZ
10
U
0X
Tπ∗+ Y
T+ θε Z
T0
h
rdS
rHdθ.
Now using (H2), Lebesgue’s dominated convergence theorem implies that (3.5) can be rewrit- ten as
E
U
0(X
Tπ∗+ Y
T) Z
T0
h
rdS
rH≤ 0 (3.6)
for every bounded predictable process h. Applying integration by parts to U
0(X
sπ∗+Y
s)
s∈[0,T]and R
s0
h
rdS
rHs∈[0,T]
, we get U
0(X
Tπ∗+ Y
T)
Z
T 0h
rdS
rH= U
0(x + Y
0) × 0 + Z
T0
U
0(X
sπ∗+ Y
s)h
sdS
sH+
Z
T 0Z
s 0h
rdS
rHU
00(X
sπ∗+ Y
s) h
(π
s∗+ Z
s)dW
sH+ (π
s∗· θ
s+ f (s, X
sπ∗, Y
s, Z
s))ds i + 1
2 Z
T0
Z
s 0h
rdS
rHU
(3)(X
sπ∗+ Y
s)|π
∗s+ Z
s|
2ds +
Z
T 0U
00(X
sπ∗+ Y
s)h
s· (π
s∗+ Z
s)ds.
By definition of the driver f , the previous expression reduces to U
0(X
Tπ∗+ Y
T)
Z
T 0h
rdS
rH= Z
T0
U
0(X
sπ∗+ Y
s)θ
s+ U
00(X
sπ∗+ Y
s)(π
s∗+ Z
s)
· h
sds +
Z
T 0Z
s 0h
rdS
rHU
00(X
sπ∗+ Y
s)(π
s∗+ Z
s)dW
sH+ Z
T0
U
0(X
sπ∗+ Y
s)h
sdW
sH. (3.7) The next step would be to apply the conditional expectations in (3.7), however the two terms on the second line of the right hand side are a priori only local martingales. We start by showing that the first one is a uniformly integrable martingale. Indeed, from the computations which have led to (3.3) we have that
U
00(X
π∗+ Y )(π
∗+ Z) = β,
where we recall that β is the square integrable process appearing in (3.2). Using the BDG inequality we get
E
"
sup
s∈[0,T]
Z
s 0Z
r 0h
udS
uHU
00(X
rπ∗+ Y
r)(π
r∗+ Z
r)dW
rH#
≤ C E
Z
T0
Z
s0
h
rdS
rH2
|β
s|
2ds
1/2
≤ C E
sup
s∈[0,T]
Z
s 0h
rdS
rH2
!
1/2Z
T 0|β
s|
2ds
1/2
. Young’s inequality furthermore yields
E
sup
s∈[0,T]
Z
s 0h
rdS
rH2
!
1/2Z
T 0|β
s|
2ds
1/2
≤ C E
"
sup
s∈[0,T]
Z
s 0h
rdS
rH2
# + C E
Z
T 0|β
s|
2ds
≤ C 1 + E
"
sup
s∈[0,T]
Z
s 0h
rdW
rH2
#!
where we have used that h and θ are bounded. Applying once again the BDG inequality, we obtain
E
"
sup
s∈[0,T]
Z
s 0h
rdW
rH2
#
≤ 4 E Z
T0
|h
r|
2dr
< ∞.
Putting together the previous steps, we have that E
"
sup
s∈[0,T]
Z
s 0Z
r 0h
udS
uHU
00(X
rπ∗+ Y
r)(π
∗r+ Z
r)dW
rH#
< ∞, thus we get
E Z
T0
Z
s 0h
rdS
rHU
00(X
sπ∗+ Y
s)(π
s∗+ Z
s)dW
sH= 0.
Note that R
t0
U
0(X
sπ∗+ Y
s)h
sdW
sHt∈[0,T]
is a square integrable martingale. Indeed U
0(X
π∗+ Y ) = α is a square integrable martingale and thus
E Z
T0
U
0(X
sπ∗+ Y
s)h
s2
ds
< ∞.
Similarly,
E
U
0(X
Tπ∗+ Y
T) Z
Tt
h
rdS
Hr< ∞.
Taking expectation in (3.7) we obtain for every n ≥ 1 that E
U
0(X
Tπ∗+ Y
T) Z
T0
h
rdS
rH= E Z
T0
U
0(X
sπ∗+ Y
s)θ
s+ U
00(X
sπ∗+ Y
s)(π
∗s+ Z
s)
· h
sds
, (3.8)
which in conjunction with (3.6) leads to E
Z
T 0U
0(X
sπ∗+ Y
s)θ
s+ U
00(X
sπ∗+ Y
s)(π
s∗+ Z
s)
· h
sds
≤ 0 for every bounded predictable process h. Replacing h by −h, we get
E Z
T0
U
0(X
sπ∗+ Y
s)θ
s+ U
00(X
sπ∗+ Y
s)(π
s∗+ Z
s)
· h
sds
= 0. (3.9)
Now fix i in {1, . . . , d
1}. Let A
is:= U
0(X
sπ∗+ Y
s)θ
s+ U
00(X
sπ∗+ Y
s)(π
s∗i+ Z
si) and h
s:=
(0, . . . , 0, 1
Ais>0
, 0, . . . , 0) where the non-vanishing component is the i-th component. From (3.9) we get that
E Z
T0
1
Ais>0
[U
0(X
sπ∗+ Y
s)θ
si+ U
00(X
sπ∗+ Y
s)(π
∗si+ Z
si)]ds
= 0.
Hence, A
i≤ 0, dP ⊗ dt − a.e.. Similarly by choosing h
s= (0, . . . , 0, 1
Ais<0, 0, . . . , 0) we deduce that
U
0(X
π∗+ Y )θ
i+ U
00(X
π∗+ Y
t)(π
t∗i+ Z
ti) = 0, d P ⊗ dt − a.e.
This concludes the proof since i ∈ {1, . . . , d
1} is arbitrary. 2 The verification theorem above can also be expressed in terms of a fully-coupled Forward- Backward system.
Theorem 3.2. Under the assumptions of Theorem 3.1, the optimal strategy π
∗for (2.2) is given by
π
∗ti= −θ
itU
0(X
t+ Y
t)
U
00(X
t+ Y
t) − Z
ti, t ∈ [0, T ], i = 1, . . . , d
1,
where (X, Y, Z) ∈ R × R × R
dis a triple of adapted processes which solves the FBSDE
X
t= x − R
t 0θ
sUU000(X(Xss+Y+Yss))+ Z
sdW
sH− R
t 0θ
sUU000(X(Xss+Y+Yss))+ Z
s· θ
Hsds Y
t= H − R
Tt
Z
sdW
s− R
T t−
12|θ
sH|
2U(3)(Xs+Ys)|U0(Xs+Ys)|2 (U00(Xs+Ys))3
+|θ
sH|
2U0(Xs+Ys)
U00(Xs+Ys)
+ Z
s· θ
sH−
12|Z
sO|
2UU(3)00(X
s+ Y
s)
ds,
(3.10)
with the notation Z = (Z
1, . . . , Z
d1| {z }
=:ZH
, Z
d1+1, . . . , Z
d| {z }
=:ZO
). In addition, the optimal wealth process X
π∗is equal to X.
Proof. From Theorem 3.1 we know that the optimal strategy is given by π
∗ti= −θ
tiU
0(X
tπ∗+ Y
t)
U
00(X
tπ∗+ Y
t) − Z
ti, t ∈ [0, T ], i ∈ {1, . . . , d
1}
where (Y, Z) is a solution to the BSDE (3.3) with driver f like in (3.4). Now plugging the expression of π
∗in relation (3.4) yields
X
tπ∗= x − R
t 0θ
sU0(Xπ∗ s +Ys)
U00(Xsπ∗+Ys)
+ Z
sdW
sH− R
t 0θ
sU0(Xπ∗ s +Ys)
U00(Xsπ∗+Ys)
+ Z
s· θ
sHds Y
t= H − R
Tt
Z
sdW
s− R
Tt
−
12|θ
Hs|
2U(3)(Xπ∗
s +Ys)|U0(Xsπ∗+Ys)|2 (U00(Xsπ∗+Ys))3
+|θ
Hs|
2U0(Xsπ∗+Ys)
U00(Xsπ∗+Ys)
+ Z
s· θ
Hs−
12|Z
sO|
2UU(3)00(X
sπ∗+ Y
s)
ds.
(3.11)
Recalling that X
π:= x + R
·0
π
s(dW
sH+ θ
Hsds) for any admissible strategy π, we get the
forward part of the FBSDE. 2
Remark 3.3. Using Itˆ o’s formula and the FBSDE (3.10), we have that U
0(X + Y ) = U
0(x + Y
0) +
Z
· 0−θ
HsU
0(X
s+ Y
s)dW
sH+ Z
·0
U
00(X
s+ Y
s)Z
sOdW
sO.
Remark 3.4. Note that using the system (3.10), for α := U
0(X
π∗+Y ), integration by parts yields for every t in [0, T ]
U
0(X
tπ∗+ Y
t)(X
tπ− X
tπ∗)
= Z
t0
(X
sπ− X
sπ∗)dα
s+ Z
t0
α
s(π
s− π
∗s)dW
sH+
Z
t 0α
sθ
sH+ U
00(X
sπ∗+ Y
s)(Z
sH+ π
s∗)
· (π
s− π
s∗)ds
= Z
t0
(X
sπ− X
sπ∗)dα
s+ Z
t0
α
s(π
s− π
∗s)dW
sHshowing that U
0(X
π∗+ Y )(X
π− X
π∗) is a local martingale for every π in Π
x.
The converse implication of Theorems 3.1 and 3.2 constitutes the second main result.
Theorem 3.5. Let (H1) be satisfied for U . Let (X, Y, Z) be a triple of predictable processes which solves the FBSDE (3.10) satisfying: Z is in H
2(R
d), E[|U (X
T+H)|] < ∞, E[|U
0(X
T+ H)|
2] < ∞, and U
0(X + Y ) is a positive martingale. Moreover, assume that there exists a constant κ > 0 such that
− U
0(x) U
00(x) ≤ κ for all x ∈ R. Then
π
t∗i:= − U
0(X
t+ Y
t)
U
00(X
t+ Y
t) θ
ti− Z
ti, t ∈ [0, T ], i ∈ {1, . . . , d
1}, is an optimal solution of the optimization problem (2.2).
Proof. Note first that by definition of π
∗, X = X
π∗. Since the risk tolerance −
UU000(x)(x)is bounded and since Z is in H
2( R
d), we immediately get E
h R
T0
|π
s∗|
2ds i
< ∞, thus, π ∈ Π
x. By assumption, U
0(X + Y ) is a positive continuous martingale, hence there exists a continuous local martingale L such that U
0(X + Y ) = E(L). And we know from Remark 3.3 that
L = log(U
0(x + Y
0)) + Z
·0
−θ
HsdW
sH+ Z
·0
U
00(X
s+ Y
s)
U
0(X
s+ Y
s) Z
sOdW
sO. Define the probability measure Q ∼ P by
d Q
d P := U
0(X
T+ H) E [U
0(X
T+ H)] .
Girsanov’s theorem implies that ˜ W := ˜ W
H+ ˜ W
O= (W
1+ θ
1·dt, . . . , W
d1+θ
d1·dt, W
d1+1−
U00(X+Y)
U0(X+Y)
