Forward-backward systems for expected utility maximization

SFB 649 Discussion Paper 2011-061

Forward-backward systems for expected utility

maximization

Ulrich Horst*, Ying Hu**, Peter Imkeller*, Anthony Réveillac***

and Jianing Zhang****

* Humboldt-Universität Berlin, Germany

** Université de Rennes, France

*** Université Paris Dauphin, France

**** Weierstraß Institute for Applied Analysis and Stochastics, Germany

This research was supported by the Deutsche

Forschungsgemeinschaft through the SFB 649 "Economic Risk".

http://sfb649.wiwi.hu-berlin.de ISSN 1860-5664

SFB 649, Humboldt-Universität zu Berlin

S FB

6 4 9

E C O N O M I C

R I S K

B E R L I N

Forward-backward systems for expected utility maximization

Ulrich Horst^∗, Ying Hu^†, Peter Imkeller^‡, Anthony R´eveillac^§and Jianing Zhang^¶ October 4, 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∈AE[U(X_T^π+H)] (1.1)

for a real-valued functionU, whereAdenotes the set ofadmissible trading strategies,T <∞ is the terminal time,X_T^π is the wealth of the agent when he follows the strategyπ∈ Aand his initial capital at the initial time zero isx >0, andH 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,T^π +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 Xt,T denotes 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 strategyπ^∗can essentially be addressed usingconvex 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 onU 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 andH∈L², orU(x) := ^x_γ^γ forγ ∈(0,1) orU(x) = lnxandH= 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 T

t

Z_sdW_s− Z T

t

f(s, Z_s)ds, t∈[0, T], (1.2) where the driverf(t, z) is a predictable process of quadratic growth in thez-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 asV(t, x) =g(x)Vtwhere gis a deterministic function andV 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-independent¹. 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 T

t

ϕ(s, x)dWs− Z T

t

|ϕ_x(s, x)|²

Vxx(s, x) ds, t∈[0, T] (1.3) where ϕx denotes the partial derivative of ϕ with respect to x and Vxx the 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 bondS⁰ with interest rate zero and of d≥1 stocks given by

dS˜_tⁱ:= ˜S_tⁱdW_tⁱ+ ˜Sⁱ_tθ_tⁱdt, i∈ {1, . . . , d}

where W is a standard Brownian motion on R^d defined 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 := (θ¹, . . . , 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 := dS˜_tⁱ S˜_tⁱ .

We denote byα·βthe inner product inR^dof vectorsαandβ and by| · |the usual associated L²-norm on R^d. In all the paperC will denote a generic constant which can differ from line to line. We also define the following spaces:

S²(R^d) :=

(

β : Ω×[0, T]→R^d, predictable, E[ sup

t∈[0,T]

|β_t|²]<∞ )

,

H²(R^d) :=

β : Ω×[0, T]→R^d, predictable, E Z T

0

|β_t|²dt

<∞

.

Since the market price of risk θis assumed to be bounded, the stochastic process E(−θ·W)t:= exp

− Z t

0

θsdWs−1 2

Z t 0

|θ_s|²ds

has finite moments of order p for any p > 0. We assume d1 + d2 = d and that the agent can invest in the assets ˜S¹, . . . ,S˜^d1 while the stocks ˜S^d1+1, . . . ,S˜^d2 cannot be in- vested into. Denote S^H := (S¹, . . . , S^d1,0. . . ,0), W^H := (W¹, . . . , W^d1,0. . . ,0), W^O :=

(0, . . . ,0, W^d1+1, . . . , W^d2),andθ^H := (θ¹, . . . , θ^d1,0. . . ,0) (the notationHrefers to “hedge- able” and O to “orthogonal”). We define the set Π^x of admissible strategies with initial capital x >0 as

Π^x :=

π: Ω×[0, T]→R^d1, E Z T

0

|π_t|²dt

<∞, πis self-financing

(2.1) where for π in Π^x the associated wealth processX^π is defined as

X_t^π :=x+ Z t

0

π_rdS_r^H=x+

d1

X

i=1

Z t 0

π_rⁱdS_rⁱ, t∈[0, T].

Everyπ in Π^x is extended to an R^d-valued process by

˜

π := (π¹, . . . , π^d1,0, . . . ,0).

In the following, we will always write π in place of ˜π, i.e. π is anR^d-valued process where the last d2 components are zero. Moreover, we consider a utility functionU :I →Rwhere I is an interval of R such that U is strictly increasing and strictly concave. We seek for a strategy π^∗ in Π^x satisfying E[U(X_T^π∗+H)]<∞ such that

π^∗ = argmax_π∈Πx,E[|U(X_T^π+H)|]<∞{E[U(X_T^π+H)]} (2.2) where H is a random variable inL²(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 thatU is strictly increasing and strictly concave and that the agent is endowed with a claim H∈L²(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 elementπ^∗ in Π^x, ifE[|U⁰(X_T^π∗+H)|²]<∞ and if for every bounded predictable process h: [0, T]→R, the family of random variables

Z T 0

h_rdS_r^H Z 1

0

U⁰

X_T^π∗+H+εr Z T

0

h_rdS_r^H

dr

ε∈(0,1)

is uniformly integrable.

Before presenting the first main result of this section, we prove that condition (H2) is satisfied for every strategyπ^∗ such thatE[|U⁰(X_T^π∗+H)|]<∞when one has an exponential growth condition on the marginal utility of the form:

U⁰(x+y)≤C 1 +U⁰(x)

(1 + exp(αy)) for someα∈R. Indeed, let G:=RT

0 h_rdS_r^H and d >0. We will show that the quantity q(d) := sup

ε∈(0,1)

E

G Z 1

0

U⁰(X_T^π∗+H+εrG)dr

1|GR1

0 U⁰(X_T^π∗+H+εrG)dr|>d

vanishes when d goes to infinity. For simplicity we write δ_ε,d := 1|^GR₀^1U0(X_T^{π∗+H+εrG)dr}|^>d. By the Cauchy-Schwarz inequality

q(d)≤ sup

ε∈(0,1)E

(1 +U⁰(X_T^π∗+H))

G(1 + Z 1

0

exp(αεrG))dr

δ_ε,d

≤CE

h|U⁰(X_T^π∗+H)|²i1/2

sup

ε∈(0,1)E

"

G Z 1

0

exp(αεrG)dr

2

δ_ε,d

#1/2

. Since E

|U⁰(X_T^π∗+H)|²

is 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,d i1/2

.

Applying successively the Cauchy-Schwarz inequality and the Markov inequality, it holds that

q(d)≤CE

h|G(2 + exp(αG))|⁴i1/4

sup

ε∈(0,1)E[δ_ε,d]^1/4

≤CE h

|G(2 + exp(αG))|⁴i1/4

d^−1/4 sup

ε∈(0,1)

E

|G|

Z 1 0

U⁰(X_T^π∗+H+εrG)dr 1/4

≤CE

h|G(2 + exp(αG))|⁴i1/4

d^−1/4 E

|G(2 + exp(αG))|²1/8

. Let p≥2. Sinceh and θ are bounded it is clear thatE

|G|^2p

<∞ and E[|G(2 + exp(αG))|^p]

≤E

|G|^2p1/2

E h

|2 + exp(αG)|^2pi1/2

≤C 2 +E

h|exp(αG)|^2pi1/2

=C

2 +E

exp Z T

0

2pαhrdW_r^H−1 2

Z T 0

|2pαh_r|²dr

exp 1

2 Z _T

0

|2pαh_r|²+ 2pαh_r·θ_rdr

^1/2

≤C.

Hence limd→∞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. Letπ^∗∈Π^x be an optimal solution to the problem (2.2)which satisfies assumption(H2). Then there exists a predictable processY withYT =H such that U⁰(X^π∗+Y) is a martingale inL²(Ω,F_T,P) and

π^∗_tⁱ =−θ_tⁱU⁰(X_t^π∗+Y_t)

U⁰⁰(X_t^π∗+Yt) −Z_tⁱ, t∈[0, T], i= 1, . . . , d₁ where Zt:= ^dhY,Wi_dt ^t :=

_dhY,Wii

t

dt , . . . ,^dhY,W_dt^dit

.

Proof. We first prove the existence ofY. SinceE[|U⁰(X_T^π∗+H)|²]<∞, the stochastic process αdefined asα_t:=E[U⁰(X_T^π∗+H)|F_t], fortin [0, T] is a square integrable martingale. Define Y_t:= (U⁰)⁻¹(α_t)−X_t^π∗. ThenY is (F_t)_t∈[0,T]-predictable. Now Itˆo’s formula yields

Yt+X_t^π∗ =YT +X_T^π∗− Z T

t

1

U⁰⁰(U⁰⁻¹(αs))dαs+ 1 2

Z T t

U⁽³⁾(U⁰⁻¹(αs))

(U⁰⁰(U⁰⁻¹(αs)))³dhα, αi_s. (3.1) By definition, α is the unique solution of the zero driver BSDE

α_t=U⁰(X_T^π∗+Y_T)− Z T

t

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

Yt+X_t^π∗ =X_T^π∗+H− Z T

t

1

U⁰⁰(X_s^π∗+Ys))βsdWs+1 2

Z T t

U⁽³⁾(X_s^π∗+Y_s)

(U⁰⁰(X_s^π∗+Ys))³|β_s|²ds.

Setting ˜Z := ¹

U⁰⁰(X^π∗+Y))β, we have Y_t+X_t^π∗ =X_T^π∗+H−

Z _T

t

Z˜_sdW_s+1 2

Z _T

t

U⁽³⁾

U⁰⁰ (X_s^π∗+Y_s)|Z˜_s|²ds.

Now by putting Zⁱ := ˜Zⁱ−π^∗i, i = 1, . . . , d, we have shown that Y is a solution to the BSDE

Yt=H− Z T

t

ZsdWs− Z T

t

f(s, X_s^π∗, Ys, Zs)ds, t∈[0, T], (3.3) where f is given by

f(s, X_s^π∗, Y_s, Z_s) :=−1 2

U⁽³⁾

U⁰⁰ (X_s^π∗+Y_s)|π_s^∗+Z_s|²−π^∗_s·θ_s. (3.4) Finally, by construction we haveU⁰(X_t^π∗+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^d1 be a bounded predictable process. We extend h into R^d by setting ˜h :=

(h¹, . . . , h^d1,0, . . . ,0) and use the convention that ˜h is again denoted by h. Thus for every εin (0,1) the perturbed strategyπ^∗+εhbelongs to Π^x. Sinceπ^∗ is optimal it is clear that for every suchh it holds that

l(h) := lim

ε→0

1 εE

U(x+

Z T 0

(π_r^∗+εhr)dS_r^H+YT)−U(x+ Z T

0

π^∗_rdS_r^H+YT)

≤0. (3.5) Moreover we have

1 ε

U(x+

Z T 0

(π_r^∗+εhr)dS_r^H+YT)−U(x+ Z T

0

π^∗_rdS_r^H+YT)

= Z T

0

hrdS_r^H Z 1

0

U⁰

X_T^π∗+YT +θε Z T

0

hrdS_r^H

dθ.

Now using (H2), Lebesgue’s dominated convergence theorem implies that (3.5) can be rewrit- ten as

E

U⁰(X_T^π∗+YT) Z T

0

hrdS_r^H

≤0 (3.6)

for every bounded predictable processh. Applying integration by parts toU⁰(X_s^π∗+Y_s)_s∈[0,T] and Rs

0 hrdS_r^H

s∈[0,T], we get U⁰(X_T^π∗+YT)

Z T 0

hrdS_r^H

=U⁰(x+Y0)×0 + Z T

0

U⁰(X_s^π∗+Ys)hsdS_s^H +

Z T 0

Z s 0

h_rdS_r^HU⁰⁰(X_s^π∗+Y_s)h

(π_s^∗+Z_s)dW_s^H+ (π_s^∗·θ_s+f(s, X_s^π∗, Y_s, Z_s))dsi +1

2 Z T

0

Z s 0

hrdS_r^HU⁽³⁾(X_s^π∗+Ys)|π^∗_s+Zs|²ds +

Z T 0

U⁰⁰(X_s^π∗+Y_s)h_s·(π_s^∗+Z_s)ds.

By definition of the driver f, the previous expression reduces to U⁰(X_T^π∗+Y_T)

Z T 0

h_rdS_r^H

= Z T

0

U⁰(X_s^π∗+Ys)θs+U⁰⁰(X_s^π∗+Ys)(π_s^∗+Zs)

·hsds +

Z T 0

Z s 0

h_rdS_r^HU⁰⁰(X_s^π∗+Y_s)(π_s^∗+Z_s)dW_s^H+ Z T

0

U⁰(X_s^π∗+Y_s)h_sdW_s^H. (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

U⁰⁰(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 0

Z r 0

h_udS_u^HU⁰⁰(X_r^π∗+Y_r)(π_r^∗+Z_r)dW_r^H

#

≤CE





Z _T

0

Z _s

0

h_rdS_r^H

2

|β_s|²ds

1/2



≤CE



 sup

s∈[0,T]

Z s 0

hrdS_r^H

2!1/2

Z T 0

|β_s|²ds ^1/2



. Young’s inequality furthermore yields

E



 sup

s∈[0,T]

Z s 0

h_rdS_r^H

2!1/2

Z T 0

|β_s|²ds ^1/2





≤CE

"

sup

s∈[0,T]

Z s 0

h_rdS_r^H

2# +CE

Z T 0

|β_s|²ds

≤C 1 +E

"

sup

s∈[0,T]

Z s 0

hrdW_r^H

2#!

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 0

h_rdW_r^H

2#

≤4E Z T

0

|h_r|²dr

<∞.

Putting together the previous steps, we have that E

"

sup

s∈[0,T]

Z s 0

Z r 0

h_udS_u^HU⁰⁰(X_r^π∗+Y_r)(π^∗_r+Z_r)dW_r^H

#

<∞, thus we get

E Z T

0

Z s 0

hrdS_r^HU⁰⁰(X_s^π∗+Ys)(π_s^∗+Zs)dW_s^H

= 0.

Note that Rt

0U⁰(X_s^π∗+Y_s)h_sdW_s^H

t∈[0,T] is a square integrable martingale. Indeed U⁰(X^π∗+Y) =α is a square integrable martingale and thus

E Z T

0

U⁰(X_s^π∗+Ys)hs

2

ds

<∞.

Similarly,

E

U⁰(X_T^π∗+Y_T) Z T

t

h_rdS^H_r

<∞.

Taking expectation in (3.7) we obtain for everyn≥1 that E

U⁰(X_T^π∗+Y_T) Z T

0

h_rdS_r^H

=E Z T

0

U⁰(X_s^π∗+Y_s)θ_s+U⁰⁰(X_s^π∗+Y_s)(π^∗_s+Z_s)

·h_sds

, (3.8)

which in conjunction with (3.6) leads to E

Z T 0

U⁰(X_s^π∗+Y_s)θ_s+U⁰⁰(X_s^π∗+Y_s)(π_s^∗+Z_s)

·h_sds

≤0 for every bounded predictable process h. Replacing h by −h, we get

E Z T

0

U⁰(X_s^π∗+Ys)θs+U⁰⁰(X_s^π∗+Ys)(π_s^∗+Zs)

·hsds

= 0. (3.9)

Now fix i in {1, . . . , d₁}. Let Aⁱ_s := U⁰(X_s^π∗ +Y_s)θ_s+U⁰⁰(X_s^π∗ +Y_s)(π_s^∗i +Z_sⁱ) and h_s :=

(0, . . . ,0,1_Aⁱ

s>0,0, . . . ,0) where the non-vanishing component is the i-th component. From (3.9) we get that

E Z T

0

1_Aⁱ

s>0[U⁰(X_s^π∗+Y_s)θ_sⁱ+U⁰⁰(X_s^π∗+Y_s)(π^∗_sⁱ+Z_sⁱ)]ds

= 0.

Hence, Aⁱ ≤ 0, dP⊗dt −a.e.. Similarly by choosing hs = (0, . . . ,0,1_Aⁱ_s<0,0, . . . ,0) we deduce that

U⁰(X^π∗+Y)θⁱ+U⁰⁰(X^π∗+Yt)(π_t^∗i+Z_tⁱ) = 0, dP⊗dt−a.e.

This concludes the proof sincei∈ {1, . . . , d₁} 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 =−θⁱ_tU⁰(X_t+Y_t)

U⁰⁰(X_t+Y_t)−Z_tⁱ, t∈[0, T], i= 1, . . . , d₁,

where (X, Y, Z)∈R×R×R^d is a triple of adapted processes which solves the FBSDE



















X_t = x−Rt 0

θ_sU^U00^0(X(X^ss^+Y+Y^ss⁾) +Z_s

dW_s^H−Rt 0

θ_sU^U00^0(X(X^ss^+Y+Y^ss⁾)+Z_s

·θ^H_s ds Yt = H−RT

t ZsdWs−RT t

−¹₂|θ_s^H|^{2U(3)(Xs+Ys)|U}

0(Xs+Ys)|² (U⁰⁰(Xs+Ys))³

+|θ_s^H|^2U

0(Xs+Ys)

U⁰⁰(Xs+Ys) +Zs·θ_s^H−¹₂|Z_s^O|^2U_U⁽³⁾00(Xs+Ys)

ds,

(3.10)

with the notation Z = (Z¹, . . . , Z^d1

| {z }

=:Z^H

, Z^d1+1, . . . , Z^d

| {z }

=:Z^O

). 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_tⁱU⁰(X_t^π∗+Yt)

U⁰⁰(X_t^π∗+Y_t) −Z_tⁱ, t∈[0, T], i∈ {1, . . . , d₁}

where (Y, Z) is a solution to the BSDE (3.3) with driverf like in (3.4). Now plugging the expression of π^∗ in relation (3.4) yields



















X_t^π∗ = x−Rt 0

θ_s^U0(Xπ

∗ s +Ys)

U⁰⁰(X_s^π∗+Ys)+Z_s

dW_s^H−Rt 0

θ_s^U0(Xπ

∗ s +Ys)

U⁰⁰(X_s^π∗+Ys) +Z_s

·θ_s^Hds Yt = H−R_T

t ZsdWs−R_T

t

−¹₂|θ^H_s |^2U(3)(Xπ

∗

s +Ys)|U⁰(X_s^π∗+Ys)|² (U⁰⁰(X_s^π∗+Ys))³

+|θ^H_s |^2U

0(X_s^π∗+Ys)

U⁰⁰(X_s^π∗+Ys)+Z_s·θ^H_s −¹₂|Z_s^O|^2U_U⁽³⁾00 (X_s^π∗+Y_s)

ds.

(3.11)

Recalling that X^π := x+R·

0πs(dW_s^H+θ^H_s ds) for any admissible strategy π, we get the

forward part of the FBSDE. 2

Remark 3.3. Using Itˆo’s formula and the FBSDE (3.10), we have that U⁰(X+Y) =U⁰(x+Y₀) +

Z · 0

−θ^H_s U⁰(X_s+Y_s)dW_s^H+ Z ·

0

U⁰⁰(X_s+Y_s)Z_s^OdW_s^O.

Remark 3.4. Note that using the system (3.10), for α:=U⁰(X^π∗+Y), integration by parts yields for every t in [0, T]

U⁰(X_t^π∗+Yt)(X_t^π−X_t^π∗)

= Z t

0

(X_s^π−X_s^π∗)dαs+ Z t

0

αs(πs−π^∗_s)dW_s^H +

Z t 0

αsθ_s^H+U⁰⁰(X_s^π∗+Ys)(Z_s^H+π_s^∗)

·(πs−π_s^∗)ds

= Z t

0

(X_s^π−X_s^π∗)dα_s+ Z t

0

α_s(π_s−π^∗_s)dW_s^H

showing that U⁰(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 inH²(R^d),E[|U(XT+H)|]<∞,E[|U⁰(XT+ H)|²]<∞, and U⁰(X+Y) is a positive martingale. Moreover, assume that there exists a constant κ >0 such that

−U⁰(x) U⁰⁰(x) ≤κ for all x∈R. Then

π_t^∗i :=−U⁰(X_t+Y_t)

U⁰⁰(Xt+Yt)θ_tⁱ−Z_tⁱ, t∈[0, T], i∈ {1, . . . , d₁}, 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 −_U^U00^0(x)(x) is bounded and sinceZis inH²(R^d), we immediately getE

hRT

0 |π_s^∗|²dsi

<∞, thus,π ∈Π^x. By assumption,U⁰(X+Y) is a positive continuous martingale, hence there exists a continuous local martingale Lsuch that U⁰(X+Y) =E(L). And we know from Remark 3.3 that

L= log(U⁰(x+Y₀)) + Z ·

0

−θ^H_s dW_s^H+ Z ·

0

U⁰⁰(X_s+Y_s)

U⁰(X_s+Y_s)Z_s^OdW_s^O. Define the probability measure Q∼P by

dQ

dP := U⁰(XT +H) E[U⁰(X_T +H)].

Girsanov’s theorem implies that ˜W := ˜W^H+ ˜W^O = (W¹+θ¹·dt, . . . , W^d1+θ^d1·dt, W^d1+1−

U⁰⁰(X+Y)

U⁰(X+Y)Z^d1+1·dt, . . . , W^d2−^U_U⁰⁰0(X+Y^(X+Y)⁾Z^d2·dt) is a standard Brownian motion underQ. Thus X^π is a local martingale underQfor everyπin Π^x. Now fixπ in Π^x withE[|U(X_T^π+H)|]<

∞. Let (τ_n)_n be a localizing sequence for the local martingale X^π −X^π∗. Since U is a concave, we have

U(X_T^π+H)−U(X_T^π∗+H)≤U⁰(X_T^π∗+H)(X_T^π−X_T^π∗). (3.12)

Taking expectations in (3.12) we get E[U(X_T^π+H)−U(X_T^π∗+H)]

E[U⁰(X_T +H)] ≤E_Q[X_T^π −X_T^π∗]

=E_Q

n→∞lim Z T∧τn

0

(π_s−π^∗_s)dW˜_s^H

= lim

n→∞E_Q

Z T∧τ_n 0

(π_s−π^∗_s)dW˜_s^H

= 0,

which eventually follows as a consequence of Lebesgue’s dominated convergence theorem.

To this end we prove that E_Q

"

sup

t∈[0,T]

Z t 0

(π_s−π^∗_s)dW˜_s^H

#

<∞.

Indeed the BDG inequality and the Cauchy-Schwarz inequality imply that EQ

"

sup

t∈[0,T]

Z _t

0

(π_s−π^∗_s)dW˜_s^H

#

≤CEQ



 Z T

0

|π_s−π^∗_s|²ds

1 2





=CE





U⁰(XT +H) E[U⁰(X_T +H)]

Z T 0

|π_s−π_s^∗|²ds

1 2





≤CE

"

U⁰(XT +H) E[U⁰(X_T +H)]

2#¹₂ E

Z T 0

|π_s−π^∗_s|²ds ¹₂

<∞.

2 We have proved in Theorem 3.2 that if (2.2) exhibits an optimal strategyπ^∗∈Π^x, then there exists an adapted solution to the FBSDE (3.10). As a byproduct we showed the optimization procedure singles out a “pricing measure” under which the asset prices and marginal utilities are martingales. In that sense, the processY captures the impact of future trading gains on the agent’s marginal utilities. If we assume additional conditions on the utility functionU, we get the following regularity properties of the solution (X, Y, Z).

Proposition 3.6. Assume that for H ∈L^∞(Ω,F_T,P) and that the FBSDE (3.10) admits an adapted solution (X, Y, Z) such that Y is bounded. Let

ϕ₁(x) := U⁰(x)

U⁰⁰(x), ϕ₂(x) := U⁽³⁾(x)|U⁰(x)|²

(U⁰⁰(x))³ , ϕ₃(x) := U⁽³⁾(x)

U⁰⁰(x) , x∈R.

Assume that U is such that ϕi, i= 1,2,3 are bounded and Lipschitz continuous functions.

Then (X, Y, Z) is the unique solution of (3.10) in S²(R)×S^∞(R)×H²(R^d). In addition, Z·W is a BMO-martingale.

Proof. Let (X, Y, Z) be a solution to (3.10) such that Y is bounded. Then, using the usual theory on quadratic growth BSDEs (see for example [20, Theorem 2.5 and Lemma 3.1]) we have only from the backward part of the FBSDE that Z is in H²(R^d) and that Z ·W is a BMO-martingale. In addition there exists a unique solution to the backward component in this space for a given process X. Now the previous regularity properties of the processes (Y, Z) imply that X is in S²(R). We turn to the uniqueness of the X process. Assume that there exists another solution (X⁰, Y⁰, Z⁰) of (3.10). Hence, Theorem 3.5 implies that π^∗0 := −_U^U00^0(X(X^00+Y+Y⁰⁰⁾)θⁱ+Z⁰ⁱ, i ∈ {1, . . . , d₁} is an optimal solution to our original problem (2.2) and X⁰ is the optimal wealth process. However, by strict concavity of U and by convexity of Π^x the optimal strategy has to be unique. So X and X⁰ are the wealth processes of the same optimal strategy, thus, they have to coincide (for instance X_T =X_T⁰ ,P−a.s.) which implies (Y⁰, Z⁰) = (Y, Z). 2 In the complete case we are able to construct the solution (X, Y, Z). This is the subject of the following Section.

3.2 Characterization and verification: complete markets

In this section we consider the benchmark case of a complete market. We assumed= 1 for simplicity. H denotes a square integrable random variable measurable with respect to the Brownian motion W.

In the complete case we can give sufficient conditions for the existence of a solution to the system (3.10). Our construction relies on the following remark.

Remark 3.7. Using (3.10)the martingaleU⁰(X^π∗+Y) becomes more explicit, because Itˆo’s formula applied to U⁰(X^π∗+Y) yields

U⁰(X_t^π∗+Yt) =U⁰(x+Y0) + Z t

0

U⁰⁰(X_s^π∗+Ys)(π_s^∗+Zs)dWs

=U⁰(x+Y0)− Z t

0

U⁰(X_s^π∗+Ys)θsdWs,

where we have replaced π^∗ by its characterization in terms of (X, Y, Z) from Theorem 3.1.

Hence,

U⁰(X_t^π∗+Yt) =U⁰(x+Y0)E(−θ·W)t, t∈[0, T]. (3.13) This remark will allow us to prove existence of a solution to the system (3.10) under a condition on the risk aversion coefficient −^U_U⁰⁰0 of U. To this end, we give a sufficient condition on U for the system (3.10) to exhibit a solution. We have the following remark.

Remark 3.8. If (X, Y, Z) is an adapted solution to the system (3.10), then P :=X+Y is solution of the forward SDE

P_t=x+Y₀− Z _t

0

θ_sU⁰(Ps)

U⁰⁰(P_s)dW_s− Z _t

0

1

2|θ_s|²U⁽³⁾(Ps)|U⁰(Ps)|²

(U⁰⁰(P_s))³ ds, t∈[0, T]. (3.14) In addition, if (X, Y, Z) is in S²(R)×S²(R)×H²(R^d), then P ∈S²(R). Thus a necessary condition for the FBSDE (3.10)to have a solution is that the SDE (3.14)admits a solution.