Optimal investment under behavioural criteria in incomplete diffusion market models

0.22 MB·English

by  Miklós Rásonyi

Optimal investment under behavioural criteria in
incomplete diffusion market models
5 ∗
1
0
2 M. Ra´sonyi†and J.G. Rodr´ıguez-Villarreal‡

n
August 10, 2016
a
J

7

]
M
1 Introduction
P
. Themostcommonlyacceptedmodelforinvestors’preferencesisexpectedutility
n
i theory,goingback to [2,20]. Accordingto the tenets ofthis theory,an investor
f
- prefers a random return X to Y if Eu(X) Eu(Y) for some utility function
q u : R R that is usually assumed non-incre≥asing and concave. More recently,
[ →
other theories have emerged and pose new challenges to mathematics.

1 The present paper treats preferences of cumulative prospect theory (CPT),
v [11, 19], where an “S-shaped” u is considered (i.e. convex up to a certain point
4
andconcavefromthereon). Also,distortedprobabilitymeasuresareappliedfor
0
calculating the utility of a given position with respect to a (possibly random)
5
1 benchmark G. We remark that techniques of the present paper easily carry
0 over to other types of preferences, too, such as rank-dependent utility [15] or
. acceptability indices [8].
1
0 The theory of optimal portfolio choice for CPT preferences is in its infancy
5 yet. Continuous-time studies almost always assume a complete market model,
1 [3, 10, 7, 5, 17]. Only very specific types of incomplete continuous-time models
:
v have been treated to date (finite mixtures of complete models; the case where
i thepriceisamartingaleunderthephysicalmeasure;thecasewherethemarket
X
price of risk is deterministic), see [18, 16]. In the present paper we make a step
r
a forwardand consider incomplete models of a diffusion type where the return of
the investment in consideration depends on some economic factors. Our main
result asserts, under mild assumptions, the existence of an optimal strategy
when the driving noise of the economic factors is independent of that of the
investment and the rate of return is non-negative. The independence condition
is, admittedly, rather stringent and does not allow a leverage effect (see [4]).
∗The second author gratefully acknowledges support from grants of CONACYT, Mexico
and of the University of Edinburgh. Part of this work was done while the first author was
affiliatedwiththeUniversityofEdinburgh.
†Alfr´ed R´enyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest and
P´azm´anyP´eter CatholicUniversity,Budapest
‡UniversityofEdinburgh
1
We are also able to accomodate models of a specific type where the factor may
have non-zero correlation with the investment. We think that our results open
the door for further generalizations.
2 Optimal investment model under behavioural
criteria
In this section definitions and notation related to the problem of behavioural
optimal investment are presented, based on [12], [6].
Unfortunately, most of the techniques developed in the literature for find-
ing optimal policies rely on either the Markovian nature of the problem or on
convex duality. These are no longer applicable under behavioural criteria. For
this reason we shall consider a weak-type formulation of the control problem
associated with optimal investment (Subsection 2.1). Introducing a relaxation
of the problem for which results in [12] apply, we can prove the existence of an
optimal investment strategy (Subsection 2.3).
2.1 The setting: market and preferences
Fix a finite horizon T >0. We consider a financial market consisting of a risky
asset, whose discounted price (S ) depends on economic factors. These
t 06t6T
factors are described by a d-dimensional stochastic processes Y . Without
{ t}t>0
entering into rigorous definitions at this point, our market model is described
by the equations
dY =ν (Y)dt+κ (Y)dB , and Y =y, (1)
t t · t · t 0
dS =θ (Y)S dt+λ (Y)S dW and S =s>0, (2)
t t · t t · t t 0
with B,W independent standardBrownianmotions ofappropriatedimensions.
Wealsoassumethatthereisarisklessassetofconstantpriceequalto1. We
shall be more specific later in this section. Stochastic volatility models provide
prime examples of financial market models with dynamics (1) and (2), see [9].
The investor trades in the risky and riskless assets, investing a proportion
φ [0,1]ofhiswealthintotheriskyassetattimet. Thisleadstothefollowing
t
∈
equation for the wealth of the investor at time t:
dX =φ θ (Y)X dt+φ λ (Y)X dW and X =x, (3)
t t t · t t t · t t 0
where x>0 is the investor’s initial capital.
Borrowing and short selling are not allowed, hence φ is a process taking
t
values in [0,1]. We note that, in this model, the risky asset’s price has no
influence on the economic factors. We will see in Section 3 below how this
assumption can be weakened.
We willneedcertainclosednessresultsonthe lawsofItoˆ processesfrom[12]
hence it is necessary to work in the ’weak’ setting of stochastic control theory,
where the underlying probability space is not fixed.
2
We first set out the requirements for the coefficients in (1), (3).
Let C([0,T];Rn) denote the family of Rn-valued continuous functions on
[0,T].
Denote by p : C [0,T],Rd Rd the projections p (x) = x and define
t t · t
the σ-algebras N =σ( p :s6t→), and N =σ( p :s6T ).
t s s
(cid:0) { (cid:1) } { }
Definition 2.1. Let ν(t,y) : [0,T] C [0,T],Rd Rd be such that the
·
× →
restriction of ν to [0,t] C [0,T];Rd is ([0,t]) N -measurable, for any
× (cid:0)B (cid:1)⊗ t
06t6T. We shall denote this functional by either ν (y) or ν(t,y).
t · ·
(cid:0) (cid:1)
Similarly,wedefinethecoefficientsθ,λ,κwiththesamemeasurabilityprop-
erties as ν, but with values in R, R and Sd, respectively, where Sd denotes the
+ +
set of real, symmetric and positive semidefinite d d matrices.
×
Definition 2.2. An investment stategy π is given by the following collection:
π := Ω, , ,P,X ,Y ,φ ,(B ,W ),(x,y) ,
F {Ft}06t6T t t t t t
(cid:16) (cid:17)
with x>0 and y Rd, where
∈
(a) Ω, , ,P is a complete filtered probability space whose filtra-
F {Ft}06t6T
(cid:16)tion satisfies the us(cid:17)ual conditions;
(b) theprocess(B ,W ) isastandardd+1-dimensional -Wienerprocess;
t t t>0 Ft
(c) φ :Ω [0,T] [0,1] is ([0,T])-measurable and -adapted;
t t
× → F ⊗B F
(d) on the filtered probability space X ,Y are ([0,T])-measurable and
t t
F ⊗B
-adapted processes such that
t
F
t t
Y =y+ ν (Y)ds+ κ (Y)dB , (4)
t s · s · s
Z0 Z0
t t
X =x+ φ θ (Y)X ds+ φ λ (Y)X dW , (5)
t s s · s s s · s s
Z0 Z0
for 06t6T.
In other words, Ω, , ,P,X ,Y ,(B ,W ),(x,y) is a weak so-
F {Ft}06t6T t t t t
lution of the system(cid:16)of equations (1), (3). The process φt rep(cid:17)resents a ratio
of investment in the risky asset, it is measurable and -adapted. We do not
t
F
consider the price process S from (2) at all since it is enough to work with the
t
’controlled dynamics’ X .
t
When needed, we will use the notation Xπ,Yπ, etc. to indicate that the
object we mean belongs to π. Let Π = Π(x,y) denote the collection of all
strategies.
Assumption 2.1. The functional θ is non-negative i.e. θ(t,y) > 0 for all
·
t R and y C [0,T];Rd .
+
∈ ∈
(cid:0) (cid:1)
3
Remark2.3. Inotherwords,thereturnoftheriskyassetmustbenon-negative.
This looks rather a harmless assumption. On the other hand, as mentioned
before, (b) in Definition 2.2 is stringent. It excludes the ’leverage effect’ where
the volatility and the stock prices have (negative) correlation. This condition
can be relaxed, see Section 3.
We now present the framework of optimal investment under CPT, as pro-
posed in [19]. We follow [16] and [6].
The investor assesses strategies by means of utilities on gains and losses,
which are described in terms of functions u : R R , by a reference point
± + +
→
G and functions w : [0,1] [0,1]. The latter functions w are introduced
± ±
→
with the aim of explaining the distortions of her perception on the “likelihood”
of her gains and losses.
AccordingtothetenetsofCPT,investorsusebenchmarkstoassesthe port-
foliooutcomes,this is modelledby areal-valuedrandomvariableG. The quan-
tity G depends on economic factors as follows: let us denote by F a fixed
deterministic functional F : C [0,T];Rd R which is N -measurable. As
+ T
→
the probability space is not fixed, for each π Π we define the corresponding
(cid:0) (cid:1) ∈
reference point by Gπ := F (Yπ). That is, we assume that the benchmark is a
·
non-negativefunctionalofthe economicfactors. Results caneasilybe extended
to the slightly more general case where Gπ := F(Ypi,Bπ) for some functional
· ·
F.
For any strategy π Π, we define the functionals
∈
∞
V (π):= w Pπ u (Xπ Gπ) >t dt, (6)
+ + + T − +
Z0
(cid:0) (cid:0) (cid:0) (cid:1) (cid:1)(cid:1)
and
∞
V (π):= w Pπ u (Xπ Gπ) >t dt. (7)
− − − T − −
Z0
(cid:0) (cid:0) (cid:0) (cid:1) (cid:1)(cid:1)
TheoptimalportfolioproblemforaninvestorunderCPTconsistsinmaximising
the following performance functional:
V (π):=V (π) V (π), (8)
+ −
−
whichisdefinedprovidedthatatleastoneofthesummandsisfinite. Fixx>0,
y R. Set Π′ := π Π(x,y):V (π)< and define
−
∈ { ∈ ∞}
V := sup V (π). (9)
π∈Π′
The value V represents the maximal satisfaction achievable by investing in the
stock and riskless asset in a CPT framework. Our purpose is to prove the
existence of πˆ Π′ such that V(πˆ)=V.
∈
2.2 Main result
We make the following assumptions. Recall the notation y⋆ =sup y .
t s6t| s|
4
Assumption 2.2. The functionals κ, λ, θ and ν are uniformly bounded on
[0,T] C [0,T];Rd . Furthermore, for fixed t > 0 and functions yn,z
× ∈
C [0,T];Rd such that (yn z)⋆ 0, n we have κ (yn) κ (z)
(cid:0) (cid:1) − t → → ∞ t · → t ·
and the same holds for the functionals λ,θ and ν. We will refer to this as the
(cid:0) (cid:1)
coefficients being path-continuous at any time t [0,T].
∈
Assumption 2.3. A (weak) solution of equation (4) exists and it is unique in
law.
Assumption 2.4. We assume that u : R R and w :[0,1] [0,1] are
± + + ±
→ →
continuous, non-decreasing functions with u (0) = 0, w (0) = 0, w (1) = 1,
± ± ±
and
u (x)6k (xα+1),for all x R , (10)
+ + +
∈
w (p)6g pγ, for all p [0,1], (11)
+ +
∈
with γ, α>0, k ,g >0 fixed constants.
+ +
We denote by Lp(Ω,P) the usual space of p-integrable random variables on
a probability space (Ω, ,P).
F
Assumption 2.5. There is ϑ > 0 such that ϑγ > 1 and Gπ Lϑγ(Ω,Pπ) for
∈
all π Π.
∈
Note that, under Assumption 2.3, the law of Gπ is independent of π and
hence Assumption 2.5 holds iff Gπ Lϑγ(Ω,Pπ) for one particular π.
∈
In order to ensure that the functional V and the optimisation problem in
(9) are defined overa non-empty set, we introduce the following assumptionon
u , the distortion function w and the reference point Gπ.
− −
Assumption 2.6. The functions w ,u are such that, for all π Π,
− −
∈
∞
w (Pπ(u (Gπ)>y))dy < . (12)
− −
∞
Z0
ThisassumptionensuresthatthesetΠ′isnotempty. Indeed,let(Ω, , ,P)
F {Ft}06t6T
be a filteredprobabilityspacewhere (1) hasa solutionY . Thensetting φ :=0
t t
and X :=x for all t,
t
Ω, , ,P,x,Y ,0,(B ,W ),(x,y)
F {Ft}06t6T t t t
(cid:16) (cid:17)
belongs to Π′. Fix x>0 and y R. Our main result can now be stated.
∈
Theorem 2.4. Under Assumptions 2.1, 2.2, 2.3, 2.4, 2.5 and 2.6 the problem
(9)is well-posed, i.e. V < . Moreover, thereexists an optimal strategy πˆ Π′
∞ ∈
attaining the supremum in (9), i.e. V =V (πˆ).
5
2.3 A relaxation of the set of controls
We introduce a relaxation of the problem by extending the class of investment
strategiesgiveninDefinition2.2,weshallcallthisextensiontheclassofauxiliary
controls. This relaxation is introduced in order to ensure the closedness of the
set of laws of the processes (Y,X).
· ·
We follow the martingale problem formulation, thus we refer to a and b
t t
as the drift/diffusion coefficients of the process (Y ,X ), as they appear in the
t t
martingale problem formulation of equations (4) and (5). In order to use [12],
these coefficients musttake values in a family ofconvex subsets ofSd+1 Rd+1
+ ×
hence we shall consider a ’convex extension’ of the set in which the coefficients
in equations (4) and (5) take values.
Definition2.5. DenoteA=Sd+1 Rd+1. Foranypairofcontinuousfunctions
+ ×
(x,y) C([0,T];R Rd) and for any t [0,T] we define
· ·
∈ × ∈
1κκ⋆(t,y) 0 ν(t,y)
A (x,y) = (a,b) A (a,b)= 2 · , · ,
t · · (cid:26) ∈ (cid:12) (cid:18)(cid:18) 0 21mλ2(t,y·)x2t (cid:19) (cid:18) lθ(t,y·)xt (cid:19)(cid:19)
06m61(cid:12)(cid:12),
06l6√(cid:12)m (13)
(cid:27)
Remark 2.6. Notice that, for any investment strategy π as in Definition 2.2,
κ(t,y) 0 ν(t,y)
if σ = · and b = · then, defining
t 0 φ λ(t,y)x t φ θ(t,y)x
t · t t · t
(cid:18) (cid:19) (cid:18) (cid:19)
a = 1σ σ⋆, the pair (a ,b ) belongs to A (x,y).
t 2 t t t t t · ·
Thefollowingdefinitiondescribesthefamiliyofauxiliarycontrolsusedthrough-
out this work. It stresses the fact of having Itoˆ processes whose coefficients
belong to the convex sets A (x,y) in ’a measurable way’ as t,x and y vary.
t · · · ·
Definition 2.7. We defineafamiliy ofauxiliarycontrolsΠ=Π(x,y). Namely,
an auxiliary control π Π consists of a collection
∈
π := Ω, , ,P,X ,Y ,(B ,W ),(x,y)
F {Ft}06t6T t t t t
(cid:16) (cid:17)
where x>0, y R,
∈
(a) Ω, , ,P is a complete filtered probability space whose filtration
F {Ft}t>0
(cid:16)satisfies the usua(cid:17)l conditions;
(b) ξ :=(B ,W ) is an Rd+1-valued standard -Brownian motion;
t t t t
F
(c) thereexistsanA-valued, ([0,T])-measurableand -adaptedprocess,
t
F⊗B F
denoted by (a ,b ), such that (d) and (e) below hold;
t t
(d) X and Y are ([0,T])-measurable and -adapted such that a.s. for
t t t
F ⊗B F
all t>0;
Y y t t
t = + √2a dξ + b ds; (14)
X x s s s
(cid:18) t (cid:19) (cid:18) (cid:19) Z0 Z0
6
(e) for almost all (ω,t) Ω [0,T], we have (a ,b ) A (X,Y).
t t t · ·
∈ × ∈
We will often write Xπ, Yπ to indicate that we mean X, Y belonging to π.
For each π Π, we can define V (π) as before and we can set V(π) :=
±
∈ ′
V (π) V (π) for π Π := π Π:V (π)< .
+ − −
− ∈ { ∈ ∞}
Remark 2.8. For a pair of processesprocesses a and b in A (X,Y) one can
t t t · ·
define the corresponding real-valued processes l and m with 0 6 m 6 1,0 6
t t t
lt 6√mt setting
l :=bd+11 /(X θ (t,Y )), m :=ad+1,d+11 /(X2λ2(t,Y )).
t t θt(t,Yt)6=0 t t t t λt(t,Yt)6=0 t t
Conditions (c),(d) in Definition 2.7 together with Assumption 2.2 imply that
l ,m can be chosen ([0,T]) measurable and -adapted.
t t t
F ⊗B F
Equation (14) can be rewritten as the set of equations below. Denote
κ(t,Y) 0
σ = ·
t 0 √mtλ(t,Y·)Xt
(cid:18) (cid:19)
and
ν(t,Y)
b = · .
t l θ(t,Y)X
t · t
(cid:18) (cid:19)
Setting a := 1σ σ⋆,
t 2 t t
t t
Y =y+ ν (Y)ds+ κ (Y)dB , (15)
t s · s · s
Z0 Z0
t t
X =x+ l θ (Y)X ds+ √m λ (Y)X dW . (16)
t s s · s s s · s s
Z0 Z0
Definition 2.9. Let π Π be a relaxedcontrol. We say that Xπ is a portfolio
∈ t
value process if lt =√mt, i.e.
dX =√m θ(t,Y)X dt+√m λ(t,Y)X dW . (17)
t t · t t · t t
Remark 2.10. If Xtπ is a portfolio value process then, taking φt = √mt, we
can see that
Ωπ, π, π ,Pπ,Xπ,Yπ,φ , Bπ,Wπ ,(x,y)
F Ft t>0 t t t t t
(cid:16) (cid:8) (cid:9) (cid:0) (cid:1) (cid:17)
belongs to Π.
Remark2.11. Supposethatwearegivenaπ Πi.e. thereisastandardd+1-
∈
dimensionalBrownianmotion(B,W)on Ωπ, π, π ,Pπ andprocesses
F Ft t>0
Xtπ,Ytπ,mπt,ltπ such that equations (15) (cid:16)and (16) (cid:8)hold(cid:9). Define(cid:17)the continuous
semimartingale Mtπ := 0t mπstλs(Y·)dWs + 0tlsπtθs(Y·)ds. Then we can
rewrite equation (16) as
R p R
t
X =x+ X dMπ. (18)
t s s
Z0
7
Equation(18)hasauniquestrongsolutiononthegivenprobabilityspace,given
the stochastic exponential
t t 1
Xπ =xexp √m λ Yπ dWπ + l θ Yπ m λ2 Yπ ds ,
t s s · s s s · − 2 s s ·
(cid:26)Z0 Z0 (cid:20) (cid:21) (cid:27)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (19)
and this process is positive Pπ-a.s.
2.4 Krylov’s theorem and related results
Lemma 2.12. Let M =max κ , λ , θ , ν . The set A (x,y) is
∞ ∞ ∞ ∞ t · ·
{k k k k k k k k }
convex, closed and bounded, where the bound depends on M and x only.
t
Proof. Notation will refer to Euclidean norms of varying dimensions. For
|·|
simplicity, we assume d=1. Notice that
(σ ,b ) = κ2(y)+m2λ2(y)x2+ν2(y)+l2θ2(y)x2 1/2,
| t t | t · t · t t · t · t
(cid:0) (cid:1)
hence
(σ ,b ) 6√2M(1+ x ). (20)
t t t
| | | |
It is clear that the set is closed. For a fixed t, x and y the set is bounded.
· ·
Indeed, let (a ,b ) A (x,y). Then we have
t t t · ·
∈
(a ,b ) = 1 κ2(y) 2+ 1 m λ2(y) x2 2+ν2(y)+l2θ2(y)x2 1/2,
| t t | 4 · t · 4 · · t · · t t · t · t
(cid:18) (cid:19)
(cid:0) (cid:1) (cid:0) (cid:1)
so
1/2
(a ,b ) 6 1M4+ 1 M4 x2 2+M2+M2x2 ,
| t t | 4 4 · · t t
(cid:18) (cid:19)
(cid:0) (cid:1)
which leads to (a ,b ) 6 1(M +1)2+M2x2.
| t t | 2 t
In particular,
A (x,y) :=max (a ,b ) :(a ,b ) A (x,y) 6K 1+ x 2 , (21)
t · · t t t t t · · t
k k {| | ∈ } | |
(cid:16) (cid:17)
for some K 0.
≥
The set A (x,y) is also convex. Indeed, let (α,b),(γ,c) A (x,y) then,
t · · t · ·
∈
for 0 µ 1,
≤ ≤
µ(α,b)+(1 µ)(γ,c)=
−
1κ2(y) 0 ν (y)
2 t · , t ·
0 1(µm+(1 µ)n)λ2(y)x2 (µl+(1 µ)p)θ (y)x
(cid:18)(cid:18) 2 − t · t (cid:19) (cid:18) − t · t (cid:19)(cid:19)
with 0 6 m,n 6 1, 0 6 l 6 √m and 0 6 p 6 √n. Clearly, µl+(1 µ)p 6
−
µm+(1 µ)n, by concavity of the square root function.
−
p
8
In order to deal with (semi)continuity issues related to the family of sets
defined in Definitions 2.5 and (13), the support functions of sets A (x,y) are
t · ·
now considered. We denote for all u R(d+1)(d+1), v Rd+1 and t [0,T],
∈ ∈ ∈
F (x,y)(u,v)=max a u + b v : (a,b) A (x,y) . (22)
t · ·  ij ij j j ∈ t · · 
Xi,j Xj 
Under Assumption 2.2, for fixed t > 0 and (u,v), the support function
(x,y) F (x,y)(u,v) is continuous, since we are fixing t, restricting the
· · t · ·
→
trajectories to [0,t], and thus the max is taken over a compact set by Lemma
2.12. In particular, the set A (x,y) is upper-semicontinuous in the sense of
t · ·
Assumption3.1iii)in[12]. Itisalsoclearthat,forfixedu,v A, F (u,v,x,y)
t · ·
∈
is a Borel function on [0,T] C [0,T];Rd+1 .
×
We nowpresentsomemomentestimateswhichwill,inparticular,guarantee
(cid:0) (cid:1)
tightness for the family of the laws of (Xπ,Yπ), π Π in C [0,T];Rd+1 .
∈
Proposition 2.13. For the ease of reference we denote ζ (cid:0)= (Y ,X ). U(cid:1)nder
t t t
Assumption 2.2, for any m>0,
supE sup ζπ m < . (23)
π | t | ∞
π∈Π (cid:20)t6T (cid:21)
Proposition 2.14. Under Assumption 2.2, let π Π and Yπ,Xπ its associ-
∈ t t
ated processes solving (15) and (16). Then, there exists a constant K > 0 not
(cid:0) (cid:1)
depending on π Π, such that for any η >0 and s,t [0,T],
∈ ∈
Eπ ζt ζs η 6K t s η2. (24)
k − k | − |
See the Appendix for a standard proof of both propositions above. A well-
known result on tightness of measures on C [0,T];Rd+1 gives the following
corollary. This could also be obtained by the method of Theorem 3.2 in [13].
(cid:0) (cid:1)
Corollary 2.15. Let Assumption 2.2 be in force. Let π Π. The set of
n
{ } ⊂
laws of the process ζπn on C [0,T];Rd+1 is relatively weakly compact. (cid:3)
·
Now we restate Theorem(cid:0)3.2 of [12] in(cid:1)our setting, which will provide weak
compactness of the distributions of weak controls.
Theorem 2.16. Let Assupmtion 2.2 be in force. Denote by Qπ the distribution
of ζπ on C [0,T];Rd+1 . Then the set Qπ : π Π is sequentially weakly
· ∈
compact: for any sequenceπ Πthereis asubsequencen(m) as m
(cid:0) (cid:1) n ∈ (cid:8) (cid:9) →∞ →∞
and a π Π such that for any real-valued, bounded, continuous function H(x)
·
∈
on C [0,T];Rd+1 we have
(cid:0) (cid:1) lim EνmH(ζνm)=EπH(ζπ), (25)
· ·
m→∞
where ν =π .
m n(m)
9
Proof. It follows from the above discussions that Assumption 3.1 ii) and iii)
in [12] hold in the present case. One does not have Assumption 3.1 i) of [12]
though (linear growth condition on A (X,Y) ), there is a quadratic growth
t · ·
k k
instead, see (21). But, as Corollary 2.15 shows, this is still sufficient to get
tightness (and hence relative weak compactness) of the sequence Qπn in our
setting. Then one cancheck that the proofofTheorem3.2 in[12] goes through
and we can conclude.
The next lemma shows that, to any auxiliary control π in the sense of Def-
inition 2.7, we can associate an investment strategy (in the sense of Definition
2.2) with higher value function.
Lemma 2.17. Let
π = Ωπ, π, π ,Pπ, Xπ,Yπ , Bπ,Wπ ,(x,y) Π.
F Ft t>0 ∈
(cid:16) (cid:8) (cid:9) (cid:0) (cid:1) (cid:0) (cid:1) (cid:17)
Then a solution to
dY =ν (Y)dt+κ (Y)dB , Y =y, (26)
t t · t · t 0
dXˆ =√m θ (Y)Xˆ dt+√m λ (Y)Xˆ dW , X =x, (27)
t t t · t t t · t t 0
exists on the same filtered probability space and Xˆ Xπ a.s. Furthermore, Xˆ
T ≥ T t
is a portfolio value process.
Proof. Let us define
t
Z :=exp lπ mπ Xπ,Yπ θ Yπ ds
t − s − s · · ·
(cid:18) Z0 (cid:16) p (cid:17)(cid:0) (cid:1) (cid:0) (cid:1) (cid:19)
and set Xˆ := Z Xπ. Itoˆ’s formula shows that Xˆ indeed verifies (27). Since
t t t t
θ 0 was assumed, we get that Z 1 hence Xˆ Xπ, for all t.
t ≥ t ≥ t ≥ t
2.5 Proof of Theorem 2.4
Proof. Let t>0. By (11) and (10),
t γ
w Qπ u (Xπ Gπ) >t 6g Qπ (Xπ Gπ)α > 1 .
+ + T − + + T − + k −
(cid:20) (cid:18) + (cid:19)(cid:21)
(cid:0) (cid:0) (cid:0) (cid:1) (cid:1)(cid:1)
Hence,
γ
V (π) 6g ∞ Qπ (Xπ Gπ)α > t 1 =
+ + 0 T − + k+ −
=g 1R+ h∞ Q(cid:16)π (Xπ Gπ)α > t (cid:17)i1 γ ,
+ k+ T − + k+ −
∞ Qπ (Xπ Gπ)α(cid:16)> tR 1h γ(cid:16)dy 6k ∞ Qπ (Xπ(cid:17)iG(cid:17)π)α >s γdx.
T − + k − + T − +
Zk+ (cid:20) (cid:18) + (cid:19)(cid:21) Z0
(cid:2) (cid:0) (cid:1)(cid:3)(28)
10