INTERNATIONAL CENTRE FOR ECONOMIC RESEARCH
WORKING PAPER SERIES
Michael Mania and Marina Santacroce
EXPONENTIAL UTILITY MAXIMIZATION UNDER PARTIAL INFORMATION
Working Paper no. 24/2008
APPLIED MATHEMATICS AND QUANTITATIVE METHODS WORKING PAPER SERIES
Exponential Utility Maximization
under Partial Information
Michael Mania1 and Marina Santacroce2
1A. Razmadze Mathematical Institute, M. Aleksidze St. 1, Tbilisi, Georgia and Georgian-American
University, 3, Alleyway II, Chavchavadze Ave. 17, A, Tbilisi, Georgia. e-mail: misha.mania@gmail.com
2 Department of Mathematics, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino,
Italy. e-mail: marina.santacroce@polito.it
Abstract
We consider the exponential utility maximization problem under partial information. The underlying asset price process follows a continuous semimartingale and strategies have to be con-structed when only part of the information in the market is available. We show that this problem is equivalent to a new exponential optimization problem, which is formulated in terms of observable processes. We prove that the value process of the reduced problem is the unique solution of a backward stochastic differential equation (BSDE), which characterizes the optimal strategy. We examine two particular cases of diffusion market models, for which an explicit solution has been provided. Finally, we study the issue of sufficiency of partial information.
Key words: Backward stochastic differential equation; semimartingale market model; exponential utility maximization problem; partial information; sufficient filtration.
MSC: 90A09; 60H30; 90C39.
JEL Classification: C61; G11.
1
Introduction
Investors acting in the market often have only limited access to the information flow. Besides, they may not be able or may not want to use all available information even if they have access to the full market flow. In such cases investors take their decisions using only part of the market information. We study the problem of maximizing the expected exponential utility of terminal net wealth, when the asset price process is a continuous semimartingale and the flow of observable events does not necessarily contain all information on the underlying asset’s prices.
We assume that the dynamics of the price process of the asset traded on the market is described by a continuous semimartingale S = (St, t ∈ [0, T]) defined on a filtered probability space (Ω,A,A =
(At, t∈[0, T]), P), satisfying the usual conditions, whereA =AT andT <∞is a fixed time horizon. Suppose the interest rate to be equal to zero and the asset price process to satisfy the structure condition, i.e., the process S admits the decomposition
St=S0+Nt+ Z t
0
whereN is a continuousA-local martingale and λis aA-predictable process. LetG be a filtration smaller than A
Gt⊆ At, for every t∈[0, T].
The filtration G represents the information that the investor has at his disposal. Hence, hedging strategies have to be constructed using only information available inG.
LetU(x) =−e−αx be an exponential utility function, where α >0 is a fixed constant. We consider the exponential utility maximization problem with random payoffH at time T,
to maximize E[−e−α(x+
RT
0 πudSu−H)] over all π ∈Π(G), which is equivalent to the problem (without loss of generality we can take x= 0)
to minimize E[e−α(R0TπudSu−H)] over all π∈Π(G), (2) where Π(G) is a certain class of G-predictable S-integrable processes (to be specified later) and (Rt
0πudSu, t∈[0, T]) represents the wealth process related to the self-financing strategyπ.
The utility maximization problem with partial information has been considered in the literature under various setups. In most papers (see, e.g., [14, 11, 25, 31]) the problem was studied for market models where only stock prices are observed, while the drift can not be directly observed, i.e. under the hypothesis FS⊆G.
We also consider the case when the flow of observable events G does not necessarily contain all information on prices of the underlying asset, i.e., when S is not a G-semimartingale in general. We show that the initial problem is equivalent to a certain problem of maximizing the filtered terminal net wealth and apply the dynamic programming method to the reduced problem. Such an approach, in the context of mean variance hedging, was considered in [18] (for the mean-variance hedging problem under partial information see also [27, 4, 24]).
Let us introduce an additional filtrationF = (Ft, t∈[0, T]), which is the augmented filtration
gen-erated byFS and G. The price processSis also aF-semimartingale and the canonical decomposition of S with respect to the filtration F is of the form (see, e.g., [13])
St=S0+
Z t
0
b
λFudhMiu+Mt, (3)
whereλbFu denotes theF-predictable projection of λand
Mt=Nt+ Z t
0
[λu−bλFu]dhNiu (4)
is aF-local martingale. BesideshMi=hNi and these brackets areFS-predictable. Let us consider the following assumptions:
A)hMiis G-predictable and dhMitdP a.e. λbF =λbG, hence for each t
B) anyG-martingale is anF-local martingale, C) the filtrationGis continuous,
D) for anyG-local martingale m(g),hM, m(g)i isG-predictable, E)H is anAT-measurable bounded random variable, such thatP- a.s.
E[eαH|FT] =E[eαH|GT]. (6)
Let us make some remarks on conditions A)-E). It is evident, that if FS ⊆ G, then G = F and conditions A), B), D) and equality (6) of condition E) are satisfied. Condition B) is satisfied if and only if the σ-algebras FtS∨Gt and GT are conditionally independent given Gt for all t∈ [0, T] (see
Theorem 9.29 in Jacod (1979)). Recall that Condition C) means that all G-local martingales are continuous. Under condition B) the continuity of F implies the continuity of G, but not vice-versa. So, the filtration F may be not continuous in general. Equality (6) is satisfied if, e.g., H is of the form f(η, ξ), where η is a GT-measurable random variable and ξ is AT-measurable random variable
independent of FT.
We shall use the notationYbtfor theG-optional projection ofYt(note that under the presence
con-ditions for all processes we considered the optional projection coincides with the predictable projection and therefore we use for them the same notation). Under condition A) S admits the decomposition
St=S0+
Z t
0
b
λudhMiu+Mt, (7)
whereλbt=bλGt. Moreover, condition A) implies that
b St=E(St|Gt) =S0+ Z t 0 b λudhMiu+Mct, (8)
whereMct is the G-local martingale E(Mt|Gt).
Under these assumptions, we show in Proposition 3.1 that the initial optimization problem (2) is equivalent to the problem
to minimize E[e−α( RT 0 πudSbu−He)+α 2 2 RT 0 π 2 u(1−κ2u)dhMiu] (9) over all π∈Π(G), where
κ2t = dhMcit dhMit and He = 1 αlnE(e αH|G T).
To prove the main result we will also make the following assumption:
F)R0T bλ2tdhMit≤C, P −a.s..
We prove (Theorem 1) that under assumptions A)-F) the value process V of the problem (9) is the unique bounded strictly positive solution of the following BSDE
Yt=Y0+ 1 2 Z t 0 (ψuκ2u+bλuYu)2 Yu dhMiu+ Z t 0 ψudMcu+Lt, YT =E[eαH|GT], (10)
whereLis aG-local martingale orthogonal toMc. To show the existence of a solution of this equation
we use recent results of [30] and [21] on BSDEs with quadratic growth and driven by martingales. Moreover the optimal strategy exists and is equal to
πt∗ = 1
α(λbt+ ψtκ2t
Yt
). (11)
We also examine two particular market diffusion models.
In section 4 we consider a market model with one risky asset, when the drift of the return process S
changes value from 0 toµ6= 0 at some random timeτ
dSt=µI(t≥τ)dt+σdWt.
The change-pointτ admits a known prior distribution, although the variableτ itself is unknown and it cannot be directly observed. Agents in the market have only knowledge of the measurement processS
and not of the Brownian motion and of the random variableτ. In this case we give an explicit solution of problem (2) in terms of the a posteriori probability process pt =P(τ ≤ t|FtS) which satisfies the
stochastic differential equation (SDE)
pt=p0+ µ σ Z t 0 pu(1−pu)dfWu+γ Z t 0 (1−pu)du.
For this and for the formulation of the change-point problem, the reader is referred to [28].
In section 5, we consider a diffusion market model consisting of two risky assets with the following dynamics
dSt=µ(t, η)dt+σ(t, η)dWt1,
dηt=b(t, η)dt+a(t, η)dWt,
where W1 and W are standard Brownian motions with correlation ρ. η represents the price of a nontraded asset (e.g. an index) andS denotes the process of returns of the tradable one. We consider problem (2): an agent is hedging a contingent claimH trading with the liquid assetS and using only the information onη. Under suitable conditions onµ, σ, b, a andH, namely 1)-4) of section 5, we give an explicit expression of the optimal amount of money which should be invested in the liquid asset.
In section 6 we study the issue of sufficiency of partial information of the optimization problem (2). The filtration Gis said to be sufficient (for A) if
inf π∈Π(A)E[e −α(RT 0 πudSu−H)] = inf π∈Π(G)E[e −α(RT 0 πudSu−H)]. (12) We give conditions (Proposition 5 and Theorem 2) which guarantee the sufficiency of filtration G.
2
Main definitions and auxiliary facts
Denote byMe(F) the set of equivalent martingale measures forS, i.e., the set of probability measures
Q equivalent to P such that S is a F-local martingale under Q. For any Q∈ Me(F), let Z
the density process (with respect to the filtration F) ofQ relative toP.
It follows from (7) that the density process Zt(Q) of any element Q of Me(F) is expressed as an
exponential martingale of the form
Et(−bλ·M+L),
whereLis aF-local martingale strongly orthogonal toM andEt(X) is the Dol´eans-Dade exponential
of X. If condition F) is satisfied, the local martingale Ztmin = Et(−bλ·M) is a true martingale,
dQmin/dP =ZTmin defines the minimal martingale measure forS on F. Let us define the classes of admissible strategies
Π(F) ={π :F−predictable, π·M ∈BM O(F)},
Π(G) ={π :G−predictable, π·M ∈BM O(F)}.
It is evident that Π(G)⊆Π(F).
Remark 1 Note that if π ∈Π(G), thenπ·Mc∈BM O(G). Indeed, sinceκ2 ≤1 (see, e.g., [18]) for
any G-stopping timeτ (which is alsoF-stopping time)
E(hπ·MciT − hπ·Mciτ|Gτ) =E( Z T τ πu2κ2udhMiu|Gτ)≤ ≤E(E( Z T τ π2udhMiu|Fτ)|Gτ)≤ ||π·M||2BM O(F).
Under conditions A)-C) the process Mct=E(Mt|Gt) admits the representation (see, e.g., Proposition
2.2 of [18]) c Mt= Z t 0 \ (dhM, m(g)iu dhm(g)iu )dmu(g) +Lt(g), (13)
where m(g) is any G-local martingale and L(g) is a G-local martingale orthogonal to m(g). In particular, if condition D) is also satisfied, then
c Mt= Z t 0 dhM, m(g)iu dhm(g)iu dmu(g) +Lt(g), (14) and hM, m(g)it=hM , mc (g)it (15)
for any G-local martingale m(g).
Lemma 1 Let conditions A)-D) be satisfied. Then
E(Et(M)|Gt) =Et(Mc), (16)
Proof. The process E(Et(M)|Gt) is a strictly positive G-local martingale and there exists aG-local
martingale Mfsuch that
E(Et(M)|Gt) =Et(Mf). Therefore f Mt= Z t 0 1 E(Eu(M)|Gu) dE(Eu(M)|Gu). (17)
From (13), applied to the continuous F-local martingale
Et(M) = 1 + Z t 0 Eu(M)dMu we have E(Et(M)|Gt) = 1 + Z t 0 E Eu(M)dhM, m(g)iu dhm(g)iu |Gu dmu(g) +Let,
whereLe is aG-local martingale orthogonal to m(g). Therefore, by condition D)
E(Et(M)|Gt) = 1 + Z t 0 E Eu(M)|Gu dhM, m(g)iu dhm(g)iu dmu(g) +Let and from (17) we obtain that
f Mt= Z t 0 dhM, m(g)iu dhm(g)iu dmu(g) + Z t 0 1 E(Eu(M)|Gu) dLet.
Taking the mutual characteristics with respect tom(g) of the both sides of this equality we have hM , mf (g)it=hM, m(g)it
and by (15)
hM , mf (g)it=hM , mc (g)it
for any G-local martingale m(g). By the hypothesis hMi is G-predictable, we know that M can be localized by G-stopping times and this implies that Mc is a G-local martingale. Since also Mf is a
G-local martingale, by arbitrariness of m(g), we have that McandMf are indistinguishable.
Corollary 1 If conditions A)-D) are satisfied, then for any π ∈Π(G)
E(Et(π·M)|Gt) =Et(\π·M) =Et(π·Mc).
For all unexplained notations concerning the martingale theory used below we refer the reader to [3, 10, 15].
3
Utility Maximization Problem
Let us introduce the value process of problem (9):
Vt= ess inf π∈Π(G)E[e −α(RT t πudSbu−He)+α 2 2 RT t π2u(1−κu2)dhMiu|G t], (18)
where, we recall that,
e H= 1 αlnE[e αH|G T], κ2t = dhMcit dhMit . Let Vt(G) = ess inf π∈Π(G)E[e −α(RT t πudSu−H)|Gt] (19) be the value process of the problem (2).
Proposition 1 Let conditions A)-E) be satisfied. Then Vt=Vt(G) and the problems (2) and (9) are
equivalent. Moreover, for any π ∈Π(G)
E[e−α( RT t πudSu−H)|G t] =E[e−α( RT t πudSbu−He)+α 2 2 RT t πu2(1−κu2)dhMiu|G t].
Proof. Taking the conditional expectation with respect to FT and using condition E), we have that
E[e−αRtTπudSu+αH|Gt] =E e−αRtTπudSuE[eαH|FT]|Gt =E e−α( RT t πudSu−He)| Gt . (20)
Let s ∈ [0, T],t ≥ s and denote by Est(M) = EEt(s(MM)). Using successively decomposition (7),
condition A), Lemma 1 for the strictly positive G-local martingale {E(Est(M)|Gt), t ≥s} with s∈
[0, T],decomposition (8) ofSband equalityκ2t =dhMcit/dhMit, we have
E[e−α(RtTπudSu−He)| Gt] =E[e−α RT t πudMu−α 2 2 RT t π2udhMiu+α 2 2 RT t π2udhMiu−α RT t πubλudhMiu+αHe| Gt] =E[EtT(−απ·M)eα 2 2 RT t π 2 udhMiu−αRtTπubλudhMiu+αHe| Gt] =E[E(EtT(−απ·M)|GT)e α2 2 RT t πu2dhMiu−αRtTπubλudhMiu+αHe| Gt] =E[EtT(−απ·Mc)e α2 2 RT t π 2 udhMiu−α RT t πubλudhMiu+αHe| Gt] =E[e−α RT t πudMcu−α 2 2 RT t π2udhcMiu+α 2 2 RT t π2udhMiu−αRtTπubλudhMiu+αHe| Gt] =E[e−α RT t πudSbu+α 2 2 RT t π 2 u(dhMiu−dhMciu)+αHe| Gt] =E[e−α( RT t πudSbu−He)+α 2 2 RT t πu2(1−κu2)dhMiu|G t]
Remark 2 If conditions A)-D) are satisfied and His bounded andGT-measurable, thenHe =H and problem (2) is equivalent to minimize E[e−α(R0TπudSbu−H)+α 2 2 RT 0 π 2 u(1−κ2u)dhMiu].
Remark 3 It is evident that sinceHis bounded, there will be a positive constantC,Vt≤C. Besides
if condition F) is satisfied then we have thatVt≥cfor a positive constantcdirectly by duality issues.
Indeed, if Vt(F) = ess infπ∈Π(F)E[e−α( RT
t πudSu−H)|Ft], then it is well known that under condition F),
Vt(F)≥c,(see e.g. [2] or [17]). Therefore
Vt= ess inf π∈Π(G)E[e −α(RT t πudSu−H)|Gt] = (21) ess inf π∈Π(G)E h Ee−α(RtTπudSu−H)|Ft |Gt i ≥E(VtF|Gt)≥c. (22)
Lemma 2 The martingale part of any bounded strictly positive solution of (10) is in BM O.
Proof. LetY be a bounded strictly positive solution of (10). By Ito’s formula and the boundary condition we write
YT2−Yτ2=E2(eαH|GT)−Yτ2= Z T τ 2Yt(ψtdMct+dLt) + Z T τ (ψtκ2t +λbtYt)2dhMit +hψ·MciT − hψ·Mciτ+hLiT − hLiτ,
whereτ is aG-stopping time. Without loss of generality we may assume thatψ·Mc+Lis a square
inte-grable martingale, otherwise we can use localization arguments. After taking conditional expectation, we see that
E[hψ·MciT − hψ·Mciτ|Gτ] +E[hLiT − hLiτ|Gτ]≤C, (23)
which implies assertion of the Lemma. Theorem 1 Let H be a bounded AT-measurable random variable, hMi be G-predictable and let
con-ditions B), C) and F) be satisfied. Then the value process Vt is the unique solution of the BSDE
Yt=Y0+ 1 2 Z t 0 (ψuκ2u+bλuYu)2 Yu dhMiu+ Z t 0 ψudMcu+Lt, (24) YT =E(eαH|GT), hM , Lc i= 0
in the class of processes satisfying the two-sided inequality
c≤Yt≤C, (25)
where cand C are strictly positive constants. Besides the optimal strategy of the problem (9) exists in the class Π(G) and is equal to
πt∗ = 1
α(λbt+ κ2tψt
Yt
). (26)
If conditions A)-F) are satisfied, then Vt=Vt(G) and π∗ defined by (26) is the optimal strategy also
Proof. LetYt be a strictly positive process satisfying Equation (24).
Writing Ito’s formula for the processZt= lnYt, we find thatZ satisfies
dZt= 1 2 h ( ˜ψtκ2t+bλt)2−ψ˜2tκ2t i dhMit−1 2dh ˜ Lit+ ˜ψtdMct+dL˜t, (27)
with the boundary conditionZT = lnE(eαH|GT) =αHe. Recall that we denoted byHe = α1 lnE(eαH|GT).
In the previous equation we set ˜ψt= Y1tψtand ˜Lt= (Y1·L)t. The existence of a solution of the previous
BSDE follows from [30] or [21], where new results on the existence and uniqueness of solutions are proved for BSDEs with quadratic growth driven by continuous martingales.
Now we should show that the solution is unique and coincides with the value processV and that the strategy π∗ is optimal.
For anyπ ∈Π(G),let us denote the process e−αR0tπudSbu+α 2 2
Rt
0πu2(1−κu2)dhMiu by J
t(π).
Let us considerYt,solution of (24) satisfying (25). By using Ito’s formula for the productYtJt(π),
d(YtJt(π)) =YtJt(π) −απtdSbt+ α2 2 π 2 t(1−κ2t)dhMit+ α2 2 π 2 tdhMcit +Jt(π) ψtdMct+dLt+ 1 2 (bλtYt+ψtκ2t)2 Yt dhMit ! −Jt(π)ψtαπtκ2tdhMit and d(YtJt(π)) =YtJt(π) (ψt Yt −απt)dMct+ 1 Yt dLt+ 1 2(απt−(bλt+ ψt Yt κ2t))2dhMit) . Therefore YtJt(π) =Y0Et(( ψ Y −απ)·Mc+ 1 Y ·L)e 1 2 Rt 0(απu−(bλu+ψu Yuκ 2 u))2dhMiu). (28) In Equation (28), Y J(π) is written as the product of a strictly positive increasing process and a uniformly integrable martingale. In fact, by (23) and since π∈Π(G) implies thatπ·Mc∈BM O(G)
(see Remark 2.1), the process (Yψ −απ)·Mc+Y1 ·L) is a BMO(G)-martingale, thus, the exponential
martingale Et((Yψ −απ)·Mc+ 1
Y ·L) is a uniformly integrable martingale by [12]. Therefore, Y J(π)
is a submartingale. Using the boundary condition YT =E(eαH|GT), we find
YtJt(π)≤E(JT(π)eαHe|Gt), a.s. hence Yt≤E(e−α RT t πudSbu+α 2 2 RT t π 2 u(1−κ2u)dhMiu+αHe| Gt) a.s..
This is true for any π∈Π(G), and, recalling the definition of Vt, we immediately see that
Yt≤ ess inf π∈Π(G)E[e −αRT t πudSbu+α 2 2 RT t πu2(1−κ2u)dhMiu+αHe| Gt] =Vt a.s.. (29)
Let us take ˜πu =
b
λuYu+ψuκ2u
αYu . By Ito’s formula for YtJt(˜π),similarly to (28), we find that the process is a strictly positive local martingale hence a supermartingale. The supermartingale property and the boundary condition give
Yt≥E[e−α RT t π˜udSbu+α 2 2 RT t π˜ 2 u(1−κ2u)dhMiu+αHe| Gt] a.s.. (30)
Let us show that ˜π belongs to the class Π(G).
From condition F) follows that bλ·M is inBM O(F) and by (23) (since Y ≥c and κ2t ≤1) we have
that ψκY2 ·M is also in BM O(F). Thus 1
α(λb+ ψκ2
Y )· M ∈BM O(F)
which implies that ˜π∈Π(G). Therefore,
E[e−αRtTπ˜udSbu+α 2 2 RT t ˜πu2(1−κu2)dhMiu+αHe |Gt]≥Vta.s.. (31)
Hence from (29), (30) and (31) we have that
Yt=Vt, a.s.. (32)
Therefore, (30), (31) and (32) give the equality
E[e−αRtTπ˜udSbu+α 2 2 RT t ˜πu2(1−κ2u)dhMiu+αHe| Gt] =Vta.s..
which means that ˜π=π∗ is optimal. It follows from Proposition 3.1 that if A)-F) are satisfied, then
Vt=Vt(G) and π∗ is the optimal strategy of the problem (2).
Remark 4 In terms of Equation (27) the optimal strategy is expressed as
πt∗= 1
α(bλt+ψetκ
2
t). (33)
Remark 5 One can see from the proof of the unicity that condition F) can be replaced by the weaker condition bλ·M is in BM O(F).
If FtS ⊆ Gt, then hMi is G-predictable, since it is FS-predictable. Besides, Gt = Ft ≡FtS ∨Gt
and conditions A), B), D) and equality (6) of condition E) are satisfied. So, in this case, Mct = Mt
and κ2t = 1 for all t∈[0, T]. SinceFS ⊆G,S is aG-semimartingale with canonicalG-decomposition
St=S0+
Z t
0
b
λudhMiu+Mt, M ∈ Mloc(G) (34)
and for any π∈Π(G)
E e−α( RT t πudSu−H)|Gt =E e−α( RT t πudSu−He)| Gt ,
withHe = α1 lnE(eαH|GT), whereHe =H ifH isGT-measurable.
So, taking the G-decomposition (34) in mind, problem (2) is equivalent to the problem to minimize E(e−α(
RT
0 πudSu−He)
) over all π∈Π(G). (35) Thus, we have the following corollary of Theorem 1:
Corollary 2 Let FS ⊆ G ⊆ A and let conditions C), F) be satisfied and H be a bounded AT -measurable random variable. Then, the value process V is the unique solution of the BSDE
Yt=Y0+ 1 2 Z t 0 (ψu+λbuYu)2 Yu dhMiu+ Z t 0 ψudMu+Lt, YT =E(eαH|GT), (36)
satisfying 0< c≤Yt≤C. Moreover, the optimal strategy is equal to
π∗t = 1
α(bλt+ ψt
Yt
). (37)
Remark 6 Note that in the case of full information Gt=At, we additionally have
c
Mt=Mt=Nt, λbt=λt, YT =eαH
and Equation (36) takes the form
Yt=Y0+ 1 2 Z t 0 (ψu+λuYu)2 Yu dhNiu+ Z t 0 ψudNu+Lt, YT =eαH. (38)
Equations of type (38) (or equivalent to (38)) were derived in [8, 9, 29] for BSDEs driven by Brownian motion and in [17] for BSDEs driven by martingales.
4
Application to the disorder problem
As an example we consider a market with one risky asset, where the drift of this asset changes value (from 0 to µ,µ6= 0) at a random time, which cannot be directly observed.
Let (Ω,A,A = (At, t ∈ [0, T]), P), be a filtered probability space, where A = AT, hosting a
Brownian motion W and a random variable τ with distribution
P(τ = 0) =p and P(τ > t|τ >0) =e−γt, for all t∈[0, T]
for some known constants p∈[0,1[ andγ >0. The Brownian motion W and the random variable τ
are assumed to be independent.
The dynamics of the asset priceSeis determined by the SDE
dSet=Set(µI(t≥τ)dt+σdWt),
whereµ6= 0 and σ2>0. By dSt= dSet
e
St ,we introduce the return process which satisfies
dSt=µI(t≥τ)dt+σdWt. (39)
So, agents in the market do not observe the Brownian motion and the random instantτ, but only the measurement process S (orSe). The filtration Gis generated by the observations process Gt=FtS =
σ(Su,0≤u≤t). So, we have
We consider the optimization problem (2), where the strategyπtis interpreted as a dollar amount
invested in the stock at time t. Assume that the contingent claim H is of the form H = g(ST, ζ),
where g is a positive bounded function of two variables and ζ is an AT-measurable random variable independent of FS
T. ζ can be the terminal value of a non traded asset.
Let pt=P(τ ≤t|FtS) be the a posteriori probability process. With respect to the filtrationG=FS
the processS admits the decomposition
St=S0+µ Z t 0 pudu+σfWt, (40) where fWt= 1 σ(St−S0−µ Rt
0pudu) is a Brownian motion, an innovation process, with respect to the filtrationFS (see, e.g., [16]).
The measure Qdefined by
dQ=ET(−µ
σp·Wf)dP
is a martingale measure for S on FS. It is evident that F) is satisfied.
It follows from [28] that the process pt satisfies the stochastic differential equation
pt=p0+ µ σ Z t 0 pu(1−pu)dfWu+γ Z t 0 (1−pu)du. (41)
Moreover, (pt, FtS) is a strong Markov process.
In this case Nt=σWt, Mt=Mct=σWft, κ2t = 1 λt= µ σ2I(τ≤t), bλt= µ σ2pt. By Corollary 2, the value process of problem (35) satisfies the following BSDE
Yt=Y0+ 1 2 Z t 0 (σψu+µσpuYu)2 Yu du+ Z t 0 σψudfWu, YT =E(eαH|FTS), (42)
withLt= 0, since anyFS-local martingale is representable as a stochastic integral with respect toWf
(see, e.g., Theorem 5.17 in [16]). Besides, the process Zt= lnYtis the solution of the linear equation
(here we set ψe= ψY) Zt=Z0+ Z t 0 (µpuψeu+ µ2 2σ2p 2 u)du+ Z t 0 σψeudWfu, ZT = lnE(eαH|FTS). (43)
Since µptdt+σdWft=dSt, (43) can be written in the following equivalent form
Zt=Z0+ 1 2 µ2 σ2 Z t 0 p2udu+ Z t 0 e ψudSu, ZT = lnE(eαH|FTS). (44)
The solution of this equation is expressed as
Zt=E EtT(− µ σp·fW)(lnE(e αH|FS T)− µ2 2σ2 Z T t p2udu)|FtS. (45)
Besides, takingt=T in Equation (44), one can find the integrandψeby the martingale representation property lnE(eαH|FTS)− µ 2 2σ2 Z T 0 p2udu=c+ Z T 0 e ψudSu. (46)
Since the left side of (46) is the difference of twoFTS-measurable random variables, by the martingale representation theorem, there exist two predictableS-integrable processes ψe(1) andψe(2), such that
lnE(eαH|FTS) =c1+ Z T 0 e ψu(1)dSu (47) − µ 2 2σ2 Z T 0 p2udu=c2+ Z T 0 e ψu(2)dSu. (48)
Thus, the logarithm of the value process is the sum of two parts Zt(1) and Zt(2)
Zt=EQ lnE(eαH|FTS)|FtS −EQ µ 2 2σ2 Z T t p2udu|FtS =Zt(1) +Zt(2).
With regard toZt(1), we first observe that
E(eαH|FTS) =E(eαg(ζ,ST)|FTS) =f(ST), where f(S) =E(eαg(ζ,ST)|ST =S) = Z ∞ 0 eαg(u,S)dFζ(u).
In the last equality we used the independence ofζ and ST. Note thatf >1, sinceg is positive. Since
St=S0+σWtandWt=Wft+µσ Rt
0puduis aQ-Brownian motion, by the Markov property ofS under (Q, FS) we easily see that
EQ lnE(eαH|FTS)|FtS=EQ lnf(ST)|St and G(t, x)≡EQ(lnf(ST)|St=x) = 1 p 2πσ2(T−t) Z R lnf(y)e− (y−x)2 2σ2(T−t)dy. (49) Since G∈C1,2([0, T[×R) andZ
t(1) is aQ-martingale, by Ito’s formula
Zt(1) =G(t, St) =G(0, S0) + Z t
0
Gx(t, St)dSt. (50)
Comparing (50) and (47) we see that ψet(1) =Gx(t, St), dP dt a.e..
While, with regard to Zt(2), let us observe that under the measure Q the process pt satisfies the
SDE pt=p0+ µ σ Z t 0 pu(1−pu) 1 σdSu+ Z t 0 (1−pu)(γ− µ2 σ2p 2 u)du, (51)
ThereforeZt(2) can be represented as Zt(2) =− µ2 2σ2U(t, pt), where U(t, x) =EQ( Z T t p2udu|pt=x). (52)
LetR(t, x) be a solution of the linear PDE
Rt(t, x) + µ2 2σ2x 2(1−x)2R xx(t, x) + γ− µ2 σ2x 2 (1−x)Rx(t, x) +x2= 0, R(T, x) = 0, (53)
where Rt, Rx and Rxx are partial derivatives ofR. The existence of a solution of (53) from the class
∈C1,2([0, T]×(0,1)) follows from [23].
It follows from the Ito formula forR(t, pt), taking in mind thatR(t, x) solves equation (53), that
the process R(t, pt) + Z t 0 p2udu=R(0, p0) + µ σ2 Z t 0 Rx(u, pu)dSu (54)
is a martingale under Q. Using the martingale property and the boundary condition
R(t, pt) =EQ( Z T
t
p2udu|Ftp)
and from (52) we obtain that R(t, pt) =U(t, pt). Comparing the Qmartingale parts of (48)and (54)
we obtain that dP dta.e..
e
ψt(2) =−
µ3
2σ4pt(1−pt)Rx(t, pt). Thus, we proved
Proposition 2 The value process of problem (35) is equal to
Vt=eG(t,St)−
µ2 2σ2R(t,pt)
and the optimal strategy π∗ is equal to
π∗t = µpt ασ2(1− µ2 2σ2(1−pt)Rx(t, pt)) + 1 αGx(t, St),
where G(t, x) is defined by (49),R(t, x) satisfies the linear PDE (53) andptis a solution of SDE (51).
Remark 7 The optimal wealth process is the sum of two components, a hedging fund 1
α
Z t
0
Gx(u, Su)dSu
(which is zero ifH = 0) and an investment fund
µ ασ2 Z t 0 pu 1− µ2 2σ2(1−pu)Rx(u, pu) dSu.
This fact is a consequence of wealth independent risk aversion of the exponential utility function, which makes the investors behavior not depending on his initial endowment.
Remark 8 IfH= 0 andτis deterministic,τ =t0, thenRx = 0 and the optimal strategy is ασµ2I(t≥t0). If t0 = 0, then we obtain Merton’s optimal strategy π∗= ασµ2,(see [19]).
5
Diffusion model with correlation.
In this section we deal with a model with two assets one of which has no liquid market. First we consider an agent who is finding the best strategy to hedge a contingent claim H, trading with the liquid asset but using just the information on the non tradable one. Then we specialize the result to the case when the model is Markovian andH depends only on the nontraded asset’s price at time T. Finally, we solve the problem for full information whenH is function only of the nontraded asset and compare the obtained result.
Let S and η denote respectively the return of the tradable and the price of the non tradable assets. We assume S and η have the following dynamics
dSt=µ(t, η)dt+σ(t, η)dWt1, (55)
dηt=b(t, η)dt+a(t, η)dWt, (56)
whereW1 and W are correlated Brownian motions with constant correlation ρ∈(−1,1). The coefficients µ,σ,a andb are non anticipative functionals such that
1)R0T µσ22((t,ηt,η))dt is bounded, 2)σ2 >0, a2 >0,
3) equation (56) admits a unique strong solution,
4) H is bounded and measurable with respect to the σ-algebra FTη ∨σ(ξ), where ξ is a random variable independent of S.
Note that under conditions 2) and 3),FS,η=FW1,W,Fη =FW. So, we will have
Ft=FtS,η⊆ At and Gt=Ftη.
Here FW1,W (resp. FW) is an augmented filtration generated by W1 and W (resp. W). So the filtrationFtη is continuous and the condition C) is satisfied.
Since FT =FTS,η, assumption 4) implies that condition E) is fulfilled.
Condition F) is here represented by 1). We consider the optimization problem
to minimize E[e−α(
RT
0 πudSu−H)] over all π ∈Π(Fη), (57) where π represents the dollar amount the agent invests in the stock and, constructing the optimal strategy, he uses only the information based onη.
Let us observe that
Mt= Z t 0 σ(u, η)dWu1, hMit= Z t 0 σ2(u, η)du, λt= µ(t, η) σ2(t, η).
We denote the market price of risk by
θt=
µ(t, η)
σ(t, η). (58)
It is convenient to think of W as a linear combination of two independent Brownian motions W0 and
W1, thus
Wt=ρWt1+ p
1−ρ2W0
t. (59)
It is evident thatW is a Brownian motion also with respect to the filtration FW0,W1 (which is equal
toFW1,W) and condition B) is satisfied. Therefore, by Proposition 2.2 in [18],
c Mt=ρ Z t 0 σ(u, η)dWu. Hence, hMcit=ρ2 Rt 0σ 2(u, η)du=ρ2hMi
t and κ2t =ρ2, therefore Sbsatisfies
dSbt=µ(t, η)dt+ρσ(t, η)dWt.
Since any Fη local martingale is represented as a stochastic integral with respect to W and σ(t, η) is
Ftη-adapted, condition D) is also satisfied.
Therefore, it follows from Proposition 1, that under conditions 1)-4) the optimization problem (57) is equivalent to the problem
to minimize E[e−α( RT 0 πudSbu−He)+α 2 2 (1−ρ 2)RT 0 π 2
uσ2(u,η)du] over all π ∈Π(Fη), (60) whereHe = 1
αlnE(e αH|Fη
T).
We shall show that in this case problem (60) admits an explicit solution. LetQe be the measure defined by
dQe
dP =ET(−ρ θ·W). (61)
Note that Qe is a martingale measure forSb(onFη)and by Girsanov’s theorem, under the measurefQ,
f Wt=Wt+ρ Z t 0 θudu is a Brownian motion.
Proposition 3 Let us assume conditions 1)-4). Then, the value process related to problem (60) is equal to Vt= EQe [e(1−ρ2)(αHe−1 2 RT t θ2udu)|Fη t] 1 1−ρ2 . (62)
Moreover, the optimal strategy π∗ is identified by
π∗t = 1 ασ(t, η) θt+ ρht (1−ρ2)(c+Rt 0hudWfu) , (63)
where ht is the integrand of the integral representation
e(1−ρ2)(αHe−1 2 RT 0 θt2dt)=c+ Z T 0 htdfWt. (64)
Proof. Conditions 1)-4) imply that the assumptions A)-F) are satisfied. Thus, according to Theorem 1, the value process Vtrelated to the problem (60) is the unique bounded solution of Equation (24). In
this case, after the change of variables x≡ρσψ, the BSDE (24) can be written
Yt=Y0+ 1 2 Z t 0 [(θuYu+xuρ) 2 Yu ]du+ Z t 0 xudWu, YT =E(eαH|FTη). (65)
Besides, the equation for the logarithm of the value process takes the following form
dZt=− 1 2[ϕ 2 t(1−ρ2)−2ϕtρθt−θt2]dt+ϕtdWt, ZT =αH.e (66) Note thatϕt= xYtt.
Under Qe, Equation (66) becomes
dZt=−
1 2[ϕ
2
t(1−ρ2)−θt2]dt+ϕtdWft, ZT =αH.e (67)
From (67), using the boundary condition ZT =αHe, we have
Z0+ Z T 0 ϕtdfWt− 1−ρ2 2 Z T 0 ϕ2tdt=−1 2 Z T 0 θt2dt+αH.e (68)
Multiplying both parts of this equation by 1−ρ2 and taking exponentials we obtain
cET((1−ρ2)ϕ·fW) =e(1−ρ 2)(α e H−1 2 RT 0 θ2tdt), (69) wherec=e(1−ρ2)Z0. SinceRT 0 θ 2 tdtand He areF η
T-measurable andHe is bounded, by the martingale representation theorem
and Bayes rule, there exists aFη-predictable functionh, such that the martingaleEQe
e(1−ρ2)(αHe−1 2 RT 0 θ 2 udu)|Fη t
admits the representation
EQe [e(1−ρ2)(αHe−1 2 RT 0 θ 2 udu)|Fη t] =c+ Z t 0 hudWfu. (70)
Therefore, taking conditional expectations in (69),
Et((1−ρ2)ϕ·fW) = 1 +c−1 Z t 0 hudWfu. (71) Thus, (ϕ·Wf)t= (1−ρ2)−1 Z t 0 (c+ Z s 0 hudWfu)−1hsdWfs and dP dta.e. ϕt= (1−ρ2)−1(c+ Z t 0 hudWfu)−1ht. (72)
Note that, since we changed the variables (x ≡ ψρσ) in (65), the processes ψe from (27) and ϕ are
related by the equality ψe=ϕ/ρσ. Therefore, it follows from (33) that the optimal hedging strategy
Using the martingale property of ϕ·Wf and the boundary condition, from (67) we obtain Zt=EQe αHe + 1 2 Z T t (1−ρ2)ϕ2u−θu2du|Ftη . (73)
Now let us express Ztin a more explicit form.
From (64), we have (1−ρ2)(αHe − 1 2 Z T 0 θt2dt) = ln(c+ Z T 0 hudWfu)
and, by Ito’s formula,
dln(c+ Z t 0 hudWfu) = ht c+Rt 0 hudfWu dWft− 1 2 h2t (c+Rt 0hudWfu)2 dt.
From this equation, using (72), we have (1−ρ2)−1 ln(c+ Z T 0 hudfWu)−ln(c+ Z t 0 hudWfu) = Z T t ϕudWfu− 1 2(1−ρ 2) Z T t ϕ2udu.
Taking conditional expectation with respect to the measure Qe, we find that
(1−ρ2) 2 E e Q( Z T t ϕ2udu|Ftη) =EQe (−αHe + 1 2 Z T 0 θ2udu|Ftη) + ln(c+ Z t 0 hudWfu)(1−ρ 2)−1 (74) If we substitute (74) in (73), we have that
Zt= 1 2 Z t 0 θu2du+ ln(c+ Z t 0 hudfWu)(1−ρ 2)−1
which implies that the value process of problem (60) is
Vt=e 1 2 Rt 0θ 2 udu(c+ Z t 0 hudfWu)(1−ρ 2)−1 , which is equal to (62), by (70).
Remark 9 If H = 0, like in the pure investment problem, and µ, σ are constants (or if the mean variance tradeoff RT 0 θ 2 tdtis deterministic), thenh= 0 by (70), Vt=e− 1 2θ2(T−t) and π∗ t = µ ασ2
is the optimal investment strategy for maximizing exponential utility (as in the Merton’s model), which keeps a constant dollar amount invested in the stock.
Remark 10 If H 6= 0, µ, σ are constants, but ρ = 0, it follows from (63) that the same strategy is optimal. In this case He = f(η) is independent of the traded asset and the risk is “completely
Remark 11 Ifρ2= 1, then the solution of Equation (66) gives the value process is Vt=eE e Q(α e H−1 2 RT t θ 2 udu|F η t).
The same result is obtained by taking the limit as ρ→ ±1 in (62), since for any t, we have lim ρ→±1 EQe [e(1−ρ2)(αHe−1 2 RT t θ 2 udu)|Fη t] 1 1−ρ2 =eEQe(α e H−1 2 RT t θ 2 udu|F η t).
Note that ifρ2= 1, to avoid arbitrage opportunities, the coefficients of (55), (56) must be related by
µ(t, η) =±b(t, η)
a(t, η)σ(t, η), if ρ=±1.
We move on to considering a contingent claimH =f(ηT) and we assume also the Markov structure
of the coefficients. Introduce the function
R(t, x) =EQe
[e(1−ρ2)(αf(ηT)−12 RT
t θ2(u,ηu)du)|ηt=x].
Then we can express R(t, x) as a solution of a linear PDE andh in terms of the first derivative of R. Classical results yield that, under some regularity conditions on the coefficients, R∈C1,2([0, T]×
R)
and satisfies the following PDE (see, e.g.,[5])
Rt(t, x) + 1 2a 2(t, x)R xx(t, x) +Rx(t, x)(b(t, x)−ρθ(t, x)a(t, x))−R(t, x) 1−ρ2 2 θ 2(t, x) = 0, (75)
with terminal condition R(T, x) = eα(1−ρ2)f(x), where Rt, Rx and Rxx are partial derivatives of
R. On the one hand, let us observe that
R(t, ηt) =c e
1−ρ2 2
Rt
0θ2(u,ηu)duEt((1−ρ2)ϕ·Wf), hence R(t, ηt) is solution of the following backward equation
dR(t, ηt) =R(t, ηt)[(1−ρ2)ϕtdWft+ (1−ρ2) 2 θ 2(t, η t)dt], R(T, ηT) =eα(1−ρ 2)f(ηT) . (76)
Writing the Ito formula forR(t, ηt) and comparing with (76) we obtain that
ϕt=
Rx(t, x)a(t, x)
(1−ρ2)R(t, x). Using representation (64), in an analogous way, one can write
R(t, ηt) =e 1−ρ2 2 Rt 0θ 2(u,ηu)du (c+ Z t 0 hudWfu). Thus, ht=e− 1−ρ2 2 Rt 0θ2(u,ηu)duRx(t, ηt)a(t, ηt).
The optimal strategyπ∗ in this case takes the following form π∗t = 1 ασ(t, ηt) θ(t, ηt) + Rx(t, ηt) (1−ρ2)R(t, η t) ρa(t, ηt) ,
whereR(t, x) satisfies the PDE (75).
Now we deal with the case of full information. To this end, we consider the same market model (55)-(56) consisting of two risky assets, one of which is non traded, and denote by Set the asset price
of the traded one. Thus, the dynamics of the prices are respectively:
dSet=Set(µ(t, η)dt+σ(t, η)dWt1) (77)
dηt=b(t, η)dt+a(t, η)dWt. (78)
Let the coefficients µ, σ, a and b satisfy conditions 1)-3) and let H be a bounded FTη-measurable random variable.
We consider the problem
to minimizeEe−α(
RT
0 πtdSet−H) over all π ∈Π(FS,ηe
), (79) whereπt represents the number of stocks held at timetand is adapted to the filtrationF
e
S,η t .
We assume an agent is trading with a portfolio of stocks Se in order to hedge a contingent claim
H = f(η), written on the non traded asset. Notice that the agent builds his strategy using all market information, and Gt=F
e
S,η
t =At. The market is incomplete, since η is non tradable and the
contingent claim H is not attainable by the wealth process Xtπ = x+R0tπudSeu, by means of a self
financing strategyπ.
This problem was earlier studied in [1, 6, 7, 20, 22], for the model (77)-(78) (which is sometimes also called the “basis risk model”) with constant coefficients µ and σ. In the case of constant coefficients
µ, σ, assuming the Markov structure of b and a, in [22] (see also [6]) an explicit expression for the value function related to the problem (79) is given, which is
v(t, y) = EQ[e(1−ρ2)(αf(ηT)−12(T−t)µ 2 σ2|ηt=y] 1 1−ρ2 , (80)
where the measureQ is defined by
dQ
dP =ET(−θ·W
1). (81)
Note that, although the measuresQand Qe, defined in (61), are different, the processηt has the same
law under both of them and the value process Vt of Equation (62) (in case H =f(η)) and the value
function (80) are related by the equality
Vt=v(t, ηt).
Using Theorem 1, we shall show that if H is FTη-measurable, then Vt, defined by (62), coincides
with the value process related to problem (79) also in a non Markovian setting and that the optimal amount of money depends only on the observation coming from the non traded asset.
Under assumptions 1)-3), straightforward calculations yield that
Mt=Mct= Rt 0 Seuσ(u, η)dWu1, hMit=hMcit= Rt 0Seu2σ2(u, η)du and λt = µ(t,η) e
Stσ2(t,η), besides the orthogonal part will be expressed as a stochastic integral with respect to
W0.
Thus, according to Theorem 1, the value process related to problem (79) is the unique bounded solution of the BSDE
Yt=Y0+ 1 2 Z t 0 (ψu+θuYu)2 Yu du+ Z t 0 ψudWu1+ Z t 0 ψu⊥dWu0, YT =eαH (82)
and the optimal strategy is of the form
πt∗ = 1 ασ(t, η)Set θt+ ψt Yt , (83) whereθt is defined in (58).
Proposition 4 The process
Yt= EQ[e(1−ρ2)(αH−12 RT t θu2du)|Fη t] 1 1−ρ2 (84)
is the unique bounded strictly positive solution of Equation (82). Besides, the optimal strategy is
e π∗t = 1 ασ(t, η)Set θt+ ρht (1−ρ2)(c+Rt 0 hudfWu) ! , (85) where h is defined by (64). Proof.
Using (64) and Ito’s formula, we find that
dYt=d e12 Rt 0θ 2 udu(EQ[e(1−ρ2)(αH−12 RT 0 θ 2 udu)|Fη t]) 1 1−ρ2 =d e12 Rt 0θu2du(c+ Z t 0 hudWfu) 1 1−ρ2 =e12 Rt 0θ 2 udu(c+ Z t 0 hudWfu) 1 1−ρ2 1 2θ 2 tdt+e 1 2 Rt 0θ 2 udu× × 1 1−ρ2(c+ Z t 0 hudfWu) ρ2 1−ρ2 h tdWft+ 1 2 ρ2 (1−ρ2)2(c+ Z t 0 hudWfu) ρ2 1−ρ2−1h2 tdt = 1 2[ (θtYt+φtρ)2 Yt ]dt+φt(ρdWt1+ p 1−ρ2dW0 t), (86)
where in the last equality, we introduced the symbolφ to denote φt=e 1 2 Rt 0θ 2 udu 1 1−ρ2(c+ Z t 0 hudfWu) ρ2 1−ρ2 h t. (87)
Comparing (86) with (82), we see that Yt is a bounded solution of (82), where ψt = φtρ and ψt⊥ =
φt p
1−ρ2.By the uniqueness of (82), we obtain (84).
To obtain the optimal strategy, we substitute the expressions ofψt=φtρ, whereφis specified by (87),
and of Yt, written in terms of the representation (64), in (83).
Remark 12 Note that Yt satisfies also the BSDE (65), related to problem (57), which is simpler
than (82). This follows either from the proof of Proposition 3 or directly from (86), since ρdW1
t + p
1−ρ2dW0
t =Wt.
So, the optimization problems (57) and (79) are equivalent since the corresponding value processes coincide and the optimal strategies of these problems are related by the equality
πt∗=eπ∗tSet.
This means that, if H =f(η), the optimal dollar amount invested in the assets is the same in both problems and is based only on the information coming from the non traded asset η. We deduce that if the contingent claim does not depend on the stock but only on the non traded assetη, we need the same optimal amount of money to hedge with the stock S in both situations: if we consider just the information on η and if we use all the information of the market.
Remark 13 If we consider the optimization problem to minimizeE e−α( RT 0 πtdSet−H) over all π ∈Π(Fη), (88) where πt represents the number of stocks held at time t and is adapted to the filtration Ftη, then
this problem is not equivalent to (57) and (79). Indeed, the optimal quantity of assets in (79) is not
Fη-predictable and then it is evident that
πt∗6=eπ∗tSet,
whereπ∗ and eπ
∗ denote respectively the optimal strategies of (57) and (88).
Theorem 1 cannot be applied directly to problem (88). In fact, for the processSe, condition A) is not
satisfied and the equivalent Fη-adapted problem for (88) is more complicated.
6
Sufficiency of filtrations
In this section we study the issue of sufficiency of partial information for the optimization problem (2).
LetVt(A) andVt(G) be the value processes of the problem corresponding respectively to the cases
of full and partial information
Vt(A) = ess inf π∈Π(A)E[e −α(RT t πudSu−H)|At], Vt(G) = ess inf π∈Π(G)E[e −α(RT t πudSu−H)|Gt]. It is convenient to express the optimization problem as the set
OΠ ={(Ω,A,A, P), S, H}
formed by the filtered probability space, the asset price process S and the terminal reward H.
Definition 1 The filtration G is said to be sufficient for the optimization problem OΠ if V0(G) =
V0(A), i.e., if inf π∈Π(A)E[e −α(RT 0 πudSu−H)] = inf π∈Π(G)E[e −α(RT 0 πudSu−H)], (89)
where Π(A) (respectively Π(G)) is the class ofA- predictable (respectivelyG- predictable) processes
π such thatπ·N ∈BM O(A).
Note that for the model (55)-(56) satisfying conditions 1)-3) the filtrationGt=Ftη is sufficient, if
Atcoincides with the filtrationFtS,ηand ifHisF η
T-measurable. Our aim is to give sufficient conditions
for (89) for a more general model.
In this section we shall use the following assumptions:
A0)hNi isG-predictable,
B0) anyG-martingale is an A-local martingale,
C0) the filtrationG is continuous,
D0) λand hN, m(g)i (for anyG-local martingale m(g)) are G-predictable,
E0) H is aGT-measurable bounded random variable,
F0) R0Tλ2tdhNit≤C, P −a.s..
Remark 14 These assumptions are similar to A)-F), but there are several differences. Here we don’t use the auxiliary filtrationF and conditionsA0)−F0) are formulated in terms of filtrationA, e.g.,D0) and E0) are conditions onA-decomposition terms ofS and in assumptions B0) andE0) the filtration
F is replaced by A) (so, they are stronger). On the other hand, condition A0) is the first part of A) and C0) is the same as C).
Throughout this section we shall assume that conditionsA0), B0), C0) andF0) are satisfied. Under conditions A0), B0), C0) b St=E(St|Gt) =S0+ Z t 0 b λudhNiu+Nt, (90)
whereNt admits the representation (the proof is similar to [15] (Theorem 1 of Ch.4.10 )) Nt= Z t 0 \ (dhN, m(g)iu dhm(g)iu )dmu(g) +Lt(g), (91)
for any G-local martingale m(g). Here L(g) is a G-local martingale orthogonal to m(g). Note that
Nt is equal to E(Nt|Gt) if λisG-predictable.
It follows from (91) that
hN, m(g)iG=hN , m(g)i (92)
for any G-local martingale m(g), where by AG we denote the dual G-predictable projection of the processA.
Note that kt2 =dhNit/dhNit ≤1 and, like in section 2, one can show that π·N ∈BM O(G) for
any π ∈Π(G).
Lemma 3 Let V0(A) =V0(G). Then Vt(G) =E(Vt(A)|Gt). If in addition FS ⊆G, and conditions
B0), E0) are satisfied, thenVt(A) =Vt(G) for all t∈[0, T].
Proof. IfV0(A) =V0(G) then the optimal strategyπ∗ forV0(A) isG-predictable. Therefore
Vt(A) =E e−α( RT t π ∗ udSu−H)|A t (93) and taking conditional expectations with respect to Gt we obtain the equalityVt(G) =E(Vt(A)|Gt).
If FS ⊆G andH isGT-measurable then
E e−α( RT t π ∗ udSu−H)|A t =E e−α( RT t π ∗ udSu−H)|G t =Vt(G), (94)
since by condition B0) the σ-algebras At and GT are conditionally independent with respect to Gt.
Thus, (93) and (94) imply the equalityVt(A) =Vt(G).
Proposition 5 Let conditions A0), B0), C0) and F0) be satisfied. Then Vt(A) =Vt(G) if and only if
H is GT-measurable and the process Vt(A) satisfies the BSDE
Yt=Y0+ 1 2 Z t 0 (ψuku2+bλuYu)2 Yu dhNiu+ Z t 0 ψudNu+Net, YT =eαH, (95)
where Ne is a G-local martingale orthogonal to N and kt2=dhNit/dhNit.
Proof. Let Vt(A) =Vt(G). This implies thateαH =E(eαH|GT) and H is GT-measurable. According
to Corollary 2 (more exactly, by (36)) Vt(A) is the unique (bounded strictly positive) solution of the
BSDE Yt=Y0+ 1 2 Z t 0 Yu λu+ ϕu Yu 2 dhNiu+ Z t 0 ϕudNu+Lt, YT =eαH (96)
and the optimal strategy is of the form
π∗t = 1
α(λt+ ϕt
Yt
Since Vt(A) =Vt(G), the optimal strategy π∗ isG-predictable, henceπ∗ =bπ ∗, i.e., λt+ ϕt Vt(A) =bλt+ b ϕt Vt(A) . (98)
Therefore (96), (98) and conditionA0) imply that the martingale partmt= Rt
0 ϕudNu+Ltis aG-local martingale. By the Galtchouk-Kunita-Watanabe (G-K-W) decomposition of mwith respect to N
Z t 0 ϕudNu+Lt= Z t 0 ψudNu+Net, (99)
whereψis aG-predictableN-integrable process andNe is aG-local martingale orthogonal toN (since
mt= Rt
0 ϕudNu+Lt is G-adapted). It is evident that
ψt= dhm, Nit hNit and ψtk2t = dhm, Nit hNit . (100) Taking first the mutual characteristics (with respect to N) and then G-dual predictable projections of both parts in (99) Z t 0 b ϕudhNiu= Z t 0 dhm, Niu dhNiu dhNu, NiGu +hN,NeiGt. (101)
It follows from (92) thathN , NiG=hNi and hN, e NiG=hN , e Ni= 0. Therefore Z t 0 b ϕudhNiu =hN , mit and b ϕt= dhm, Nit dhNit =ψtκ2t. (102)
Substituting (98), (99) and (102) in (96) we obtain thatVt(A) satisfies (95).
Let us assume now, thatVt(A) satisfies equation (95). It follows from the proof of Theorem 1 that
the unique bounded strictly positive solution of equation (24) is the process
Vt= ess inf π∈Π(G)E[e −α(RT t πudSbu−H)+α 2 2 RT t πu2(1−ku2)dhNiu|G t]. (103)
Note that here we don’t need condition D0), which is needed just to show the equivalence Vt(G) =Vt
in Proposition 1.
Thus Vt(A) = Vt. This implies that Vt(A) is G-adapted and the optimal strategy π∗ for Vt(A) is
G-predictable. Therefore Vt(A) = ess inf π∈Π(G)E[e −α(RT t πudSu−H)|At] =E[e−α( RT t π ∗ udSu−H)|A t]
and taking conditional expectations (with respect toGt) in this equality we have
Vt(A) =E[e−α( RT t π ∗ udSu−H)|G t]≥ ess inf π∈Π(G)E[e −α(RT t πudSu−H)|Gt] =Vt(G). (104) On the other hand Vt(G) ≥ E(Vt(A)|Gt) (this is proved similarly to (21)) which together with
Corollary 3 Let conditions C0) and F0) be satisfied and let FS ⊆G. Then a)Vt(A) =Vt(G) if and only if H is GT-measurable and λis G-predictable.
b) If in addition condition B0) is satisfied and H is GT-measurable, then V0(A) = V0(G) if and
only if λis G-predictable.
Proof. Let H be a GT-measurable and λ is G-predictable. Since FS ⊆G the square bracket hNi is
G-predictable and from (1) follows thatN isG-adapted. Therefore,N =N and k2 = 1. This implies that equations (95) and (96) coincide (note thatλ=λb), henceVt(A) satisfies (95) andVt(A) =Vt(G)
by Proposition 5.
LetVt(A) =Vt(G). Since hm, Ni=hV(G), Si beG-predictable, ϕ= dhdm,NhNii isG-predictable and
it follows from equality (98) thatλis alsoG-predictable.
The proof of the part b) follows from part a) and Lemma 3.
Theorem 2 Let conditionsA0)–F0)be satisfied. ThenVt(A) =Vt(G)and the filtrationGis sufficient.
Proof. It follows from the proof of Theorem 1 (and Proposition (1)) that under conditions A0)−F0) the value processV(G) coincides withV (defined by (103)) and satisfies equation (95). It follows from (92) and conditionD0) thathN, Ni=hNi, which implies that
Nt= Z t
0
kt2dNt+Let, (105)
where Le is orthogonal to N. On the other hand under conditions A
0
)−F0) and equality (92) also imply thathN , Ne i=hN , Ne i= 0 (Ne is defined by (95)), i.e., a local martingale orthogonal toN is also
orthogonal toN. Therefore plugging (105) into (95), by a change of variablesϕ=ψk2, we obtain that equations (95) and (96) are equivalent (note that λ=bλby D
0
)). This implies that Vt(G) =Vt(A).
Remark 15 Note that under conditions A)–F) the filtrationGtis sufficient for the auxiliary filtration
Ft=FtS∨Gt.
Remark 16 Of course, condition E0) is not necessary for the filtration G to be sufficient, i.e. the filtration G can be sufficient for some claims H which are not GT-measurable. One can show that
Theorem 2 remains true if we replace condition E0) by assuming that in the G-K-W decomposition
H =EH+RT
0 ϕ
H
udNu+LHT the processesϕH,hLHiandhLH, m(g)i(for anyG-local martingalem(g))
areG-predictable.
Acknowledgements
This work has been partially supported by the MiUR Project “Stochastic Methods in Finance”. M. Mania also gratefully acknowledges financial support from ICER,Torino.
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