¯ Gt
´
= 11{τ >t}eΓ(t)−2Γ(T )EP
³ Z T t
e−θ2(T −s)γ(s)eΓ(s)ds
¯¯
¯ Ft
´
= 11{τ >t}eΓ(t)−2Γ(T )
Z T
t
e−θ2(T −s)γ(s)eΓ(s)ds.
One can check that, at time 0, the value function is indeed smaller that the one obtained with F-adapted portfolios.
Case of an Attainable Claim
Assume now that a claim X is FT-measurable. Then Xt = EQ(X | Gt) is the price of X, and it satisfies dXt= bxtdWtQ. The optimal strategy is, in a feedback form,
πt∗= σ−1¡ b
xt− θ(Vt− Xt)¢ and the associated wealth process satisfies
dVt= πt∗(νdt + σdWt) = π∗tσ dWtQ= σ−1¡
σbxt− ν(Vt− Xt)¢ dWtQ. Therefore,
d(Vt− Xt) = −θ(Vt− Xt) dWtQ.
Hence, if we start with an initial wealth equal to the arbitrage price EQX of X, then we that Vt= Xt
for every t ∈ [0, T ], as expected.
Hodges Price
Let us emphasize that the Hodges price has no real meaning here, since the problem min EP((VTv)2) has no financial interpretation. We have studied in Bielecki et al. [176] a more pertinent problem, with a constraint on the expected value of VTv under P. Nevertheless, from a mathematical point of view, the Hodges price would be the value of p such that
(v2− (v − p)2) = Z T
0
eθ2sEP(γsxe2se−Γs)11{τ >t}ds
In the case of the example studied in Section 6.4.2, the Hodges price would be the non-negative value of p such that
2vp − p2= e−2ΓT Z T
0
eθ2sγseΓsds.
Let us also mention that our results are different from results of Lim [151]. Indeed, Lim studies a model with Poisson component, and thus in his approach the intensity of this process does not vanish after the first jump.
6.4.3 Jump-Dynamics of Price
We assume here that the price process follows
dSt= St−(νdt + σdWt+ ϕdMt), S0> 0
where the constant ϕ satisfy ϕ > −1 so that the price St is strictly positive. Hence, the primary market, where the savings account and the asset S are traded is arbitrage free, but incomplete (in general). It follows that the wealth process follows
dVtv(π) = πt(νdt + σdWt+ ϕdMt), V0v(π) = v.
As in the previous subsection, our aim is, for a given initial endowment v, solve the minimization problem:
minπ EP((VTv(π) − X)2).
In order to characterize the value function we proceed analogously as before. That is, we are looking for processes X, Θ and Ψ such that the process (for simplicity we write Vtin place of Vtv(π))
J(t, Vt) = (Vt− Xt)2Θt+ Ψt
is a submartingale for any π and a martingale for some π∗, and such that ΨT = 0, XT = X, ΘT = 1.
(Note that Mania and Tevzadze[156] did a similar approach for continuous processes, with a value function of the form Jt = Φ0(t) + Φ1(t)Vt+ Φ2(t)Vt2.) Let us assume that the dynamics of these processes are of the form
dXt = ftdt + bxtdWt+ extdMt, (6.27) dΘt = Θt(ϑtdt + bϑtdWt+ eϑtdMt) (6.28) dΨt = ψtdt + bψtdWt+ eψtdMt (6.29) where the drifts ft, ϑt and ψt have to be determined.
From Itˆo’s formula we obtain
d(Vt− Xt)2= 2(Vt− Xt)(πtσ − bxt)dWt
+ £
(Vt+ πtϕ − Xt− ext)2− (Vt− Xt)2¤ dMt
+ ¡
2(Vt− Xt)(πtµ − ft) + (πtσ − bxt)2 + ξt
£(Vt+ πtϕ − Xt− ext)2− (Vt− Xt)2− 2(Vt− Xt)(πtϕ − ext)¤¢
dt.
Process Θt(Vt− Xt)2+ Ψtis a (local) martingale iff k(πt, ft, ϑt, ψt) = 0 for all t, where k(π, ϑ, f, ψ) = ψ + Θt
£ϑt(Vt− Xt)2 + 2(Vt− Xt)
³
(πµ − f ) + bϑt(πσ − bxt) − ξt(πϕ − ext)
´
+ (πσ − bxt)2 + ξt(eϑt+ 1)¡
(Vt+ πϕ − Xt− ext)2− (Vt− Xt)2¢i .
In the first step, we find π] such that the maximum of k(π) is obtained. Then, one defines (f∗, ϑ∗, ψ∗) such that k(π], f∗, ϑ∗, ψ∗) = 0. This implies that, for any π, k(π, f∗, ϑ∗, ψ∗) ≤ 0, and that k(π], f∗, ϑ∗, ψ∗) = 0.
The optimal π] is the solution of
(Vt− Xt)(µ − ξtϕ + bϑtσ) + σ(πσ − bxt) + ξt(eϑt+ 1)ϕ(Vt+ πϕ − Xt− ext) = 0 . hence
π]t = 1
σ2+ ϕ2ξt(eϑt+ 1)
³
(σbxt+ ξtϕ(eϑt+ 1)ext) − (µ + bϑtσ + ξtϕeϑt)(Vt− Xt)´
= At− Bt(Vt− Xt) with
At = ³
σbxt+ ξtϕ(eϑt+ 1)ext
´
∆−1t Bt =
³
µ + bϑtσ + ξtϕeϑt
´
∆−1t
∆t = σ2+ ϕ2ξt(eϑt+ 1) .
After some computations the drift term of Θt(Vt− Xt) + Ψtis found to be Θt(Vt− Xt)2(ϑt− B2t∆t) + 2Θt(Vt− Xt)
³
AtBt∆t− bϑtxbt− ξtϑetxet− ft
´
+ Θtξt(eϑt+ 1)(Atϕ − ext)2+ Θt(Atσ − bxt)2+ ψt. Then, we choose
ϑ∗t = Bt2∆t
ft∗ = AtBt∆t− bϕtxbt− ξtϑetxet
ψt∗ = −Θtξt(eϑt+ 1)(Atϕ − ext)2− Θt(Atσ − bxt)2.
Let us suppose that with this choice of drifts equations (6.28)–(6.29) admit solutions (we shall discuss this issue below). Next, let us denote these solutions as (Θ∗, bϑ∗, eϑ∗), (X∗, bx∗, ex∗) and (Ψ∗, bψ∗, eψ∗);
the corresponding processes A, B and ∆ will be denoted as A∗, B∗ and ∆∗. Consequently, the drift term of Θ∗t(Vt∗(π) − Xt∗) + Ψ∗t is non-positive for any admissible π and it is equal to 0 for π∗= A∗t− Bt∗(Vtv,∗(π∗) − Xt∗).
The three dimensional process (Θ∗, bϑ∗, eϑ∗) is supposed to satisfy the BSDE
dΘt = Θt
Ã
(µ + bϑtσ + ξtϕeϑt)2
σ2+ ϕ2ξt(eϑt+ 1) dt + bϑtdWt+ eϑtdMt
!
(6.30)
ΘT = 1.
We shall discuss this equation later.
The three dimensional process (X∗, bx∗, ex∗) is a solution of the linear BSDE dXt = 1
∆t
(κ1,tbxt+ κ2,txet) dt + bxtdWt+ extdMt
XT = X
where
κ1,t= σµ + σϕξtϑet− ϕ2ϑbtξt(1 + eϑt), κ2,t= ϕξt(1 + eϑt)(µ + σ bϑt) − σ2ξtϑet. Thus,
Xt∗= EQκ(X|Gt), where dQκ|Gt = L(κ)t dP|Gt and
dL(κ)t = −L(κ)t−(κ1,t
∆tdWt+ κ2,t
ξ∆tdMt) . The three dimensional process (Ψ∗, bψ∗, eψ∗) is solution of
dΨt = −Θt³
ξt(eϑt+ 1)(Atϕ − ext)2+ (Atσ − bxt)2´
dt + bψtdWt+ eψtdMt ΨT = 0.
Thus, noting that
Ψ∗t+ Z t
0
Θs
³
ξs(eϑs+ 1)(Asϕ − exs)2+ (Asσ − bxs)2
´ ds is a G-martingale, we obtain that
Ψ∗t = E ÃZ T
t
Θs
³
ξs(eϑs+ 1)(Asϕ − exs)2+ (Asσ − bxs)2
´ ds|Gt
!
. (6.31)
Discussion of equation (6.30): Duality approach
Our aim is here to prove that the BSDE (6.30) has a solution. We take the opportunuity to correct a mistake in Bielecki et al [176] where we claim that, in the particular case where the intensity γt
is constant, we get a solution of the form eθt constant. The solution that appear in Bielecki et al. is valid only in the case P (τ < T ) = 1. We proceed using duality approach.
The set of equivalent martingale measure is determined by the set of densities. From Kusuoka [140] representation theorem, it follows that any strictly positive martingale in the filtration G can be written as
dLt= Lt−(`tdWt+ χtdMt) (6.32) for a G-predictable process χ satisfying χt> −1. In order that L corresponds to the Radon-Nikodym density of an emm, a relation between ` and χ has to be satisfied in order to imply that process LtSt is a P (local) martingale. (Recall that r = 0.) Straightforward application of integration by parts formula proves that the drift term of LS vanishes iff
ϕχtξt+ σ`t+ ν = 0
Recall that by definition the variance optimal measure for L is a probability measure Q∗such that it minimizes EQ∗(L2T). At this moment we are unable to verify existence/uniqueness of such a measure in the context of our model. We thus assume that the measure exists,
Hypothesis: We assume that the variance optimal measure exists.
In what follows we shall use the same argument as in Bobrovnytska and Schweizer [29]. Towards this end we denote by L∗ the Radon-Nikodym density of the variance optimal martingale measure.
Let Z be the martingale Zt = EQ∗(L∗T|Gt) and U = L∗/Z. It is proved in Delbaen and Shacher-mayer [62] (Lemma 2.2) that, if the variance optimal martingale measure exists, then there exists a predictable process bz such that
dZt/Zt−= bztdSt= zt(σdWt+ ϕdMt+ νdt)
where zt= bztSt−(in the proof of lemma 2.2, the hypothesis of continuity of the asset is not required).
The process L∗ is a (P, G) martingale, hence there exist ` and χ such that dL∗t = L∗t−(`tdWt+ χtdMt)
From Itˆo’s calculus, setting U = L∗/Z, we obtain dUt= Ut−
µ
Atdt + (`t− ztσ)dWt+
µ 1
1 + ztϕ(χt+ 1) − 1)
¶ dMt
¶
, UT = 1,
where
At = z2tσ2+ ξt(1 + χt)(ztϕ + 1
1 + ztϕ− 1)
= z2tσ2+ ξt(1 + χt) zt2ϕ2 1 + ztϕ
= z2t µ
σ2+ ξt(1 + χt) ϕ2 1 + ztϕ
¶ .
We recall that ϕχtξt+ σ`t+ ν = 0 . Hence, letting b
ut = `t− ztσ e
ut = 1
1 + ztϕ(χt+ 1) − 1 ,
we get
zt= − ν + σbut+ ϕξteut
σ2+ ϕ2ξt(1 + eut). It follows that
At = zt2 µ
σ2+ ξt(1 + χt) ϕ2 1 + ztϕ
¶
= zt2 µ
σ2+ ξt(1 + ztϕ)(1 + eut) ϕ2 1 + ztϕ
¶
= zt2¡
σ2+ ξt(1 + eut)ϕ2¢
= (ν + σbut+ ϕξtuet)2 σ2+ ϕ2ξt(1 + eut) so that process U is a solution of
dUt= Ut−
µ(ν + σbut+ ϕξteut)2
σ2+ ϕ2ξt(1 + eut) dt + butdWt+ eutdMt
¶
, UT = 1,
which establishes that the BSDE (6.30) has a solution as long as the variance optimal martingale measure exists in our set-up.