# A Maximum Principle Approach to Risk Indifference Pricing with Partial Information

Authors: Ta Thi Kieu An; Bernt Øksendal; Frank Proske
Journal: Journal of Applied Mathematics and Stochastic Analysis
 (2008)

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## Abstract

We consider the problem of risk indifference pricing on an incomplete
market, namely on a jump diffusion market where the
controller has limited access to market information. We use
the maximum principle for stochastic differential games to derive
a formula for the risk indifference pricepriskseller(G,ℰ) of a European-type claim G.


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## Body

## 1. Introduction

Suppose the value of a portfolio(π(t),S0(t)) is given by
(1.1)Xx(π)(t)=x+π(t)S(t)+S0(t),where x is the initial capital, S(t) is a semimartingle price process of a
risky asset,
π(t) is the number of risky assets held at time t,
and S0(t) is the amount invested in the risk-free asset
at time t.
Then, the cumulative cost at time t is given by(1.2)P(t)=Xx(π)(t)−∫0tπ(u−)dS(u). If P(t)=p-constant for all t,
then the portfolio strategies (π(t),S0(t)) is called self-financing. A contingent
claim with expiration date T is a nonnegative ℱT-measurable random variable G that represents the time T payoff from seller to buyer. Suppose that for
a contingent claim G there exists a self-financing strategy such
that Xx(π)(T)=G,
that is,(1.3)p+∫0Tπ(u)dS(u)=G.Then, p is the price of G in the complete market, that
is,(1.4)p=EQ[G],where Q is any martingale measure equivalent to P on the probability space (Ω,ℱt,P).In an incomplete market, an exact replication of a
contingent claim is not always possible. One of the approaches to solve the
replicating problems in an incomplete market is the utility indifference
pricing. See, for example, Grasselli and Hurd [1] for the case of stochastic
volatility model, Hodges and Neuberger [2] for the financial model with constraints, and Takino
[3] for model with
incomplete information. The utility indifference price p of a claim G is the
initial payment that makes the seller of the contract utility indifferent to
the two following alternatives: either selling the
contract with  initial payment p and with the obligation to pay out G at
time T or not selling the contract and hence receiving no initial payment.Recently, several papers discuss risk measure pricing
rather than utility pricing in incomplete markets. Some papers related to risk
measure pricing are the following: Xu [4] propose risk measure pricing and hedging in
incomplete markets; Barrieu and El Karoui [5] study a minimization problem for risk measures subject
to dynamic hedging; Klöppel and Schweizer [6] study the indifference pricing of a payoff with a
minus dynamic convex risk measure. See also the references in these papers.In our paper, we study a pricing formula based on the
risk indifference principle in a jump-diffusion market. The same problem was
studied by Øksendal and Sulem [7] with the restriction to Markov controls. So the
problem is solved by using the Hamilton-Jacobi-Bellman equation. In our paper,
the control process is required to be adapted to a given subfiltration of the
filtration generated by the underlying Lévy processes. This makes the control
problem non-Markovian. Within the non-Markovian setting, the dynamic
programming cannot be used. Here we use the maximum principle approach to find
the solution for our problem.The paper is organized as follows. In Section2, we
will implement the option pricing method in an incomplete market. In Section 3,
we present our problem in a jump-diffusion market. In Section 4, we use a
maximum principle for a stochastic differential game to find the relation
between the optimal controls of the stochastic differential game and of a
corresponding stochastic control problem. Using this result, we derive the
relationship between the two value functions of the two problems above, and
then find the formulas for the risk indifference prices for the seller and the
buyer.


## 2. Statement of the Problem

Assume that a filtered probability space(Ω,ℱ,{ℱt}0≤t≤T,P) is given.Definition 2.1.
A
nonnegative random variableG on (Ω,ℱt,P) is called a European contingent claim.From now on, we
consider a European-type option whose payoff at timet is some nonnegative random variable G=g(S(t)).
In the rest of the paper, we will identify a contingent claim with its payoff
function g.Let𝔽 be the space of all equivalence classes of
real-valued random variables defined on Ω.Definition 2.2 (see [8, 9]).
A convex risk
measureρ:𝔽→ℝ∪{∞} is a mapping satisfying the following
properties, for X,Y∈𝔽:(i)
(convexity)(2.1)ρ(λX+(1−λ)Y)≤λρ(X)+(1−λ)ρ(Y),λ∈(0,1);(ii)
(monotonicity) ifX≤Y,
then ρ(X)≥ρ(Y);If an investor sells a liability to pay out the amountg(S(T)) at time T and receives an initial payment p for such a contract, then the minimal risk
involved for the seller is(2.2)ΦG(x+p)=inf⁡π∈Pρ(Xx+p(π)(T)−g(S(T))),where 𝒫 is the set of self-financing strategies such that Xx(π)(t)≥c,
for some finite constant c and for 0≤t≤T.If the investor has not issued a claim (and hence no
initial payment is received), then the minimal risk for the investor
is(2.3)Φ0(x)=inf⁡π∈P⁢ρ(Xx(π)(T)).Definition 2.3.
The seller'srisk indifference price, p=priskseller,
of the claim G is the solution p of the equation (2.4)ΦG(x+p)=Φ0(x). Thus, priskseller is the initial payment p that makes an investor risk indifferent
between selling the contract with liability payoff G and not selling the contract.In view of the general representation formula for
convex risk measures (see [10]), we will assume that the risk measure ρ,
which we consider, is of the following type.Theorem 2.4 (representation theorem [8, 9]).
A mapρ:𝔽→ℝ is a convex risk measure if and only if there
exists a family ℒ of measures Q≪P on ℱT and a convex “penalty” function ζ:ℒ→(−∞,+∞) with infQ∈ℒζ(Q)=0 such that(2.5)ρ(X)=sup⁡Q∈ℒ{EQ[−X]−ζ(Q)},X∈F.By
this representation, we see that choosing a risk measure ρ is equivalent to choosing the family ℒ of measures and the penalty function ζ.Using the representation (2.5), we can write (2.2) and (2.3)
as follows:(2.6)ΦG(x+p)=inf⁡π∈P(sup⁡Q∈ℒ{EQ[−Xx+p(π)(T)+g(S(t))]−ζ(Q)}),(2.7)Φ0(x)=inf⁡π∈P(sup⁡Q∈ℒ{EQ[−Xx(π)(T)]−ζ(Q)}),for a given penalty function ζ.Thus, the problem of finding the risk indifference
pricep=priskseller given by (2.4) has turned into two stochastic
differential game problems (2.6) and (2.7). In the complete information,
Markovian setting this problem was solved in
[7] where the authors
use Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations and PDEs to find the
solution. In our paper, the corresponding partial information problem is
considered by means of a maximum principle of differential games for SDEs.


## 3. The Setup Model

Suppose in a financial market, there are two
investment possibilities:(i) a bond with unit priceS0(t)=1, t∈[0,T];(ii) a stock with price dynamics, fort∈[0,T],(3.1)dS(t)S(t−)[α(t)dt+σ(t)dBt+∫ℝ0γ(t,z)N(dt,dz)],S(0)=s>0. Here Bt is a Brownian motion and N(dt,dz)=N(dt,dz)−ν(dz)dt is a compensated Poisson random measure with
Lévy measure ν.
The processes α(t), σ(t),
and γ(t,z) are ℱt-predictable processes such that γ(t,z)>−1,
for a.s. t,z,
and(3.2)E[∫0T{|α(s)|+σ2(s)+∫ℝ0|log⁡(1+γ(s,z))|2ν(dz)}ds]<∞⁢a.s.,for all T≥0.Letℰt⊆ℱt be a given subfiltration. Denote by π(t), t≥0,
the fraction of wealth invested in S(t) based on the partial market information ℰt⊆ℱt being available at time t. Thus, we impose on π(t) to be ℰt-predictable. Then, the total wealth X(π)(t) with initial wealth x is given by the SDE(3.3)dX(π)(t)=π(t−)S(t−)[α(t)dt+σ(t)dBt+∫ℝ0γ(t,z)N(dt,dz)],X(π)(0)=x>0.In the sequel, we will call a portfolioπ∈𝒫admissible if π is ℰt-predictable, permits a strong solution of
(3.3), and satisfies(3.4)∫0T{|α(t)||π(t)|S(t)+σ2(t)π2(t)S2(t)+π2(t)S2(t)∫ℝ0γ2(t,z)ν(dz)}ds<∞,as well as(3.5)π(t)S(t)γ(t)>−1(ω,t,z)-a.s.The class of admissible
portfolios is denoted by Π.Now, we define the measuresQθ parameterized by given ℱt-predictable processes θ=(θ0(t),θ1(t,z)) such that(3.6)dQθ(ω)=Kθ(T)dP(ω)onℱT,where(3.7)dKθ(t)=Kθ(t−)[θ0(t)dB(t)+∫ℝ0θ1(t,z)N(dt,dz)],t∈[0,T],Kθ(0)=k>0,We assume that(3.8)θ1(t,z)≥−1for⁢a.a.t,z,∫0T{θ02(s)+∫ℝ0(log⁡(1+θ1(s,z)))2ν(dz)}ds<∞⁢a.s.Then, by the Itô formula, the
solution of (3.7) is given by(3.9)Kθ(t)=kexp⁡[−∫0tθ0(s)dB(s)−12∫0tθ02(s)ds⁢+∫0t∫ℝ0ln⁡(1−θ1(s,z))N(dt,dz)⁢+∫0t∫ℝ0{ln⁡(1−θ1(s,z))+θ1(s,z)}ν(dz)ds].We say that the control θ=(θ0,θ1) is admissible and write θ∈Θ if θ is adapted to the subfiltration ℰt
and satisfies
(3.8) and(3.10)E[Kθ(T)]=Kθ(0)=k>0.We set(3.11)dY(t)=[dY1(t)dY2(t)dY3(t)]=[dKθ(t)dS(t)dX(π)(t)]=[0S(t−)α(t)S(t−)π(t)α(t)]dt+[Kθ(t−)θ0(t)S(t−)σ(t)S(t−)π(t)σ(t)]dB(t)+∫ℝ0[Kθ(t−)θ1(t,z)S(t−)γ(t,z)S(t−)π(t)γ(t)]N(dt,dz),Y(0)=y=(y1,y2,y3)=(k,s,x),dY(t)=[dY1(t)dY2(t)]=[dKθ(t)dS(t)],Y(0)=y=(y1,y2)=(k,s).We now define two setsℒ,ℳ of measures as follows:(3.12)ℒ={Qθ;θ∈Θ},ℳ={Qθ;θ∈M},where(3.13)M={θ∈Θ;E[Mθ(t,y)∣ℰt]=0⁢∀t,y},Mθ(t,y)=Mθ(t,k,s)=α(t)+σ(t)θ0(t)+∫ℝ0γ(t,z)θ1(t,z)ν(dz).In particular, by the Girsanov theorem, all the
measuresQθ∈ℳ with E[Kθ(T)]=1 are equivalent martingale measures for the ℰt-conditioned market (S0(t),S1(t)), where(3.14)dS1(t)=S1(t−)[E[α(t)∣ℰt]dt+E[σ(t)∣ℰt]dBt+∫ℝ0E[γ(t,z)∣ℰt]N(dt,dz)]S1(0)=s>0(see, e.g.,
[11, Chapter
1]).We assume that the penalty functionζ has the form(3.15)ζ(Qθ)=E[∫0T∫ℝ0λ(t,θ0(t,Y(t)),θ1(t,Y(t),z),Y(t),z)ν(dz)dt+h(Y(T))],for some convex functions λ∈C1(ℝ2×ℝ0),h∈C1(ℝ),
such that(3.16)E[∫0T∫ℝ0|λ(t,θ0(t,Y(t)),θ1(t,Y(t),z),Y(t),z)|ν(dz)dt+|h(Y(T))|]<∞,for all (θ,π)∈Θ×Π.Using theY(t)-notation, problem (2.6) can be written as
follows:Problem A.
FindΦGℰ(t,y) and (θ*,π*)∈Θ×Π such that(3.17)ΦGℰ(t,y):=inf⁡π∈Π(sup⁡θ∈ΘJθ,π(t,y))=Jθ*,π*(t,y),where(3.18)Jθ,π(t,y)=J(θ,π)=Ey[−∫tTΛ(θ(u,Y(u)))du−h(Y(T))+Kθ(T)g(S(T))−Kθ(T)X(π)(T)],(3.19)Λ(θ)=Λ(θ(t,y))=∫ℝ0λ(t,θ0(t,y),θ1(t,y,z),y,z)ν(dz).We will relate Problem A to the following stochastic
control problem:(3.20)ΨGℰ=sup⁡Q∈ℳ{EQ[G]−ζ(Q)}.Using the Y(t)-notation, the problem gets the following
form.Problem B.
FindΨGℰ(t,y) and θˇ∈𝕄 such that(3.21)ΨGℰ(t,y):=sup⁡θ∈M⁢J0θ(t,y)=J0θˇ(t,y),where(3.22)J0θ(t,y)=Ey[−∫tTΛ(θ(u,Y(u)))du−h(Y(T))+Kθ(T)g(S(T))].Define theHamiltonianH:[0,T]×ℝ×ℝ×ℝ×Θ×Π×ℝ×ℝ×ℛ→ℝ for Problem A by(3.23)H(t,k,s,x,θ,π,p,q,r(⋅,z))=−Λ(t,Y(t))+sαp2+sαπp3+kθ0q1+sσq2+sσπq3+∫ℝ0{kθ1r1(⋅,z)+sγ(t,z)r2(⋅,z)+sπγ(t,z)r3(⋅,z)}ν(dz),and the Hamiltonian H:[0,T]×ℝ×ℝ×Θ×ℝ×ℝ×ℛ→ℝ for Problem B by(3.24)H(t,k,s,θ,p,q,r(⋅,z))=−Λ(t,Y(t))+sαp2+kθ0q1+sσq2+∫ℝ0{kθ1(t,z)r1(⋅,z)+sγ(t,z)r2(⋅,z)}ν(dz).Here ℛ is the set of functions r:[0,T]×ℝ→ℝ such that the integrals in (3.23) and (3.24)
converge. We assume that H and H are differentiable with respect to k, s,
and x.
The adjoint equations (corresponding to θ, π,
and Y(t)) in the unknown adapted processes p(t), q(t), r(t,z) are the backward stochastic differential
equations (BSDEs)(3.25)dp1(t)=(∂Λ∂k(t,Y(t))−θ0(t)q1(t)−∫ℝ0θ1(t,z)r1(t,z)ν(dz))dt+q1(t)dB(t)+∫ℝ0r1(t,z)N(dt,dz),p1(T)=−∂h∂k(Y(T))+g(S(T))−X(π)(T),(3.26)dp2(t)=(∂Λ∂s(t,Y(t))−α(t)p2(t)−σ(t)q2(t)−∫ℝ0γ(t,z)r2(t,z)ν(dz))dt+q2(t)dB(t)+∫ℝ0r2(t,z)N(dt,dz),p2(T)=−∂h∂s(Y(T))+Kθ(T)g′(S(T)),(3.27)dp3(t)=(−α(t)p3(t)−σ(t)q3(t)−∫ℝ0γ(t,z)r3(t,z)ν(dz))dt+q3(t)dB(t)+∫ℝ0r3(t,z)N(dt,dz),p3(T)=−Kθ(T).Similarly, the adjoint equations (corresponding toθ and Y(t)) in the unknown processes p(t), q(t), r(t,z) are given by(3.28)dp1(t)=(∂Λ∂k(t,Y(t))−θ0(t)q1(t)−∫ℝ0θ1(t,z)r1(t,z)ν(dz))dt+q1(t)dB(t)+∫ℝ0r1(t,z)N(dt,dz),p1(T)=−∂h∂k(Y(T))+g(S(T)),dp2(t)=(∂Λ∂s(t,Y(t))−α(t)p2(t)−σ(t)q2(t)−∫ℝ0γ(t,z)r2(t,z)ν(dz))dt+q2(t)dB(t)+∫ℝ0r2(t,z)N(dt,dz),p2(T)=−∂h∂s(Y(T)Lemma 3.1.
Letθ∈Θ and suppose that p(t)=(p1(t),p2(t)) is a solution of the corresponding adjoint
equations (3.28). For all π∈ℝ,
define(3.29)p1(t)=p1(t)−X(π)(t),(3.30)p2(t)=p2(t),(3.31)p3(t)=−Kθ(t). If θ∈𝕄,
then p(t)=(p1(t),p2(t),andp3(t)) is a solution of the adjoint equations (3.25),
(3.26), and (3.27). Then, the following relation
holds:
(3.32)H(t,Y(t),θ,π,p(t),q(t),r(t,z))=H(t,Y(t),θ,p(t),q(t),r(t,z))−S(t)πKθ(t)(α(t)+2θ0(t)σ(t)+2∫ℝ0θ1(t,z)γ(t,z)ν(dz)).Proof.
Differentiating both sides of (3.29),
we get(3.33)dp1(t)=dp1(t)−dX(π)(t)=(∂Λ∂k(t,Y(t))−θ0(t)q1(t)−∫ℝ0θ1(t,z)r1(t,z)ν(dz)−S(t)α(t)π(t))dt+(q1(t)−S(t)σ(t)π(t))dB(t)+∫ℝ0(r1(t,z)−S(t)π(t)γ(t,z))N(dt,dz).Comparing this with (3.25) by
equating the dt, dB(t), N(dt,dz) coefficients, respectively, we
get(3.34)∂Λ∂k(t,Y(t))−θ0(t)q1(t)−∫ℝ0θ1(t,z)r1(t,z)ν(dz)=∂Λ∂k(t,Y(t))−θ0(t)q1(t)−∫ℝ0θ1(t,z)r1(t,z)ν(dz)−S(t)α(t)π(t),(3.35)q1(t)=q1(t)−S(t)σ(t)π(t),(3.36)r1(t,z)=r1(t,z)−S(t)γ(t,z)π(t).Substituting (3.35) and (3.36) into
(3.34), we get(3.37)S(t)π(t)(α(t)+θ0(t)σ(t)+∫ℝ0θ1(t,z)γ(t,z)ν(dz))=0.Since θ∈𝕄,
(3.37) is satisfied, and hence p1(t) is a solution of (3.25).
Proceeding as above with the processesp2(t) and p3(t),
we get(3.38)q2(t)=q2(t),r2(t)=r2,(3.39)−α(t)p3(t)−σ(t)q3(t)−∫ℝ0γ(t,z)r3(t,z)ν(dz)=0,(3.40)q3(t)=−Kθ(t)θ0(t),r3(t,z)=−Kθ(t)θ1(t,z).With the values p3(t), q3(t),
and r3(t,z) defined as above, relation (3.39) is satisfied
if θ∈𝕄.
Hence, p1(t), p2(t),
and p3(t) are solutions of (3.29), (3.30), and (3.31),
respectively.
Equations (3.23) and (3.24) give the following relation
between H and H:(3.41)H(t,y,θ,π,p,q,r(⋅,z))=H(t,y,θ,p,q,r(⋅,z))+sπ(αp3+σq3+∫ℝ0γ(t,z)r3(⋅,z)ν(dz)).Hence,(3.42)H(t,Y(t),θ,π,p(t),q(t),r(t,z))=H(t,Y(t),θ,p1(t),p2(t),q1(t),q2(t),r1(t,z),r2(t,z))−S(t)π(t)(α(t)p3(t)+σq3(t)+∫ℝ0γ(t,z)r3(t,z)ν(dz))=H(t,Y(t),θ,p1(t),p2(t),q1(t),q2(t),r1(t,z),r2(t,z))−S(t)σ(t)π(t)Kθ(t)θ0(t)−∫ℝ0S(t)γ(t,z)π(t)Kθ(t)θ1(t,z)ν(dz)−S(t)π(t)Kθ(t)(α(t)+σ(t)θ0(t)+∫ℝ0γ(t,z)θ1(t)ν(dz))=H(t,Y(t),θ,p(t),q(t),r(t,z))−sπKθ(t)(α(t)+2σ(t)θ0(t)+2∫ℝ0γ(t,z)θ1(t,z)ν(dz)).Lemma 3.2.
Letp1(t), p2(t),
and p3(t) be as in Lemma 3.1. Suppose
that, for all π∈ℝ,
the function(3.43)θ⟶E[H(t,Y(t),θ,π(t),p(t),q(t),r(t,z))∣ℰt],θ∈Θ,has a maximum point at θ^=θ^(π).
Moreover, suppose that the function(3.44)π⟶E[H(t,Y(t),θ^(π),π,p(t),q(t),r(t,z))∣ℰt],π∈ℝ,has a minimum point at π^∈ℝ.
Then,(3.45)Mθ^(π^)=0.Proof.
The first-order conditions for a
maximum pointθ^=θ^(π) of the function E[H(t,Y(t),θ,π(t),p(t),q(t),r(t,z))∣ℰt] is(3.46)E[∇θ(H(t,Y(t),θ,π(t),p(t),q(t),r(t,z)))θ=θ^(π)∣ℰt]=0,where ∇θ=(∂/∂θ0,∂/∂θ1) is the gradient operator. The first-order
condition for a minimum point π^ of the function E[H(t,Y(t),θ^(π),π,p(t),q(t),r(t,z))∣ℰt] is(3.47)E[∇π(H(t,Y(t),θ^(π),π(t),p(t),q(t),r(t,z)))π=π^∣ℰt]=0,that is,(3.48)E[∇θ(H(t,Y(t),θ,π^,p(t),q(t),r(t,z)))θ=θ^(π^)(dθ^(π)dπ)π=π^⁢+∇π(H(t,Y(t),θ,π,p(t),q(t),r(t,z)))π=π^,θ=θ^(π^)∣ℰt]=0.Choose π=π^.
Then, by (3.46) and (3.48), we have(3.49)E[∇π(H(t,Y(t),θ,π,p(t),q(t),r(t,z)))π=π^,θ=θ^(π^)∣ℰt]=0,that is,(3.50)E[S(t)α(t)p3(t)+S(t)σ(t)q3(t)+∫ℝ0S(t)γ(t,z)r3(t,z)ν(dz)∣ℰt]=0.Substituting the values p3(t), q3(t),
and r3(t,z) as in Lemma 3.1 into (3.50), we
get(3.51)E[S(t)Kθ(t){α(t)+σ(t)θ0(t)+∫ℝ0γ(t,z)θ1(t,z)ν(dz)}∣ℰt]=0.This gives,(3.52)Mθ^(π^)=0.


## 4. Maximum Principle for Stochastic Differential Games

Problem A is related to what is known as stochastic
games studied in [12].
Applying Theorem 2.1 in [12] to our setting we get the following jump-diffusion
version of the maximum principle (of Ferris and Mangasarian type [13]).Theorem 4.1 (maximum principle for stochastic differential games [12]).
Let(θ^,π^)∈Θ×Π and suppose that the adjoint equations (3.25),
(3.26), and (3.27) admit solutions (p^1(t),q^1(t),r^1(t,z)), (p^2(t),q^2(t),r^2(t,z)), and (p^3(t),q^3(t),r^3(t,z)),
respectively. Moreover, suppose that, for all t∈[0,T],
the following partial information maximum principle holds:(4.1)sup⁡θ∈Θ⁢E[H(t,Y(t),θ,π^(t),p^(t),q^(t),r^(t,z))∣ℰt]=E[H(t,Y(t),θ^(t),π^(t),p^(t),q^(t),r^(t,z))∣ℰt]=inf⁡π∈Π⁢E[H(t,Y(t),θ^(t),π,p^(t),q^(t),r^(t,z))∣ℰt].Suppose(4.2)θ⟶J(θ,π^)is⁢⁢concave,π⟶J(θ^,π)is⁢⁢convex.
Then (θ*,π*):=(θ^,π^) is an optimal control and(4.3)ΦGℰ(x)=inf⁡π∈Π(sup⁡θ∈Θ⁢J(θ,π))=sup⁡θ∈Θ(inf⁡π∈Π⁢J(θ,π))=sup⁡θ∈Θ⁢J(θ,π^)=inf⁡π∈Π⁢J(θ^,π)=J(θ^,π^).Theorem 4.2.
Letp1(t), p2(t) be, respectively, solutions of adjoint
equations (3.28), and let p1(t), p2(t), p3(t) be defined as in Lemma 3.1. Suppose θ→H(t,Y(t),θ,p(t);q(t),r(t,⋅)) is concave. Let (θ^(π^),π^) be an optimal pair for Problem A, as given in
Lemma 3.2. Then,(4.4)θˇ:=θ^(π^)is optimal for Problem B.Proof.
By Theorem4.1 for Problem B, θˇ solves Problem B under partial information ℰt if(4.5)sup⁡θ∈M⁢E[H(t,Y(t),θ,p(t),q(t),r(t,z))∣ℰt]=E[H(t,Y(t),θˇ,p(t),q(t),r(t,z))∣ℰt],that is, if there exists C=C(t) such that(4.6)E[∇θ(H(t,Y(t),θ,p(t),q(t),r(t,z)))−C(t)M(θ))θ=θˇ∣ℰt]=0,(4.7)E[Mθˇ(t)∣ℰt]=0.Let π^, θ^(π^) be as in Lemma 3.2. Then,(4.8)E[∇θ(H(t,Y(t),θ,π^(t),p(t),q(t),r(t,z))θ=θ^(π^(t))∣ℰt]=0,(4.9)E[Mθ^(π^)(t)∣ℰt]=0.Hence, by Lemma
3.1,
(4.10)0=E[∇θ{H(t,Y(t),θ,p(t),q(t),r(t,z))−S(t)π^(t)Kθ(t)(α(t)+2σ(t)θ0+2∫ℝ0γ(t,z)θ1(z)ν(dz))}θ=θ^(π^(t))∣ℰt]=E[∇θ(H(t,Y(t),θ,p(t),q(t),r(t,z))−2S(t)π^(t)Kθ(t)Mθ)θ=θ^(π^(t))∣ℰt].Therefore, if we
choose(4.11)C(t)=2S(t)π^(t)Kθ(t),we see that (4.6) holds with θˇ=θ^(π^),
as claimed.


## 5. Risk Indifference Pricing

Let(θ*,π*)=(θˇ,π^) be as in Theorem 4.2 with the corresponding
state process Y*=Yθ*,π*.
Suppose that Y=Yθ^(π^),π is the state process corresponding to an
optimal control (θ^(π^),π).
Then, the value function ΦGℰ,
which is defined by (3.17) and (3.18), becomes(5.1)ΦGℰ(t,y)=inf⁡π∈Π⁢(sup⁡θ∈Θ⁢Jθ,π(t,y))=inf⁡π∈Π(sup⁡θ∈Θ⁢Ey[−∫tTΛ(θ(u,Y(u)))du−h(Kθ(T),S(T))+Kθ(T)g(S(T))−Kθ(T)X(π)(T)])=inf⁡π∈Π(Ey[−∫tTΛ(θ*(u,Y*(u)))du−h(Kθ*(T),S(T))+Kθ*(T)g(S(T))−Kθ*(T)X(π)(T)]).We have that, for all π∈Π,(5.2)Ey[Kθ*(T)X(π)(T)]=Ey[Kθ(T)X(π)(T)]=kE(1/k)Qθˇk,s,x[X(π)(T)]=kx,since (1/k)Qθˇ is an equivalent martingale measure for ℰt-conditioned market. On the other hand, the
first part of (5.1) does not depend on the parameter π.
Hence, (5.1) becomes(5.3)ΦGℰ(t,y)=Ey[−∫tTΛ(θˇ(u,Y(u)))du−h(Kθˇ(T),S(T))+Kθˇ(T)g(S(T))]−kx=sup⁡θ∈M⁢J0θ(t,y)−kx=ΨGℰ(t,y)−kx.We have proved the following result for the relation
between the value function for Problem A and the value function for Problem B
in the partial information case that is the same as in Øksendal and Sulem
[7] for the full
information case.Lemma 5.1.
The relationship between the value
functionΨGℰ(t,y) for Problem B and the value function ΦGℰ(t,y) for Problem A is(5.4)ΦGℰ(t,y)=ΨGℰ(t,y)−kx.We now apply Lemma5.1 to find the risk indifference
price p=priskseller,
given as a solution of the equation(5.5)ΦGℰ(t,k,s,x+p)=Φ0ℰ(t,k,s,x).By Lemma 5.1, this
becomes(5.6)ΨGℰ(t,k,s)−k(x+p)=Ψ0ℰ(t,k,s)−kx,which has the
solution(5.7)p=priskseller=k−1(ΨGℰ(t,k,s)−Ψ0ℰ(t,k,s)).In particular, choosing k=1 (i.e., all measures Q∈ℒ are probability measures), we get the
following.Theorem 5.2.
Suppose that the conditions of
Theorem4.2 hold. Then, the risk indifference price for the seller of claim G, priskseller(G,ℰ),
is given by(5.8)priskseller(G,ℰ)=sup⁡Q∈ℳ{EQ[G]−ζ(Q)}−sup⁡Q∈ℳ{−ζ(Q)}.



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*Source: 821243-2008-11-18.xml*