^{1}

^{1,2}

^{1}

^{1}

^{2}

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 price

Suppose the value of a portfolio

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 [

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 [

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 [

The paper is organized as follows. In Section

Assume that a filtered probability space

A
nonnegative random variable

From now on, we
consider a European-type option whose payoff at time

Let

(convexity)

(monotonicity) if

If an investor sells a liability to pay out the amount

If the investor has not issued a claim (and hence no
initial payment is received), then the minimal risk for the investor
is

The seller's

In view of the general representation formula for
convex risk measures (see [

A map

Using the representation (

Thus, the problem of finding the risk indifference
price

Suppose in a financial market, there are two investment possibilities:

(i) a bond with unit price

(ii) a stock with price dynamics, for

Let

In the sequel, we will call a portfolio

Now, we define the measures

We set

We now define two sets

In particular, by the Girsanov theorem, all the
measures

We assume that the penalty function

Using the

Find

We will relate Problem A to the following stochastic
control problem:

Find

Define the

Similarly, the adjoint equations (corresponding to

Let

Differentiating both sides of (

Proceeding as above with the processes

Equations (

Let

The first-order conditions for a
maximum point

Problem A is related to what is known as stochastic
games studied in [

Let

Let

By Theorem

Let

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
[

The relationship between the value
function

We now apply Lemma

Suppose that the conditions of
Theorem