Generalizing earlier work of Delbaen and Haezendonck for given compound renewal process

A basic method in mathematical finance is to replace the original probability measure with an equivalent martingale measure, sometimes called a risk-neutral measure. This measure is used for pricing and hedging given contingent claims (e.g., options, futures, etc.). In contrast to the situation of the classical Black–Scholes option pricing formula, where the equivalent martingale measure is unique, in actuarial mathematics that is certainly not the case.

The above fact was pointed out by Delbaen and Haezendonck in their pioneering paper [

However, there is one vital point about the (compound) Poisson processes which is their greatest weakness as far as practical applications are considered, and this is the fact that the variance is a linear function of time

In Section

In Section

In Section

A sequence

The next lemma is a general and helpful result, as it provides a clear understanding of the structure of

Fix an arbitrary

Clearly, for

To show (a), fix an arbitrary

Since

To show (b), let

For

To check the validity of

Applying similar arguments as above we obtain (b). □

For the definition of a

For a given aggregate claims process

Fix an arbitrary

For

For

Fix an arbitrary

But since

By (

The implication

Proposition

Take

In our next example we consider a renewal process with gamma distributed interarrival times.

Assume that

We know from Proposition

Inclusion

To check the validity of the inverse inclusion, fix an arbitrary

Note that the above lemma remains true, without the assumption

Let

Before we formulate the inverse of Proposition

By

For all

The following proposition shows that after changing the measure the process

Let

Fix an arbitrary

A well-known change of measure technique for compound renewal processes is to markovize the process and then to change the measure (cf. e.g. [

The next result is the desired characterization. Its proof is an immediate consequence of Propositions

In order to formulate the next results of this section, let us denote by

The following result allows us to convert any compound renewal process into a compound Poisson one through a change of measure.

Fix an arbitrary

For

For

The following example translates the results of Corollary

Fix an arbitrary

Conversely, let

In this section we first show that a martingale approach to premium calculation principles leads in the case of CRPs to CPPs, providing in this way a method to find progressively equivalent martingale measures. Next, using our results we show that if

In order to present the results of this section we recall the following notions. For a given real-valued process

Suppose that

Fix an arbitrary

For

The above claim is well known (cf. e.g. [

For

Thus, according to the above claim statement

For

For

For

Moreover, assuming statement

In the next proposition we find out a wide class of

Fix

For

For

The next theorem connects our results with the basic notion of no free lunch with vanishing risk ((NFLVR) for short) (see [

Fix an arbitrary

It is well known that the FTAP of Delbaen and Schachermayer uses P.A. Meyer’s

We have seen that the initial probability measure

In the next Examples

Let

Let

In our next example we show how one can obtain the Esscher principle by applying Proposition

Take

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. Due to them, the presentation of the results is more readable now.