# A characterization of equivalent martingale probability measures in a mixed
renewal risk model with applications in Risk Theory

S. M. Tzaninis  Department of Statistics and Insurance Science
University of Piraeus
80 Karaoli and Dimitriou str.
185 34 Piraeus
Greece<EMAIL_ADDRESS>and  N. D. Macheras  Department of Statistics and
Insurance Science
University of Piraeus
80 Karaoli and Dimitriou str.
185 34 Piraeus
Greece<EMAIL_ADDRESS>
###### Abstract.

If a given aggregate process $S$ is a compound mixed renewal process under a
probability measure $P$, we provide a characterization of all probability
measures $Q$ on the domain of $P$ such that $Q$ and $P$ are progressively
equivalent and $S$ is converted into a compound mixed Poisson process under
$Q$. This result extends earlier works of Delbaen & Haezendonck [2], Embrechts
& Meister [5], Lyberopoulos & Macheras [11], and of the authors [14].
Implications to the ruin problem and to the computation of premium calculation
principles in an insurance market possessing the property of no free lunch
with vanishing risk are also discussed.

Key Words: Compound mixed renewal process, Change of measures, Martingale,
Martingale measures, Progressively equivalent (martingale) measures, Premium
calculation principle, Regular conditional probabilities, Ruin probability.

AMS Subject Classification (2010): Primary 60G55, 91B30 ; secondary 28A35,
60A10, 60G44, 60K05.

##  1\. Introduction

Given a price process $U_{\mathbb{T}}$ with $\mathbb{T}:=[0,T]$, where $T>0$
is a fixed time horizon, a basic method in Mathematical Finance is to replace
the initial probability measure $P$ by an equivalent one $Q$, which converts
$U_{\mathbb{T}}$ into a martingale with respect to $Q$. The new probability
measure, often called risk neutral or equivalent martingale measure, is then
used for pricing and hedging 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. In fact, if $U_{\mathbb{T}}$
represents the liabilities of an insurance company, then Delbaen & Haezendonck
[2] noticed that there exist infinitely many equivalent martingale measures
(EMM for short) for $U_{\mathbb{T}}$, so that pricing is directly linked with
an attitude towards risk (see [2], pp. 269–270, for more details). The latter
led Delbaen & Haezendonck to the problem of characterizing all those EMMs $Q$
which preserve the structure of a compound Poisson process.

The work of Delbaen & Haezendonck played a key role in understanding the
interplay between financial and actuarial pricing of insurance (see [4] for an
overview), and has influenced the studies of many researchers. The method
provided in [2] has been successfully applied to many areas of insurance
mathematics (see [14], p. 44, for more details).

Nevertheless, such a characterization of EMMs for $U_{\mathbb{T}}$ does not
always provide a viable pricing system in Actuarial practice, since it is not
appropriate to describe inhomogeneous risk portfolios. For this reason the
work of Delbaen & Haezendonck [2] was generalized by Embrechts & Meister [5]
and Lyberopoulos & Macheras [11] to mixed Poisson risk models.

However, since the (mixed) Poisson risk model presents some serious
deficiencies, as far as practical models are considered (see [14], p. 44 and
[18], pp. 1–2), it seems reasonable to investigate the existence of EMMs for
the price process $U_{\mathbb{T}}$ in the more general mixed renewal risk
model.

In [18], Corollary 4.8, a characterization of all progressively equivalent
probability measures that convert a compound mixed renewal process CMRP for
short) into a compound mixed Poisson one (CMPP for short) was proven. Relying
on the above result, we provide the aforementioned characterization for a
subclass of progressively equivalent measures, suitable for applications in
Risk Theory, which improves the structure of a CMRP by converting it into a
CMPP with better properties than those of [18], Corollary 4.8, see Theorem
3.5.

In Section 4, we first prove a martingale characterization of CMPPs in terms
of regular conditional probabilities, within the class of CMRPS, see Theorem
4.2. The previous result together with Theorem 3.5 enables us to prove the
desired characterization of progressively EMMs for CMRPs, see Theorem 4.4,
generalizing in this way the corresponding results of of [14], Proposition
4.2, and [11], Proposition 5.1(ii). Using the latter result, we find out a
wide class of canonical price processes satisfying the condition of no free
lunch with vanishing risk (NFLVR for short), see Theorem 4.6, connecting in
this way our results with this basic notion of mathematical finance.

An implication of Theorem 4.4 to the ruin problem is discussed in Section 5,
where a characterization of all those progressively equivalent to $P$
martingale measures $Q$ for the reserve process $R^{u}$ satisfying condition
$c(\varTheta)=p(Q,\varTheta)$ is given, enabling us to calculate the ruin
probability $\psi(u)$ under $P$, see Theorem 5.4.

Finally, implications of our results to the pricing of actuarial risks
(premium calculation principles) in an insurance market possessing the
property of NFLVR are discussed in Section 6.

##  2\. Preliminaries

Throughout this paper, unless stated otherwise, $(\varOmega,\varSigma,P)$ is a
fixed but arbitrary probability. The symbols $\mathcal{L}^{1}(P)$ and
$\mathcal{L}^{2}(P)$ stand for the families of all real-valued $P$-integrable
and $P$-square integrable functions on $\varOmega$, respectively. Functions
that are $P$-a.s. equal are not identified. We denote by $\sigma(\mathcal{G})$
the $\sigma$-algebra generated by a family $\mathcal{G}$ of subsets of
$\varOmega$. Given a topology $\mathfrak{T}$ on $\varOmega$ we write
${\mathfrak{B}}(\varOmega)$ for its Borel $\sigma$-algebra on $\varOmega$,
i.e. the $\sigma$-algebra generated by $\mathfrak{T}$. Our measure theoretic
terminology is standard and generally follows [1]. For the definitions of
real-valued random variables and random variables we refer to [1], p. 308. We
apply the notation $P_{X}:=P_{X}(\theta):={\bf{K}}(\theta)$ to mean that $X$
is distributed according to the law ${\bf{K}}(\theta)$, where $\theta\in
D\subseteq{\mathbb{R}}^{d}$ ($d\in{\mathbb{N}}$) is the parameter of the
distribution. We denote again by ${\bf K}(\theta)$ the distribution function
induced by the probability distribution ${\bf K}(\theta)$. Notation ${\bf
Ga}(b,a)$, where $a,b\in(0,\infty)$, stands for the law of gamma distribution
(cf. e.g. [17], p. 180). In particular, ${\bf Ga}(b,1)={\bf Exp}(b)$ stands
for the law of exponential distribution. For two real-valued random variables
$X$ and $Y$ we write $X=Y$ $P$-a.s. if $\\{X\neq Y\\}$ is a $P$-null set. If
$A\subseteq\varOmega$, then $A^{c}:=\varOmega\setminus A$, while $\chi_{A}$
denotes the indicator (or characteristic) function of the set $A$. For a map
$f:D\longrightarrow E$ and for a non-empty set $A\subseteq D$ we denote by
$f\upharpoonright A$ the restriction of $f$ to $A$. We write
$\mathbb{E}_{P}[X\mid{\mathcal{F}}]$ for a version of a conditional
expectation (under $P$) of $X\in{\mathcal{L}}^{1}(P)$ given a
$\sigma$-subalgebra $\mathcal{F}$ of $\varSigma$. For
$X:=\chi_{E}\in\mathcal{L}^{1}(P)$ with $E\in\varSigma$ we set
$P(E\mid\mathcal{F}):=\mathbb{E}_{P}[\chi_{E}\mid\mathcal{F}]$. For the
unexplained terminology of Probability and Risk Theory we refer to [17].

Given two measurable spaces $(\varOmega,\varSigma)$ and $(\varUpsilon,H)$, a
function $k$ from $\varOmega\times H$ into $[0,1]$ is a $\varSigma$-$H$-Markov
kernel if it has the following properties:

  * (k1)

The set-function $B\longmapsto k(\omega,B)$ is a probability measure on $H$
for any fixed $\omega\in\varOmega$.

  * (k2)

The function $\omega\longmapsto k(\omega,B)$ is $\varSigma$-measurable for any
fixed $B\in{H}$.

In particular, given a real-valued random variable $X$ on $\varOmega$ and a
$d$-dimensional random vector $\varTheta$ on $\varOmega$, a conditional
distribution of $X$ over $\varTheta$ is a
$\sigma(\varTheta)$-${\mathfrak{B}}$-Markov kernel denoted by
$P_{X\mid\varTheta}:=P_{X\mid\sigma(\varTheta)}$ and satisfying for each
$B\in{\mathfrak{B}}$ condition

$P_{X\mid\varTheta}(\bullet,B)=P(X^{-1}(B)\mid\sigma(\varTheta))(\bullet)\quad{P}\upharpoonright\sigma(\varTheta)-\mbox{a.s.}.$

Clearly, for every ${\mathfrak{B}}_{d}$-${\mathfrak{B}}$-Markov kernel $k$,
the map $K(\varTheta)$ from $\varOmega\times{\mathfrak{B}}$ into $[0,1]$
defined by means of

$K(\varTheta)(\omega,B):=(k(\bullet,B)\circ\varTheta)(\omega)\quad\mbox{for
any}\;\;(\omega,B)\in\varOmega\times{\mathfrak{B}}$

is a $\sigma(\varTheta)$-${\mathfrak{B}}$-Markov kernel. Then for
$\theta=\varTheta(\omega)$ with $\omega\in\varOmega$ the probability measures
$k(\theta,\bullet)$ are distributions on ${\mathfrak{B}}$ and so we may write
$\mathbf{K}(\theta)(\bullet)$ instead of $k(\theta,\bullet)$. Consequently, in
this case $K(\varTheta)$ will be denoted by $\mathbf{K}(\varTheta)$.

For any real-valued random variables $X$, $Y$ on $\varOmega$ we say that
$P_{X|\varTheta}$ and $P_{Y|\varTheta}$ are
$P\upharpoonright\sigma(\varTheta)$-equivalent and we write
$P_{X|\varTheta}=P_{Y|\varTheta}$ $P\upharpoonright\sigma(\varTheta)$-a.s., if
there exists a $P$-null set $M\in\sigma(\varTheta)$ such that for any
$\omega\notin M$ and $B\in{\mathfrak{B}}$ the equality
$P_{X|\varTheta}(\omega,B)=P_{Y|\varTheta}(\omega,B)$ holds true.

##  3\. Compound Mixed Renewal Processes and Progressively Equivalent Measures

We first recall some additional background material, needed in this section.

A family $N:=\\{N_{t}\\}_{t\in\mathbb{R}_{+}}$ of random variables from
$(\varOmega,\varSigma)$ into
$(\overline{{\mathbb{R}}},{\mathfrak{B}}(\overline{\mathbb{R}}))$ is called a
counting process if there exists a $P$-null set $\varOmega_{N}\in\varSigma$
such that the process $N$ restricted on $\varOmega\setminus\varOmega_{N}$
takes values in ${\mathbb{N}}_{0}\cup\\{\infty\\}$, has right-continuous
paths, presents jumps of size (at most) one, vanishes at $t=0$ and increases
to infinity. Without loss of generality we may and do assume, that
$\varOmega_{N}=\emptyset$. Denote by $T:=\\{T_{n}\\}_{n\in\mathbb{N}_{0}}$ and
$W:=\\{W_{n}\\}_{n\in\mathbb{N}}$ the arrival process and interarrival
process, respectively (cf. e.g. [17], Section 1.1, page 6 for the definitions)
associated with $N$. Note also that every arrival process induces a counting
process, and vice versa (cf. e.g. [17], Theorem 2.1.1).

Furthermore, let $X:=\\{X_{n}\\}_{n\in\mathbb{N}}$ be a sequence of positive
real-valued random variables on $\varOmega$, and for any $t\geq 0$ define

$S_{t}:=\begin{cases}\sum^{N_{t}}_{k=1}X_{k}&\text{if}\;\;t>0;\\\
0&\text{if}\;\;t=0.\end{cases}$

Accordingly, the sequence $X$ is said to be the claim size process, and the
family $S:=\\{S_{t}\\}_{t\in\mathbb{R}_{+}}$ of real-valued random variables
on $\varOmega$ is said to be the aggregate claims process induced by the pair
$(N,X)$. Recall that a pair $(N,X)$ is called a risk process, if $N$ is a
counting process, $X$ is $P$-i.i.d. and the processes $N$ and $X$ are
$P$-independent (see [17], Chapter 6, Section 6.1).

Throughout what follows, unless stated otherwise,
$(\varUpsilon,H):=((0,\infty),{\mathfrak{B}}(\varUpsilon))$ and $\varTheta$ is
a $d$–dimensional random vector on $\varOmega$ with values on
$D\subseteq{\mathbb{R}}^{d}$ ($d\in{\mathbb{N}}$).

For the definition of a $P$–conditionally (stochastically) independent process
as well as of a $P$–conditionally identically distributed process over
$\sigma(\varTheta)$ we refer to [18]. We say that a process is
$P$–conditionally (stochastically) independent or identically distributed
given $\varTheta$, if it is conditionally independent or identically
distributed over the $\sigma$-algebra $\sigma(\varTheta)$.

For the rest of the paper we simply write “conditionally” in the place of
“conditionally given $\varTheta$” whenever conditioning refers to $\varTheta$.

The following definition has been introduced in [10], Definition 3.1, see also
[13], Definition 3.2 (b).

######  Definition 3.1.

A counting process $N$ is said to be a $P$-mixed renewal process with mixing
parameter $\varTheta$ and interarrival time conditional distribution
$\bf{K}(\varTheta)$ (written $P$-MRP$(\bf{K}(\varTheta))$ for short), if the
induced interarrival process $W$ is $P$-conditionally independent and

$\forall\;n\in{\mathbb{N}}\qquad[P_{W_{n}\mid\varTheta}=\bf{K}(\varTheta)\quad
P\upharpoonright\sigma(\varTheta)\text{-a.s.}].$

In particular, if the distribution of $\varTheta$ is degenerate at some point
$\theta_{0}\in D$, then the counting process $N$ becomes a $P$-renewal process
with interarrival time distribution $\bf{K}(\theta_{0})$ (written
$P$-RP$(\bf{K}(\theta_{0}))$ for short).

Accordingly, an aggregate claims process $S$ induced by a $P$–risk process
$(N,X)$ such that $N$ is a $P$–MRP$(\bf{K}(\varTheta))$ is called a compound
mixed renewal process with parameters ${\bf K}(\varTheta)$ and $P_{X_{1}}$
($P$–CMRP$({\bf K}(\varTheta),P_{X_{1}})$ for short). In particular, if the
distribution of $\varTheta$ is degenerate at $\theta_{0}\in D$ then $S$ is
called a compound renewal process with parameters ${\bf K}(\theta_{0})$ and
$P_{X_{1}}$ ($P$–CRP$({\bf K}(\theta_{0}),P_{X_{1}})$ for short).

Throughout what follows we denote again by ${\bf K}(\varTheta)$ and ${\bf
K}(\theta)$ the conditional distribution function and the distribution
function induced by the conditional probability distribution ${\bf
K}(\varTheta)$ and the probability distribution ${\bf K}(\theta)$,
respectively.

The following conditions will be useful for our investigations:

  * (a1)

The pair $(W,X)$ is $P$–conditionally independent.

  * (a2)

The random vector $\varTheta$ and the process $X$ are $P$–(unconditionally)
independent.

Next, whenever condition (a1) and (a2) holds true we shall write that the
quadruplet $(P,W,X,\varTheta)$ or (if no confusion arises) the probability
measure $P$ satisfies (a1) and (a2), respectively.

Since conditioning is involved in the definition of (compound) mixed renewal
processes, it is expected that regular conditional probabilities (or
disintegrations) will play a fundamental role in their analysis. To this
purpose we recall the following definition.

######  Definition 3.2.

Let $(\varUpsilon,H,R)$ be a probability probability space. A family
$\\{P_{y}\\}_{y\in\varUpsilon}$ of probability measures on $\varSigma$ is
called a regular conditional probability (rcp for short) of $P$ over $R$ if

  * (d1)

for each $E\in\varSigma$ the map $y\longmapsto P_{y}(E)$ is $H$-measurable;

  * (d2)

$\int P_{y}(E)\,R(dy)=P(E)$ for each $E\in\varSigma$.

We could use the term of disintegration instead, but it is better to reserve
that term to the general case when $P_{y}$’s may be defined on different
domains (see[15]).

If $f:\varOmega\longrightarrow\varUpsilon$ is an inverse-measure-preserving
function (i.e. $P(f^{-1}(B))=R(B)$ for each $B\in{H}$), a rcp
$\\{P_{y}\\}_{y\in\varUpsilon}$ of $P$ over $R$ is called consistent with $f$
if, for each $B\in{H}$, the equality $P_{y}(f^{-1}(B))=1$ holds for $R$–almost
every $y\in B$.

We say that a rcp $\\{P_{y}\\}_{y\in\varUpsilon}$ of $P$ over $R$ consistent
with $f$ is essentially unique, if for any other rcp
$\\{\widetilde{P}_{y}\\}_{y\in\varUpsilon}$ of $P$ over $R$ consistent with
$f$ there exists a $R$–null set $N\in H$ such that for any $y\notin N$ the
equality $P_{y}=\widetilde{P}_{y}$ holds true.

Let ${\mathbb{T}}\subseteq{\mathbb{R}}_{+}$ with $0\in{\mathbb{T}}$. For a
process $Y_{\mathbb{T}}:=\\{Y_{t}\\}_{t\in{\mathbb{T}}}$ denote by
${\mathcal{F}}_{\mathbb{T}}^{Y}:=\\{{\mathcal{F}}^{Y}_{t}\\}_{t\in{\mathbb{T}}}$
the canonical filtration of $Y_{\mathbb{T}}$. For
${\mathbb{T}}={\mathbb{R}}_{+}$ or ${\mathbb{T}}={\mathbb{N}}_{0}$ we simply
write ${\mathcal{F}}^{Y}$ instead of ${\mathcal{F}}^{Y}_{{\mathbb{R}}_{+}}$ or
${\mathcal{F}}^{Y}_{{\mathbb{N}}_{0}}$, respectively. Also, we write
${\mathcal{F}}:=\\{{\mathcal{F}}_{t}\\}_{t\in{\mathbb{R}}_{+}}$, where
${\mathcal{F}}_{t}:=\sigma({\mathcal{F}}^{S}_{t}\cup\sigma(\varTheta))$, for
the canonical filtration of $S$ and $\varTheta$,
${\mathcal{F}}^{S}_{\infty}:=\sigma(\bigcup_{t\in{\mathbb{R}}_{+}}{\mathcal{F}}^{S}_{t})$
and
${\mathcal{F}}_{\infty}:=\sigma({\mathcal{F}}^{S}_{\infty}\cup\sigma(\varTheta))$.

For the definition of a $(P,\mathcal{Z})$–martingale in $\mathcal{L}^{1}(P)$,
where $\mathcal{Z}=\\{\mathcal{Z}_{t}\\}_{t\in{\mathbb{R}}_{+}}$ is a
filtration for $(\varOmega,\varSigma)$, we refer to [17], p. 25. If in
addition $Z_{t}\in{\mathcal{L}}^{2}(P)$ for any $t\in{\mathbb{R}}_{+}$, then
$\\{Z_{t}\\}_{t\in{\mathbb{R}}_{+}}$ is called a $(P,\mathcal{Z})$–martingale
$\\{Z_{t}\\}_{t\in{\mathbb{R}}_{+}}$ in $\mathcal{L}^{2}(P)$. A
$(P,\mathcal{Z})$–martingale $\\{Z_{t}\\}_{t\in{\mathbb{R}}_{+}}$ in
$\mathcal{L}^{1}(P)$ is $P$–a.s. positive, if $Z_{t}$ is $P$–a.s. positive for
each $t\geq 0$. For $\ell\in\\{1,2\\}$ and $\mathcal{Z}={\mathcal{F}}$ we
write “martingale in $\mathcal{L}^{\ell}(P)$” instead of
“$(P,\mathcal{Z})$–martingale in $\mathcal{L}^{\ell}(P)$”, for simplicity.

######  Notations 3.3.

Denote by $\mathfrak{M}^{k}(D)$ ($k\in{\mathbb{N}}$) the class of all
${\mathfrak{B}}(D)$–${\mathfrak{B}}({\mathbb{R}}^{k})$–measurable functions on
$D$. In the special case $k=1$ we write
$\mathfrak{M}(D):=\mathfrak{M}^{1}(D)$. By $\mathfrak{M}_{+}(D)$ will be
denoted the class of all positive ${\mathfrak{B}}(D)$–measurable functions on
$D$. Fix on arbitrary $\ell\in\\{1,2\\}$ and $\rho\in\mathfrak{M}^{k}(D)$.

(a) The class of all real–valued ${\mathfrak{B}}(\varUpsilon\times
D)$–measurable functions $\beta$ on $\varUpsilon\times D$, defined by means of

$\beta(x,\theta):=\gamma(x)+\alpha(\theta)\,\,\text{ for any
}(x,\theta)\in\varUpsilon\times D,$

where $\gamma$ is a real-valued ${\mathfrak{B}}(\varUpsilon)$–measurable
function satisfying conditions

$\mathbb{E}_{P}\left[e^{\gamma(X_{1})}\right]=1\,\,\text{ and
}\,\,\mathbb{E}_{P}\left[X^{\ell}_{1}\cdot e^{\gamma(X_{1})}\right]<\infty$

and $\alpha\in\mathfrak{M}(D)$, will be denoted by
${\mathcal{F}}^{\ell}_{P,\varTheta}:={\mathcal{F}}^{\ell}_{P,\varTheta,X_{1}}$.

(b) The class of all real–valued ${\mathfrak{B}}(D)$–measurable functions
$\xi$ on $D$ such that

$P_{\varTheta}(\\{\xi>0\\})=1\,\,\text{ and }\mathbb{E}_{P}[\xi(\varTheta)]=1$

is denoted by
$\mathcal{R}_{+}(D):=\mathcal{R}_{+}(D,\mathfrak{B}(D),P_{\varTheta})$.

For given $\beta\in{{\mathcal{F}}}^{\ell}_{P,\varTheta}$, denote by
$\mathcal{R}^{\ast,\ell}_{+}(D):=\mathcal{R}^{\ast,\ell}_{+,\beta}(D)$ the
class of all $\xi\in\mathcal{R}_{+}(D)$ such that

$\xi(\varTheta)\cdot\big{(}\frac{e^{\alpha(\varTheta)}}{\mathbb{E}_{P}[W_{1}\mid\varTheta]}\big{)}^{\ell}\in{\mathcal{L}}^{1}(P).$

Clearly $\mathcal{R}^{\ast,\ell}_{+}(D)\subseteq\mathcal{R}_{+}(D)$.

(c) A probability measure $Q$ on $\varSigma$ is called progressively
equivalent to $P$, if it is equivalent (in the sense of absolute continuity)
to $P$ on each $\sigma$-algebra $\mathcal{F}_{t}$, in symbols
$Q\upharpoonright\mathcal{F}_{t}\sim{P}\upharpoonright\mathcal{F}_{t}$ for any
$t\in{\mathbb{R}}_{+}$.

(d) The class of all probability measures $Q$ on $\varSigma$, which satisfy
conditions (a1) and (a2), are progressively equivalent to $P$, and such that
$S$ is a $Q$–CMRP$({\bf\Lambda}(\rho(\varTheta)),Q_{X_{1}})$ with
$\mathbb{E}_{Q}[X^{\ell}_{1}]<\infty$, will be denoted by
${\mathcal{M}}^{\ell}_{S,{\bf\Lambda}(\rho(\varTheta))}:={\mathcal{M}}^{\ell}_{S,{\bf\Lambda}(\rho(\varTheta)),P,{X_{1}}}$.
In the special case $d=k$ and $\rho:=id_{D}$ we write
${\mathcal{M}}^{\ell}_{S,{\bf\Lambda}(\varTheta)}:={\mathcal{M}}^{\ell}_{S,{\bf\Lambda}(\rho(\varTheta))}$
for simplicity.

Denote by ${\mathcal{M}}^{\ast,\ell}_{S,{\bf\Lambda}(\rho(\varTheta))}$ the
class of all measures
$Q\in{\mathcal{M}}^{\ell}_{S,{\bf\Lambda}(\rho(\varTheta))}$ satisfying
condition

$\big{(}1/\mathbb{E}_{Q}[W_{1}\mid\varTheta]\big{)}^{\ell}\in{\mathcal{L}}^{1}(Q),$

under the assumption $\mathbb{E}_{Q}[W_{1}|\varTheta]\in\varUpsilon\quad
Q\upharpoonright\sigma(\varTheta)$–a.s.. Clearly
${\mathcal{M}}^{\ast,\ell}_{S,{\bf\Lambda}(\rho(\varTheta))}\subseteq{\mathcal{M}}^{\ell}_{S,{\bf\Lambda}(\rho(\varTheta))}$.

(e) For given $\theta\in D$, denote by
${{\mathcal{M}}}^{\ell}_{S,{\bf\Lambda}(\rho(\theta))}$ the class of all
probability measures $Q_{\theta}$ on $\varSigma$, such that
$Q_{\theta}\upharpoonright{\mathcal{F}}_{t}\sim
P_{\theta}\upharpoonright{\mathcal{F}}_{t}$ for any $t\in{\mathbb{R}}_{+}$ and
$S$ is a $Q_{\theta}$-CRP$({\bf\Lambda}(\rho(\theta)),(Q_{\theta})_{X_{1}})$
with $\mathbb{E}_{Q_{\theta}}[X_{1}^{\ell}]<\infty$.

######  Remark 3.4.

For $\ell\in\\{1,2\\}$ the following assertions are equivalent:

  1. (i)

$P\in{\mathcal{M}}^{\ell}_{S,{\bf K}(\varTheta)}$;

  2. (ii)

there exists a $P_{\varTheta}$–null set $L_{P}\in{\mathfrak{B}}(D)$ such that
$P_{\theta}\in{\mathcal{M}}^{\ell}_{S,{\bf K}(\theta)}$ with
$(P_{\theta})_{X_{1}}=P_{X_{1}}$ for any $\theta\notin L_{P}$.

The validity of the above remark is an immediate consequence of [18],
Proposition 3.3 along with the fact that $X_{1}\in\mathcal{L}^{\ell}(P)$ if
and only if $X_{1}\in\mathcal{L}^{\ell}(P_{\theta})$ for any
$\theta\notin{L}_{P}$.

Throughout what follows, unless stated otherwise, put
$\widetilde{\varOmega}:=\varUpsilon^{{\mathbb{N}}}\times\varUpsilon^{{\mathbb{N}}}$,
$\widetilde{\varSigma}:={\mathfrak{B}}(\widetilde{\varOmega})={\mathfrak{B}}(\varUpsilon)_{{\mathbb{N}}}\otimes{\mathfrak{B}}(\varUpsilon)_{{\mathbb{N}}}$,
$\varOmega:=\widetilde{\varOmega}\times D$ and
$\varSigma:=\widetilde{\varSigma}\otimes{\mathfrak{B}}(D)$, let
$\ell\in\\{1,2\\}$, consider $\varTheta$,
$P\in\mathcal{M}^{\ell}_{S,\mathbf{K}(\varTheta)}$ and
$\\{P_{\theta}\\}_{\theta\in{D}}$ as in [18], Proposition 4.1, put
$S_{t}^{(\beta)}(\varTheta):=\sum^{N_{t}}_{j=1}\beta(X_{j},\varTheta)$ for any
$\beta\in{\mathcal{F}}^{\ell}_{P,\varTheta}$, and assume that
$\mathbb{E}_{P}[W_{1}|\varTheta]\in\varUpsilon\quad
P\upharpoonright\sigma(\varTheta)$–a.s..

In the next result we give a characterization of all elements of
${\mathcal{M}}^{\ast,\ell}_{S,\mathbf{Exp}(\rho(\varTheta))}$ in terms of the
elements of
${{\mathcal{F}}}^{\ell}_{P,\varTheta}\times\mathcal{R}^{\ast,\ell}_{+}(D)$.

######  Theorem 3.5.

If $P\in{\mathcal{M}}^{\ast,\ell}_{S,{\bf K}(\varTheta)}$ then the following
assertions hold true:

  1. (i)

for each pair
$(\rho,Q)\in\mathfrak{M}_{+}(D)\times{\mathcal{M}}^{\ast,\ell}_{S,{\bf
Exp}(\rho(\varTheta))}$ there exists an essentially unique pair
$(\beta,\xi)\in{{\mathcal{F}}}^{\ell}_{P,\varTheta}\times\mathcal{R}^{\ast,\ell}_{+}(D)$,
where $\xi$ is a Radon-Nikodým derivative of $Q_{\varTheta}$ with respect to
$P_{\varTheta}$, satisfying conditions

($\ast$) $\gamma=\ln
f\quad\text{and}\quad\alpha(\varTheta)=\ln\rho(\varTheta)+\ln\mathbb{E}_{P}[W_{1}\mid\varTheta]\quad
P\upharpoonright\sigma(\varTheta)\text{--a.s.},$

where $f$ is a $P_{X_{1}}$–a.s. positive Radon-Nikodým derivative of
$Q_{X_{1}}$ with respect to $P_{X_{1}}$, and

($RPM_{\xi}$) $Q(A)=\int_{A}M^{(\beta)}_{t}(\varTheta)\,dP\quad\text{for
all}\,\,\,0\leq u\leq t\,\,\text{and}\,\,A\in{\mathcal{F}}_{u},$

with

$M^{(\beta)}_{t}(\varTheta):=\xi(\varTheta)\cdot\frac{e^{S_{t}^{(\beta)}(\varTheta)-\rho(\varTheta)\cdot(t-T_{N_{t}})}\cdot(\rho(\varTheta)\cdot\mathbb{E}_{P}[W_{1}\mid\varTheta])^{-N_{t}}}{1-{\bf{K}}(\varTheta)(t-T_{N_{t}})}\cdot\prod_{j=1}^{N_{t}}\frac{d{\bf
Exp}(\rho(\varTheta))}{d{\bf K}(\varTheta)}(W_{j}),$

and such that the family
$M^{(\beta)}(\varTheta):=\\{M^{(\beta)}_{t}(\varTheta)\\}_{t\in{\mathbb{R}}_{+}}$
is a $P$–a.s. positive martingale in ${\mathcal{L}}^{1}(P)$ satisfying
condition $\mathbb{E}_{P}[M^{(\beta)}_{t}(\varTheta)]=1$.;

  2. (ii)

conversely, for each pair
$(\beta,\xi)\in{{\mathcal{F}}}^{\ell}_{P,\varTheta}\times\mathcal{R}^{\ast,\ell}_{+}(D)$
there exists a unique pair
$(\rho,Q)\in\mathfrak{M}_{+}(D)\times{\mathcal{M}}^{\ast,\ell}_{S,{\bf
Exp}(\rho(\varTheta))}$ determined by ($RPM_{\xi}$), satisfying condition
($\ast$ ‣ 1) and such that the family $M^{(\beta)}(\varTheta)$ involved in
($RPM_{\xi}$) is a $P$–a.s. positive martingale in $\mathcal{L}^{1}(P)$;

  3. (iii)

in both cases $(i)$ and $(ii)$, there exist an essentially unique rcp
$\\{Q_{\theta}\\}_{\theta\in D}$ of $Q$ over $Q_{\varTheta}$ consistent with
$\varTheta$ and a $P_{\varTheta}$–null set
$\widetilde{L}_{\ast\ast}\in{\mathfrak{B}}(D)$, satisfying for any
$\theta\notin\widetilde{L}_{\ast\ast}$ conditions
$Q_{\theta}\in{{\mathcal{M}}}^{\ell}_{S,{\bf Exp}(\rho(\theta))}$,

($\widetilde{\ast}$) $\gamma=\ln
f\quad\text{and}\quad\rho(\theta)=\frac{e^{\alpha(\theta)}}{\mathbb{E}_{P_{\theta}}[W_{1}]},$

where $f$ is a $P_{X_{1}}$–a.s. positive Radon-Nikodým derivative of
$Q_{X_{1}}$ with respect to $P_{X_{1}}$, and

($RPM_{\theta}$)
$Q_{\theta}(A)=\int_{A}\widetilde{M}^{(\beta)}_{t}(\theta)\,dP_{\theta}\quad\text{for
all}\,\,\,0\leq u\leq t\,\,\text{and}\,\,A\in{\mathcal{F}}_{u},$

where

$\widetilde{M}^{(\beta)}_{t}(\theta):=\frac{e^{S_{t}^{(\beta)}(\theta)-\rho(\theta)\cdot(t-T_{N_{t}})}\cdot(\rho(\theta)\cdot\mathbb{E}_{P_{\theta}}[W_{1}])^{-N_{t}}}{1-{\bf{K}}(\theta)(t-T_{N_{t}})}\cdot\prod_{j=1}^{N_{t}}\frac{d{\bf
Exp}(\rho(\theta))}{d{\bf K}(\theta)}(W_{j}),$

and such that the family
$\widetilde{M}^{(\beta)}(\theta):=\\{\widetilde{M}^{(\beta)}_{t}(\theta)\\}_{t\in{\mathbb{R}}_{+}}$,
involved in ($RPM_{\theta}$), is a $P_{\theta}$–a.s. positive martingale in
${\mathcal{L}}^{1}(P_{\theta})$ satisfying condition
$\mathbb{E}_{P_{\theta}}[\widetilde{M}^{(\beta)}_{t}(\theta)]=1$.

###### Proof.

Ad (i): Since ${\mathcal{M}}^{\ast,\ell}_{S,{\bf
Exp}(\rho(\varTheta))}\subseteq{\mathcal{M}}^{\ell}_{S,{\bf
Exp}(\rho(\varTheta))}$, we get by [18], Corollary 4.8(i) along Remark 4.9(b),
that there exists an essentially unique pair
$(\beta,\xi)\in{{\mathcal{F}}}^{\ell}_{P,\varTheta}\times\mathcal{R}_{+}(D)$,
where $\xi$ is a Radon-Nikodým derivative of $Q_{\varTheta}$ with respect to
$P_{\varTheta}$, satisfying conditions ($\ast$ ‣ 1) and ($RPM_{\xi}$). But
since $Q\in{\mathcal{M}}^{\ast,\ell}_{S,{\bf Exp}(\rho(\varTheta))}$, we have
that $\rho(\varTheta)^{\ell}\in\mathcal{L}^{1}(Q)$, implying that
$\xi(\varTheta)\cdot(e^{\alpha(\varTheta)}/\mathbb{E}_{P}[W_{1}\mid\varTheta])^{\ell}\in{\mathcal{L}}^{1}(P)$;
hence
$(\beta,\xi)\in{\mathcal{F}}^{\ell}_{P,\varTheta}\times\mathcal{R}^{\ast,\ell}_{+}(D)$.

Ad (ii): Since
${\mathcal{F}}^{\ell}_{P,\varTheta}\subseteq{\mathcal{F}}_{P,\varTheta}$ and
$\mathcal{R}^{\ast,\ell}_{+}(D)\subseteq\mathcal{R}_{+}(D)$, it follows by
[18], Corollary 4.8(ii) along with Remark 4.9(b), that there exists a unique
pair $(\rho,Q)\in\mathfrak{M}_{+}(D)\times{\mathcal{M}}^{\ell}_{S,{\bf
Exp}(\rho(\varTheta))}$ determined by ($RPM_{\xi}$) and satisfying condition
($\ast$ ‣ 1), implying along with the assumption of (ii) that
$\xi(\varTheta)\cdot(e^{\alpha(\varTheta)}/\mathbb{E}_{P}[W_{1}\mid\varTheta])^{\ell}\in{\mathcal{L}}^{1}(P)$;
hence
$\left(1/\mathbb{E}_{Q}[W_{1}\mid\varTheta]\right)^{\ell}\in{\mathcal{L}}^{1}(Q)$.
Thus, $(\rho,Q)\in\mathfrak{M}_{+}(D)\times{\mathcal{M}}^{\ast,\ell}_{S,{\bf
Exp}(\rho(\varTheta))}$.

Ad (iii): Since ${\mathcal{M}}^{\ast,\ell}_{S,{\bf
Exp}(\rho(\varTheta))}\subseteq{\mathcal{M}}^{\ell}_{S,{\bf
Exp}(\rho(\varTheta))}$, applying [18], Corollary 4.8(iii) along with Remark
4.9(b), we get that there exists a $P_{\varTheta}$–null set
$\widetilde{L}_{\ast\ast}$, such that for any
$\theta\notin\widetilde{L}_{\ast\ast}$ conditions
$Q_{\theta}\in{\mathcal{M}}^{\ell}_{S,{\bf Exp}(\rho(\theta))}$,
($\widetilde{\ast}$ ‣ 3) and ($RPM_{\theta}$) hold true. ∎

##  4\. Compound Mixed Renewal Processes and Martingales

In this section applying Theorem 3.5 we find out two canonical processes
satisfying the condition of no free lunch with vanishing risk (written NFLVR
for short) (see [3], Definition 8.1.2), connecting in this way our results
with this basic notion in mathematical finance.

In order to present the results of this section we recall the following
notions. For a given real-valued process
$Y:=\\{Y_{t}\\}_{t\in{\mathbb{R}}_{+}}$ on $(\varOmega,\varSigma)$ a
probability measure $Q$ on $\varSigma$ is called a martingale measure for $Y$,
if $Y$ is a martingale in ${\mathcal{L}}^{1}(Q)$. We will say that $Y$
satisfies condition (PEMM) if there exists a progressively equivalent to $P$
martingale measure $Q$ for $Y$. Moreover, let $T>0$, $\mathbb{T}:=[0,T]$,
$Q_{T}:=Q\upharpoonright{\mathcal{F}}_{T}$,
$Y_{\mathbb{T}}:=\\{Y_{t}\\}_{t\in\mathbb{T}}$ and
${\mathcal{F}}_{\mathbb{T}}:=\\{{\mathcal{F}}_{t}\\}_{t\in\mathbb{T}}$. We
will say that the process $Y_{\mathbb{T}}$ satisfies condition (EMM) if there
exists an equivalent martingale measures for $Y_{\mathbb{T}}$, i.e. a
probability measure $Q_{T}$ on ${\mathcal{F}}_{T}$ such that $Q_{T}\sim P_{T}$
and $Y_{{\mathbb{T}}}$ is a ${\mathcal{F}}_{\mathbb{T}}$-martingale in
${\mathcal{L}}^{1}(Q)$.

Throughout what follows assume that $W_{1}\in{\mathcal{L}}^{1}(P)$.

######  Proposition 4.1.

For $\varTheta\in{\mathcal{L}}^{1}(P)$, define the function
$v:D\longrightarrow\varUpsilon$ by means of
$v(\theta):=\frac{1}{\mathbb{E}_{P}[W_{1}]}$ for any $\theta\in D$, and
consider the process $Z:=\\{Z_{t}\\}_{t\in{\mathbb{R}}_{+}}$ with
$Z_{t}:=S_{t}-t\cdot\frac{\mathbb{E}_{P}[X_{1}]}{\mathbb{E}_{P}[W_{1}]}$ for
any $t\geq 0$. The following statements are equivalent:

  1. (i)

$Z$ is a martingale in ${\mathcal{L}}^{\ell}(P)$;

  2. (ii)

the distribution of $v(\varTheta)$ is degenerate at
$v(\theta_{0})=\frac{1}{\mathbb{E}_{P}[W_{1}]}\in\varUpsilon$ and
$P\in{\mathcal{M}}^{\ell}_{S,{\bf Exp}(v(\theta_{0}))}$.

###### Proof.

Fix on arbitrary $t\geq 0$ and let $u\in[0,t]$. Implication (ii)
$\Longrightarrow$ (i) follows by [14], Proposition 4.1.

Ad (i) $\Longrightarrow$ (ii): Because $Z$ is a martingale in
${\mathcal{L}}^{1}(P)$, it follows by [12], Theorem 3.5, that $S$ is a
$P$-$CMPP(v(\varTheta),P_{X_{1}})$. But since

$\mathbb{E}_{P}[v(\varTheta)]=\mathbb{E}_{P}\bigl{[}\frac{1}{\mathbb{E}_{P}[W_{1}]}\bigr{]}=\frac{1}{\mathbb{E}_{P}[W_{1}]}=v(\varTheta),$

we get that there exists a $\theta_{0}\in{D}$ such that
$1=P_{v(\varTheta)}(v(\theta_{0}))=\delta_{v(\theta_{0})}(v(\theta_{0}))$, or
$P_{v(\varTheta)}=\delta_{v(\theta_{0})}$, where $\delta_{v(\theta_{0})}$ is
the Dirac measure concentrated on $v(\theta_{0})$. Consequently, the
distribution of $v(\varTheta)$ is degenerate at $v(\theta_{0})$; hence
$P\in\mathcal{M}^{\ell}_{S,\mathbf{Exp}(v(\theta_{0}))}$ by e.g. [17], Theorem
4.2.6. ∎

The preceding result, together with [18], Proposition 3.9, implies that for a
given $P\in{\mathcal{M}}^{\ell}_{S,{\bf K}(\varTheta)}$, with $P_{\varTheta}$
non-degenerate, there does not exist any probability measure
$Q\in{\mathcal{M}}^{\ell}_{S,{\bf\Lambda}(\rho(\varTheta))}$ such that $Z$ is
a martingale in ${\mathcal{L}}^{\ell}(Q)$. The latter, naturally raises the
question whether we can find processes on $(\varOmega,\varSigma)$ satisfying
condition (PEMM). The next martingale characterization plays a fundamental
role in our effort to find processes on $(\varOmega,\varSigma)$ that satisfy
condition (PEMM), but is also seems to be of independent interest, since it
provides a martingale characterization of CMPPs within the class of CMRPS.

######  Proposition 4.2.

Define the processes
$Z(\varTheta):=\\{Z_{t}(\varTheta)\\}_{t\in{\mathbb{R}}_{+}}$ and
$Z(\theta):=\\{Z_{t}(\theta)\\}_{t\in{\mathbb{R}}_{+}}$ by means of

$Z_{t}(\varTheta):=S_{t}-t\cdot\frac{\mathbb{E}_{P}[X_{1}]}{\mathbb{E}_{P}[W_{1}\mid\varTheta]}\quad\mbox{
and}\quad
Z_{t}(\theta):=S_{t}-t\cdot\frac{\mathbb{E}_{P_{\theta}}[X_{1}]}{\mathbb{E}_{P_{\theta}}[W_{1}]},$

respectively. Consider the following statements:

  1. (i)

$Z(\varTheta)$ is a martingale in ${\mathcal{L}}^{\ell}(P)$;

  2. (ii)

$P\in{\mathcal{M}}^{\ast,\ell}_{S,{\bf Exp}(g(\varTheta))}$, where
$g(\varTheta):=1/\mathbb{E}_{P}[W_{1}\mid\varTheta]$;

  3. (iii)

there exists a $P_{\varTheta}$-null set $C_{\ast}\in{\mathfrak{B}}(D)$ such
that $P_{\theta}\in{\mathcal{M}}^{\ell}_{S,{\bf Exp}(g(\theta))}$ where
$g(\theta):=1/\mathbb{E}_{P_{\theta}}[W_{1}]$ for any $\theta\notin C_{\ast}$;

  4. (iv)

there exists a $P_{\varTheta}$-null set $C_{\ast}\in{\mathfrak{B}}(D)$ such
that $Z(\theta)$ is a martingale in ${\mathcal{L}}^{\ell}(P_{\theta})$ for any
$\theta\notin C_{\ast}$.

Then $(i)\Longleftrightarrow(ii)$, $(iii)\Longleftrightarrow(iv)$ and
$(ii)\Longrightarrow(iii)$. If in addition
$P\in{\mathcal{M}}^{\ast,\ell}_{S,{\bf K}(\varTheta)}$, then all items $(i)$
to $(iv)$ are equivalent.

###### Proof.

Fix on arbitrary $t\geq 0$. The equivalence (iii) $\Longleftrightarrow$ (iv)
follows by [14], Proposition 4.1.

Ad (ii) $\Longrightarrow$ (i): Let
$P\in\mathcal{M}^{\ast,\ell}_{S,\mathbf{Exp}(g(\varTheta))}\subseteq\mathcal{M}^{\ell}_{S,\mathbf{Exp}(g(\varTheta))}$.
It then follows that $g(\varTheta)^{\ell},X_{1}^{\ell}\in\mathcal{L}^{1}(P)$.
Thus, applying Wald’s identities (cf. e.g. [17], Lemma 5.2.10) and [17], Lemma
4.2.5, we get
$\mathbb{E}_{P}[S_{t}]=t\cdot\mathbb{E}_{P}[g(\varTheta)]\cdot\mathbb{E}_{P}[X_{1}]<\infty,$
and

$\displaystyle\mathbb{V}ar_{P}[S_{t}]$
$\displaystyle=\mathbb{E}_{P}[N_{t}]\cdot\mathbb{V}at_{P}[X_{1}]+\mathbb{V}ar_{P}[N_{t}]\cdot\mathbb{E}^{2}_{P}[X_{1}]$
$\displaystyle=t\cdot\mathbb{E}_{P}[g(\varTheta)]\cdot\mathbb{V}ar_{P}[X_{1}]+\left(t\cdot\mathbb{E}_{P}[g(\varTheta)]+t^{2}\cdot\mathbb{V}ar_{P}[g(\varTheta)]\right)\cdot\mathbb{E}^{2}_{P}[X_{1}]<\infty.$

Consequently, $Z_{t}(\varTheta)\in\mathcal{L}^{\ell}(P)$. The martingale
property of $Z(\varTheta)$ follows now as in [11], Proposition 5.1(ii).

Ad (i) $\Longrightarrow$ (ii): Since $P$ is a martingale measure for
$Z(\varTheta)$ we get that
$\mathbb{E}_{P}[Z_{t}(\varTheta)\mid{\mathcal{F}}_{0}]=Z_{0}(\varTheta)=0$
$P\upharpoonright{\mathcal{F}}_{0}$–a.s., implying that
$\mathbb{E}_{P}[Z_{t}(\varTheta)\mid\varTheta]=0$
$P\upharpoonright\sigma(\varTheta)$–a.s., or equivalently that

(1)
$\mathbb{E}_{P}[N_{t}\mid\varTheta]=\frac{t}{\mathbb{E}_{P}[W_{1}\mid\varTheta]}\qquad
P\upharpoonright\sigma(\varTheta)\text{--a.s.}.$

######  Claim 4.3.

The following assertions are equivalent:

  * (a)

$N$ is a MPP$(\varTheta)$;

  * (b)

$\mathbb{E}_{P}[N_{t}\mid\varTheta]=t\cdot\varTheta\qquad
P\upharpoonright\sigma(\varTheta)\text{--a.s.}$.

###### Proof.

Implication $(a)\Longrightarrow(b)$ follows by [17], Lemma 4.2.5.

Ad (b) $\Longrightarrow$ (a): The renewal function associated with the
conditional distribution ${\bf K}(\varTheta)$ is defined by

$U(u,\varTheta):=\sum_{n=0}^{\infty}{\bf K}^{\ast
n}(\varTheta)(u)\quad\text{for any}\,\,u\in{\mathbb{R}},$

where ${\bf K}^{\ast n}(\varTheta)$ denotes the n-fold convolution of ${\bf
K}(\varTheta)$. Clearly, $U(u,\varTheta)=1+\mathbb{E}_{P}[N_{u}\mid\varTheta]$
$P\upharpoonright\sigma(\varTheta)$–a.s. for any $u\in{\mathbb{R}}_{+}$; hence
by $(b)$ we get that $U(t,\varTheta)=1+t\cdot\varTheta$
$P\upharpoonright\sigma(\varTheta)$–a.s., implying that the Laplace-Stieltjes
transform $\widehat{U}(s,\varTheta)$ of $U(t,\varTheta)$ is given by

$\widehat{U}(s,\varTheta)=\int_{{\mathbb{R}}_{+}}e^{-s\cdot u}\cdot
U(du,\varTheta)=e^{-s\cdot 0}\cdot U(0,\varTheta)+\int^{\infty}_{0}e^{-s\cdot
u}\cdot U(du,\varTheta)=\frac{s+\varTheta}{s}$

for any $s>0$, where the equalities hold
$P\upharpoonright\sigma(\varTheta)-\mbox{a.s.}$. It then follows that

$\widehat{\bf
K}(\varTheta)(s)=\frac{U(s,\varTheta)-1}{U(s,\varTheta)}=\frac{\varTheta}{\varTheta+s}\quad
P\upharpoonright\sigma(\varTheta)\text{--a.s.}$

for any $s>0$, where $\widehat{\bf K}(\varTheta)$ denotes the Laplace-
Stieltjes transform of the conditional distribution of $W_{n}$ for any
$n\in{\mathbb{N}}$; hence $P_{W_{n}|\varTheta}={\bf Exp}(\varTheta)$
$P\upharpoonright\sigma(\varTheta)$–a.s. for any $n\in{\mathbb{N}}$. But since
$W$ is also $P$-conditionally independent, Proposition 4.5 from [9] implies
that $N$ is a MPP$(\varTheta)$. ∎

Putting $g(\varTheta):=\frac{1}{\mathbb{E}_{P}[W_{1}\mid\varTheta]}$, where
$g\in\mathfrak{M}_{+}(D)$, we immediately get by condition (1) that
$\mathbb{E}_{P}[N_{t}\mid\varTheta]=t\cdot g(\varTheta)\quad
P\upharpoonright\sigma(\varTheta)$–a.s., implying according to Claim 4.3 that
$N$ is a $P$–MPP$(g(\varTheta))$. The latter together with assumption
$P\in\mathcal{M}^{\ell}_{S,\mathbf{K}(\varTheta)}$ yields
$P\in\mathcal{M}^{\ell}_{S,\mathbf{Exp}(g(\varTheta))}$, while by condition
$Z_{t}(\varTheta)\in{\mathcal{L}}^{\ell}(P)$ follows
$g(\varTheta)\in\mathcal{L}^{\ell}(P)$, implying that
$P\in{\mathcal{M}}^{\ast,\ell}_{S,{\bf Exp}(g(\varTheta))}$.

Ad (ii) $\Longrightarrow$ (iii): Since $P\in{\mathcal{M}}^{\ast,\ell}_{S,{\bf
Exp}(g(\varTheta))}\subseteq{\mathcal{M}}^{\ell}_{S,{\bf Exp}(g(\varTheta))}$
and $W_{1}\in{\mathcal{L}}^{1}(P)$, it follows by Remark 3.4 and[9], Lemma
3.5, that there exist two $P_{\varTheta}$–null sets
$\widetilde{L},L_{P}\in{\mathfrak{B}}(D)$ such that
$g(\theta)=1/\mathbb{E}_{P_{\theta}}[W_{1}]$ for any
$\theta\notin\widetilde{L}$ and $P_{\theta}\in{\mathcal{M}}^{\ell}_{S,{\bf
Exp}(g(\theta))}$ for any $\theta\notin C_{\ast}:=\widetilde{L}\cup L_{P}$,
respectively.

Ad (iii) $\Longrightarrow$ (ii): Since
$P_{\theta}\in{\mathcal{M}}^{\ell}_{S,{\bf Exp}(g(\theta))}$ for any
$\theta\notin C_{\ast}$, and $W_{1}\in{\mathcal{L}}^{1}(P)$, Remark 3.4 along
with [9], Lemma 3.5, yields that $P\in{\mathcal{M}}^{\ell}_{S,{\bf
Exp}(g(\varTheta))}$. But since
$\left(1/\mathbb{E}_{P}[W_{1}\mid\varTheta]\right)^{\ell}\in{\mathcal{L}}^{1}(P)$
by assumption, we get $P\in{\mathcal{M}}^{\ast,\ell}_{S,{\bf
Exp}(g(\varTheta))}$. ∎

In the next theorem we find out a wide class of canonical processes converting
the progressively equivalent measures $Q$ of Theorem 3.5 into martingale
measures. In this way, a characterization of all progressively equivalent
martingale measures, similar to that of Theorem 3.5, is provided.

######  Theorem 4.4.

If $P\in{\mathcal{M}}^{\ast,\ell}_{S,{\bf K}(\varTheta)}$ the following hold
true:

  1. (i)

for every pair
$(\rho,Q)\in\mathfrak{M}_{+}(D)\times{\mathcal{M}}^{\ast,\ell}_{S,{\bf
Exp}(\rho(\varTheta))}$ there exists an essentially unique pair
$(\beta,\xi)\in\mathcal{F}^{\ell}_{P,\varTheta}\times\mathcal{R}^{\ast,\ell}_{+}(D)$
satisfying conditions ($\ast$ ‣ 1) and ($RPM_{\xi}$) and such that $Q$ is a
martingale measure for the process
$V(\varTheta):=\\{V_{t}(\varTheta)\\}_{t\in{\mathbb{R}}_{+}}$ defined by means
of $V_{t}(\varTheta):=S_{t}-t\cdot\frac{\mathbb{E}_{P}[X_{1}\cdot
e^{\beta(X_{1},\varTheta)}\mid\varTheta]}{\mathbb{E}_{P}[W_{1}\mid\varTheta]}$
for any $t\in{\mathbb{R}}_{+}$;

  2. (ii)

conversely, for every pair
$(\beta,\xi)\in\mathcal{F}^{\ell}_{P,\varTheta}\times\mathcal{R}^{\ast,\ell}_{+}(D)$
there exists a unique pair
$(\rho,Q)\in\mathfrak{M}_{+}(D)\times{\mathcal{M}}^{\ast,\ell}_{S,{\bf
Exp}(\rho(\varTheta))}$ determined by ($RPM_{\xi}$) and satisfying condition
($\ast$ ‣ 1) such that $Q$ is a martingale measure for $V(\varTheta)$;

  3. (iii)

in both cases (i) and (ii), there exist a an essentially unique rcp
$\\{Q_{\theta}\\}_{\theta\in D}$ of $Q$ over $Q_{\varTheta}$ consistent with
$\varTheta$ and a $P_{\varTheta}$–null set
$M_{\ast}:=C_{\ast}\cup\widetilde{L}_{\ast\ast}\in{\mathfrak{B}}(D)$
satisfying for any $\theta\notin M_{\ast}$ conditions
$Q_{\theta}\in{{\mathcal{M}}}^{\ast,\ell}_{S,{\bf Exp}(\rho(\theta))}$,
($\widetilde{\ast}$ ‣ 3) and ($RPM_{\theta}$), and such that $Q_{\theta}$ is a
martingale measure for the process
$V(\theta):=\\{V_{t}(\theta)\\}_{t\in{\mathbb{R}}_{+}}$ defined by means of
$V_{t}(\theta):=S_{t}-t\cdot\frac{\mathbb{E}_{P_{\theta}}[X_{1}\cdot
e^{\beta(X_{1},\theta)}]}{\mathbb{E}_{P_{\theta}}[W_{1}]}$.

###### Proof.

Statements (i) and (ii) follow by Theorem 3.5(i),(ii) and Proposition 4.2.

Ad (iii): Fix on arbitrary $t\geq 0$. By Theorem 3.5(iii), it follows that
there exist an essentially unique rcp $\\{Q_{\theta}\\}_{\theta\in D}$ of $Q$
over $Q_{\varTheta}$ consistent with $\varTheta$ and a $P_{\varTheta}$-null
set $\widetilde{L}_{\ast\ast}\in\mathfrak{B}(D)$ such that for any
$\theta\notin\widetilde{L}_{\ast\ast}$ conditions
$Q_{\theta}\in{{\mathcal{M}}}^{\ell}_{S,{\bf Exp}(\rho(\theta))}$,
($\widetilde{\ast}$ ‣ 3) and ($RPM_{\theta}$) are valid. Put
$M_{\ast}:=C_{\ast}\cup\widetilde{L}_{\ast\ast}\in{\mathfrak{B}}(D)$ and fix
on $\theta\notin M_{\ast}$. Since $Q_{\theta}\in{{\mathcal{M}}}^{\ell}_{S,{\bf
Exp}(\rho(\theta))}$, applying Proposition 4.2 along with condition
($\widetilde{\ast}$ ‣ 3) we obtain that $Q_{\theta}$ is a martingale measure
for the process $V(\theta)$ for any $\theta\notin C_{\ast}$. ∎

######  Remarks 4.5.

Given a probability measure $P\in{\mathcal{M}}^{\ast,\ell}_{S,{\bf
K}(\varTheta)}$, with $P_{\varTheta}$ non-degenerate, and a pair
$(\beta,\xi)\in{{\mathcal{F}}}^{\ell}_{P,\varTheta}\times\mathcal{R}^{\ast,\ell}_{+}(D)$,
Theorems 3.5 and 4.4 allow us to find progressively equivalent martingale
measures for the processes $V(\varTheta)$ and $V(\theta)$. In particular,
$V(\varTheta)$ satisfies condition (PEMM) if and only if $V(\theta)$ satisfies
condition (PEMM) for any $\theta\notin M_{\ast}$. The latter implies that
$V(\varTheta)$ is the most natural choice for a process on
$(\varOmega,\varSigma)$.

The next theorem connects our results with the basic notion of no free lunch
with vanishing risk of Mathematical Finance, see [3], Definition 8.1.2, for
the definition.

######  Theorem 4.6.

Let $P\in{\mathcal{M}}^{\ast,2}_{S,{\bf K}(\varTheta)}$. For every pair
$(\beta,\xi)\in{{\mathcal{F}}}^{2}_{P,\varTheta}\times\mathcal{R}^{\ast,2}_{+}(D)$
there exists a unique pair
$(\rho,Q)\in\mathfrak{M}_{+}(D)\times{\mathcal{M}}^{\ast,2}_{S,{\bf
Exp}(\rho(\varTheta))}$ determined by ($RPM_{\xi}$) and satisfying condition
($\ast$ ‣ 1), and an essentially unique rcp of $Q$ over $Q_{\varTheta}$
consistent with $\varTheta$ satisfying for any $\theta\notin M_{\ast}$
conditions $Q_{\theta}\in{{\mathcal{M}}}^{2}_{S,{\bf Exp}(\rho(\theta))}$,
($\widetilde{\ast}$ ‣ 3) and ($RPM_{\theta}$), such that the following holds
true:

  1. (i)

the process
$V_{\mathbb{T}}(\varTheta):=\\{V_{t}(\varTheta)\\}_{t\in\mathbb{T}}$ satisfies
condition (NFLVR);

  2. (ii)

for any $\theta\notin M_{\ast}$ the process
$V_{\mathbb{T}}(\theta):=\\{V_{t}(\theta)\\}_{t\in\mathbb{T}}$ satisfies
condition (NFLVR),

where $M_{\ast}\in{\mathfrak{B}}(D)$ is the $P_{\varTheta}$–null set of
Theorem 4.4.

###### Proof.

Due to Theorem 3.5(ii) and (iii) there exist a unique pair
$(\rho,Q)\in\mathfrak{M}_{+}(D)\times{\mathcal{M}}^{\ast,2}_{S,{\bf
Exp}(\rho(\varTheta))}$ determined by ($RPM_{\xi}$) and satisfying condition
($\ast$ ‣ 1), an essentially unique rcp $\\{Q_{\theta}\\}_{\theta\in D}$ of
$Q$ over $Q_{\varTheta}$ consistent with $\varTheta$ and a
$P_{\varTheta}$-null set $\widetilde{L}_{\ast\ast}\in\mathfrak{B}(D)$ such
that for any $\theta\notin\widetilde{L}_{\ast\ast}\subseteq M_{\ast}$
conditions $Q_{\theta}\in{\mathcal{M}}^{\ell}_{S,{\bf Exp}(\rho(\theta))}$,
($\widetilde{\ast}$ ‣ 3) and ($RPM_{\theta}$) are valid.

Ad (i): Applying Theorem 4.4 we get that for any $T>0$ the process
$V_{\mathbb{T}}(\varTheta)$ is an ${\mathcal{F}}_{\mathbb{T}}$-martingale in
${\mathcal{L}}^{2}(Q_{T})$, implying that it is an
${\mathcal{F}}_{\mathbb{T}}$-semi-martingale in ${\mathcal{L}}^{2}(Q_{T})$
(cf. e.g. [19], Chapter 1, Section 1.3, Definition on page 23). The latter
implies that $V_{\mathbb{T}}(\varTheta)$ is also an
${\mathcal{F}}_{\mathbb{T}}$-semi-martingale in ${\mathcal{L}}^{2}(P_{T})$
since $Q_{T}\sim P_{T}$ (cf. e.g. [19], Theorem 10.1.8). Because the process
$V(\varTheta)$ satisfies condition (PEMM), we deduce that the process
$V_{\mathbb{T}}(\varTheta)$ must satisfy condition (EMM). Thus, we can apply
the FTAP of Delbaen & Schachermayer for unbounded stochastic processes [3],
Theorem 14.1.1, to obtain that the process $V_{\mathbb{T}}(\varTheta)$
satisfies condition (NFLVR).

Ad (ii): Since for any $\theta\notin M_{\ast}$ we have that
$(\rho,Q_{\theta})\in\mathfrak{M}_{+}(D)\times{\mathcal{M}}^{2}_{S,{\bf
Exp}(\rho(\theta))}$, we may apply [14], Theorem 4.1, to obtain assertion
(ii).∎

##  5\. An Application to the Ruin Problem

In this section we characterize all progressively equivalent martingale
measures $Q$ that convert a CMRP under $P$ into a CMPP, in such a way that
ruin occurs a.s. under the $Q$.

In order to justify the definition of a conditional premium density, we need
the following auxiliary lemma, which extends a well known result in the case
of a kind of compound mixed Poisson processes (cf. e.g. [6], Proposition 9.1).

######  Lemma 5.1.

If $P\in{\mathcal{M}}^{1}_{S,{\bf K}(\varTheta)}$ then condition

$\lim_{t\rightarrow\infty}\frac{S_{t}}{t}=\frac{\mathbb{E}_{P}[X_{1}]}{\mathbb{E}_{P}[W_{1}\mid\varTheta]}\qquad
P\text{--a.s.}.$

holds true.

###### Proof.

Since $P\in{\mathcal{M}}^{1}_{S,{\bf K}(\varTheta)}$, it follows by Remark 3.4
that there exists a $P_{\varTheta}$–null set $L_{P}\in{\mathfrak{B}}(D)$ such
that $P_{\theta}\in{\mathcal{M}}^{1}_{S,{\bf K}(\theta)}$ for any
$\theta\notin L_{P}$. Fix on an arbitrary $\theta\notin L_{P}$.

We first show the validity of condition

(2)
$\lim_{t\rightarrow\infty}\frac{N_{t}}{t}=\frac{1}{\mathbb{E}_{P}[W_{1}\mid\varTheta]}\qquad
P\text{--a.s.}.$

In fact, since $N$ is a $P_{\theta}$–RP$({\bf K}(\theta))$, we may apply [7],
Section 2.5, Theorem 5.1, to get that

(3)
$\lim_{t\rightarrow\infty}\frac{N_{t}}{t}=\frac{1}{\mathbb{E}_{P_{\theta}}[W_{1}]}\qquad
P_{\theta}\text{--a.s.}.$

Consider the function
$v:=\chi_{\left\\{\lim_{t\to\infty}\frac{N_{t}}{t}=\frac{1}{\mathbb{E}_{P_{\theta}}[W_{1}]}\right\\}}:\varOmega\times
D\longrightarrow[0,1]$ and put
$g:=v\circ(id_{\varOmega}\times\varTheta)=\chi_{\left\\{\lim_{t\to\infty}\frac{N_{t}}{t}=\frac{1}{\mathbb{E}_{P}[W_{1}\mid\varTheta]}\right\\}}$.
Since $v\in{\mathcal{L}}^{1}(M)$, where
$M:=P\circ(id_{\varOmega}\times\varTheta)^{-1}$, we may apply [9], Proposition
3.8(i), to get that

$\mathbb{E}_{P}\left[g\mid\varTheta\right]=\mathbb{E}_{P_{\bullet}}\left[v^{\bullet}\right]\circ\varTheta\quad
P\upharpoonright\sigma(\varTheta)\text{--a.s.},$

implying

$\displaystyle
P\left(\left\\{\lim_{t\rightarrow\infty}\frac{N_{t}}{t}=\frac{1}{\mathbb{E}_{P}[W_{1}\mid\varTheta]}\right\\}\right)$
$\displaystyle=\int\mathbb{E}_{P}[g\mid\varTheta]\,dP=\int\mathbb{E}_{P_{\bullet}}\left[v^{\bullet}\right]\circ\varTheta\,dP$
$\displaystyle=\int
P_{\theta}\left(\left\\{\lim_{t\rightarrow\infty}\frac{N_{t}}{t}=\frac{1}{\mathbb{E}_{P_{\theta}}[W_{1}]}\right\\}\right)\,P_{\varTheta}(d\theta).$

The latter together with condition (3) yields (2).

Since $S$ is a $P_{\theta}$–CRP$({\bf K}(\theta),(P_{\theta})_{X_{1}})$ with
$(P_{\theta})_{X_{1}}=P_{X_{1}}$, by Remark 3.4, we may apply [7], Section
1.2, Theorem 2.3(iii), in order to get

(4)
$\lim_{t\rightarrow\infty}\frac{S_{t}}{N_{t}}=\mathbb{E}_{P_{\theta}}[X_{1}]\qquad
P_{\theta}\text{--a.s.},$

implying along with Remark 3.4 and [9], Lemma 3.5(i), that

$P\left(\left\\{\lim_{t\rightarrow\infty}\frac{S_{t}}{N_{t}}=\mathbb{E}_{P}[X_{1}]\right\\}\right)=\int
P_{\theta}\left(\left\\{\lim_{t\rightarrow\infty}\frac{S_{t}}{N_{t}}=\mathbb{E}_{P_{\theta}}[X_{1}]\right\\}\right)\,P_{\varTheta}(d\theta).$

The latter, together with condition (4) yields
$\lim_{t\rightarrow\infty}\frac{S_{t}}{N_{t}}=\mathbb{E}_{P}[X_{1}]\quad
P\text{--a.s.}$, implying along with (2) the thesis of the lemma. ∎

For any $\theta\notin L_{P}$ conditions (3) and (4) imply

$\lim_{t\rightarrow\infty}\frac{S_{t}}{t}=\frac{\mathbb{E}_{P_{\theta}}[X_{1}]}{\mathbb{E}_{P_{\theta}}[W_{1}]}\qquad
P_{\theta}\text{--a.s.},$

that is, the limit $\lim_{t\rightarrow\infty}\frac{S_{t}}{t}$ coincides with
the premium density, i.e. the monetary payout per unit time, in a
$P_{\theta}$–Sparre Andersen model (see [14], p. 54, for more details). Thus,
for any $P\in{\mathcal{M}}^{1}_{S,{\bf K}(\varTheta)}$ we may define the
conditional premium density by means of

$p(P,\varTheta):=\frac{\mathbb{E}_{P}[X_{1}]}{\mathbb{E}_{P}[W_{1}\mid\varTheta]}\quad
P\upharpoonright\sigma(\varTheta)\text{--a.s.},$

while for any $P\in{\mathcal{M}}^{\ast,1}_{S,{\bf K}(\varTheta)}$ we define
the corresponding premium density by means of
$p(P):=\mathbb{E}_{P}[p(P,\varTheta)]$.

######  Remarks 5.2.

(a) The following statements are equivalent and each one of them holds:

  1. (i)

$p(P,\varTheta)=\frac{\mathbb{E}_{P}[X_{1}]}{\mathbb{E}_{P}[W_{1}\mid\varTheta]}$
$P\upharpoonright\sigma(\varTheta)$–a.s.;

  2. (ii)

there exists a $P_{\varTheta}$–null set $L_{P,1}\in{\mathfrak{B}}(D)$ such
that
$p(P,\theta)=p(P_{\theta})=\frac{\mathbb{E}_{P_{\theta}}[X_{1}]}{\mathbb{E}_{P_{\theta}}[W_{1}]}$
for any $\theta\notin L_{P,1}$.

In fact, for any $B\in{\mathfrak{B}}(D)$ we get

$\int_{\varTheta^{-1}[B]}p(P,\varTheta)\,dP=\int_{\varTheta^{-1}[B]}\frac{\mathbb{E}_{P}[X_{1}]}{\mathbb{E}_{P}[W_{1}\mid\varTheta]}\,dP,$

and applying [9], Lemma 3.5, we equivalently get

$\int_{\varTheta^{-1}[B]}p(P,\bullet)\circ\varTheta\,dP=\int_{\varTheta^{-1}[B]}\frac{\mathbb{E}_{P}[X_{1}]}{\mathbb{E}_{P_{\bullet}}[W_{1}]\circ\varTheta}\,dP,$

or equivalently

$\int_{B\cap L^{c}_{P}}p(P,\theta)\,dP_{\varTheta}(d\theta)=\int_{B\cap
L^{c}_{P}}\frac{\mathbb{E}_{P_{\theta}}[X_{1}]}{\mathbb{E}_{P_{\theta}}[W_{1}]}\,P_{\varTheta}(d\theta),$

where $L_{P}$ is the $P_{\varTheta}$–null set of Remark 3.4. Therefore, we
equivalently get that there exists a $P_{\varTheta}$–null set
$L_{P,1}\in{\mathfrak{B}}(D)$ containing $L_{P}$ such that

$p(P,\theta)=\frac{\mathbb{E}_{P_{\theta}}[X_{1}]}{\mathbb{E}_{P_{\theta}}[W_{1}]}\quad\text{
for any }\,\,\theta\notin L_{P,1}.$

Because $P\in{\mathcal{M}}^{1}_{S,{\bf K}(\varTheta)}$, it follows Remark 3.4
that $P_{\theta}\in{\mathcal{M}}^{1}_{S,{\bf K}(\theta)}$ with
$(P_{\theta})_{X_{1}}=P_{X_{1}}$ for any $\theta\notin L_{P,1}$, implying that

$p(P,\theta)=\frac{\mathbb{E}_{P_{\theta}}[X_{1}]}{\mathbb{E}_{P_{\theta}}[W_{1}]}=p(P_{\theta})\quad\text{
for any }\,\,\theta\notin L_{P,1},$

where $p(P_{\theta})$ is the corresponding premium density of a
$P_{\theta}$–renewal risk model (see [14], p. 54, for more details).

(b) If $P\in{\mathcal{M}}^{\ast,1}_{S,{\bf K}(\varTheta)}$, then by applying
(a) we have $p(P)=\int p(P_{\theta})\,P_{\varTheta}(d\theta)$.

Let $S$ be an aggregate claims process induced by a counting process $N$ and a
claims size process $X$. Fix on arbitrary $u\in\varUpsilon$ and
$t\in{\mathbb{R}}_{+}$ and define the function $r^{u}_{t}:\varOmega\times
D\longrightarrow{\mathbb{R}}$ by means of
$r^{u}_{t}(\omega,\theta):=u+c(\theta)\cdot t-S_{t}(\omega)$ for any
$(\omega,\theta)\in\varOmega\times D$, where $c$ is a positive
$\mathfrak{B}(D)$-measurable function.

For arbitrary but fixed $\theta\in D$ the process
$r^{u}(\theta):=\\{r_{t}^{u}(\theta)\\}_{t\in{\mathbb{R}}_{+}}$, defined by
means of $r_{t}^{u}(\theta)(\omega):=r_{t}^{u}(\omega,\theta)$ for any
$\omega\in\varOmega$, is called the reserve process induced by the initial
reserve $u$, the premium intensity or premium rate $c(\theta)$ and the
aggregate claims process $S$ (cf. e.g. [17], pp. 155–156).

The function $\psi_{\theta}:\varUpsilon\longrightarrow[0,1]$ defined by means
of
$\psi_{\theta}(u):=P_{\theta}(\\{\inf_{t\in{\mathbb{R}}_{+}}r^{u}_{t}(\theta)<0\\})$
is called the probability of ruin for the reserve process $r^{u}(\theta)$ with
respect to $P_{\theta}$ (cf. e.g. [17], p. 158).

Define the real-valued functions $r^{u}_{t}(\varTheta)$ and $R^{u}_{t}$ on
$\varOmega$ by means of
$r^{u}_{t}(\varTheta)(\omega):=r^{u}_{t}(\omega,\varTheta(\omega))$ for any
$\omega\in\varOmega$, and
$R^{u}_{t}:=r^{u}_{t}\circ(id_{\varOmega}\times\varTheta)$, respectively. The
process $R^{u}:=\\{R^{u}_{t}\\}_{t\in{\mathbb{R}}_{+}}$ is called the reserve
process induced by the initial reserve $u$, the stochastic premium intensity
or stochastic premium rate $c(\varTheta)$ and the aggregate claims process
$S$.

The function $\psi:\varUpsilon\longrightarrow[0,1]$ defined by means of
$\psi(u):=P(\\{\inf_{t\in{\mathbb{R}}_{+}}R^{u}_{t}<0\\})$ is called the
probability of ruin for the reserve process $R^{u}$ with respect to $P$.

######  Lemma 5.3.

Let $P\in{\mathcal{M}}^{1}_{S,{\bf K}(\varTheta)}$. If $c(\varTheta)\leq
p(P,\varTheta)$ $P\upharpoonright\sigma(\varTheta)$–a.s., then $\psi(u)=1$ for
any $u>0$.

###### Proof.

Fix on arbitrary $u>0$. Since $c(\varTheta)\leq p(P,\varTheta)$
$P\upharpoonright\sigma(\varTheta)$–a.s., applying Remark 5.2(a) we get
$\int_{B\cap L^{c}_{P,1}}c(\theta)\,P_{\varTheta}(d\theta)\leq\int_{B\cap
L^{c}_{P,1}}p(P_{\theta})\,P_{\varTheta}(d\theta)$ for any
$B\in{\mathfrak{B}}(D)$; hence there exists a $P_{\varTheta}$–null set
$M_{1}\in{\mathfrak{B}}(D)$ containing the $P_{\varTheta}$–null set $L_{P,1}$
such that $c(\theta)\leq p(P_{\theta})$ for any $\theta\notin M_{1}$. Since
$L_{P}\subseteq M_{1}$ we get that $P_{\theta}\in{\mathcal{M}}^{1}_{S,{\bf
K}(\theta)}$ for any $\theta\notin M_{1}$; hence for any $\theta\notin M_{1}$
we may apply [17], Corollary 7.1.4, to get that $\psi_{\theta}(u)=1$. The
latter along with [18], Remark 3.4(b), yields
$\psi(u)=\int_{D}\psi_{\theta}(u)\,P_{\varTheta}(d\theta)=1$. ∎

The above lemma implies that in order to avoid ruin a.s. we have to choose the
stochastic premium rate $c(\varTheta)$ in such a way that the conditional net
profit condition

(5)
$c(\varTheta)>\frac{\mathbb{E}_{P}[X_{1}]}{\mathbb{E}_{P}[W_{1}\mid\varTheta]}\qquad
P\upharpoonright\sigma(\varTheta)\text{--a.s.}$

is fulfilled.

For given $\rho\in\mathfrak{M}_{+}(D)$ denote by
$\mathcal{M}^{ruin,\ell}_{S,\mathbf{Exp}(\rho(\varTheta))}$ the family of all
probability measures
$Q\in\mathcal{M}^{\ast,\ell}_{S,\mathbf{Exp}(\rho(\varTheta))}$ such that
$c(\varTheta)=p(Q,\varTheta)$, and by $\mathcal{F}^{ruin,\ell}_{P,\varTheta}$
the family of all functions $\beta\in\mathcal{F}^{\ell}_{P,\varTheta}$ such
that

(6)
$c(\varTheta)=\frac{\mathbb{E}_{P}[X_{1}\cdot{e}^{\beta(X_{1},\varTheta)}\mid\varTheta]}{\mathbb{E}_{P}[W_{1}\mid\varTheta]}\quad
P\upharpoonright\sigma(\varTheta)-\mbox{a.s.}.$

For given $\theta\in{D}$ denote by
$\mathcal{M}^{ruin,\ell}_{S,\mathbf{Exp}(\rho(\theta))}$ the family of all
probability measures
$Q_{\theta}\in\mathcal{M}^{\ell}_{S,\mathbf{Exp}(\rho(\theta))}$ such that
$c(\theta)=p(Q_{\theta})$.

Applying Lemma 5.3 and Theorem 4.4, we provide the following result concerning
the explicit computation of the ruin probability for the reserve process
$R^{u}$ with respect to $P$. In order to present it let us denote by $\tau$
the ruin time of the reserve process (cf. e.g. [16], p. 84 for the
definition). Note that if the conditional net profit condition (5) is
violated, then the ruin probability $\psi(u)$ under $P$ is equal to 1 by Lemma
5.3; hence we may assume that (5) is fulfilled.

######  Theorem 5.4.

If $P\in{\mathcal{M}}^{\ast,\ell}_{S,{\bf K}(\varTheta)}$ then the following
hold true:

  1. (i)

for each pair
$(\rho,Q)\in\mathfrak{M}_{+}(D)\times{\mathcal{M}}^{ruin,\ell}_{S,{\bf
Exp}(\rho(\varTheta))}$ there exists an essentially unique pair
$(\beta,\xi)\in{\mathcal{F}}^{ruin,\ell}_{P,\varTheta}\times\mathcal{R}^{\ast,\ell}_{+}(D)$
satisfying conditions ($\ast$ ‣ 1) and ($RPM_{\xi}$) and such that $Q$ is a
martingale measure for the reserve process $R^{u}$, $R^{u}-u=-V(\varTheta)$,

(7) $\psi^{Q}(u):=Q(\\{\inf_{t\in{R}_{+}}R^{u}_{t}<0\\})=1,$

and

(8) $\psi(u)=\int\frac{1}{\xi(\varTheta)}\cdot
e^{-S_{\tau}^{(\beta)}(\varTheta)+N_{\tau}\cdot\alpha(\varTheta)}\cdot\prod_{j=1}^{N_{\tau}}\frac{d{\bf
K}(\varTheta)}{d{\bf Exp}(\rho(\varTheta))}(W_{j})\,dQ;$

  2. (ii)

conversely, for every pair
$(\beta,\xi)\in\mathcal{F}^{ruin,\ell}_{P,\varTheta}\times\mathcal{R}^{\ast,\ell}_{+}(D)$
there exists a unique pair
$(\rho,Q)\in\mathfrak{M}_{+}(D)\times{\mathcal{M}}^{ruin,\ell}_{S,{\bf
Exp}(\rho(\varTheta))}$ determined by ($RPM_{\xi}$) and satisfying condition
($\ast$ ‣ 1) such that $Q$ is a martingale measure for $R^{u}$,
$R^{u}-u=-V(\varTheta)$, and conditions (7) and (8) are valid;

  3. (iii)

in both cases (i) and (ii), there exist a an essentially unique rcp
$\\{Q_{\theta}\\}_{\theta\in D}$ of $Q$ over $Q_{\varTheta}$ consistent with
$\varTheta$ and a $P_{\varTheta}$–null set
$\widetilde{M}_{\ast}\in{\mathfrak{B}}(D)$ containing $M_{\ast}$, satisfying
for any $\theta\notin\widetilde{M}_{\ast}$ conditions
$Q_{\theta}\in{{\mathcal{M}}}^{ruin,\ell}_{S,{\bf Exp}(\rho(\theta))}$,
($\widetilde{\ast}$ ‣ 3) and ($RPM_{\theta}$), and such that $Q_{\theta}$ is a
martingale measure for the reserve process $r^{u}(\theta)$,
$r^{u}(\theta)-u=-V(\theta)$,

(9)
$\psi_{Q_{\theta}}(u):=Q_{\theta}(\\{\inf_{t\in{\mathbb{R}}_{+}}r^{u}_{t}(\theta)<0\\})=1,$

and

(10) $\psi_{\theta}(u)=\int
e^{-S_{\tau}^{(\beta)}(\theta)+N_{\tau}\cdot\alpha(\theta)}\cdot\prod_{j=1}^{N_{\tau}}\frac{d{\bf
K}(\theta)}{d{\bf Exp}(\rho(\theta))}(W_{j})\,dQ_{\theta}.$

###### Proof.

Fix on arbitrary $u\in\varUpsilon$.

Ad (i): Since
$\mathcal{M}^{ruin,\ell}_{S,\mathbf{Exp}(\rho(\varTheta))}\subseteq\mathcal{M}^{\ast,\ell}_{S,\mathbf{Exp}(\rho(\varTheta))}$,
it follows by Theorem 4.4 that there exists an essentially unique pair
$(\beta,\xi)\in{\mathcal{F}}^{\ell}_{P,\varTheta}\times\mathcal{R}^{\ast,\ell}_{+}(D)$
satisfying conditions ($\ast$ ‣ 1), ($RPM_{\xi}$) and such that $Q$ is a
martingale measure for $R^{u}$, and $R^{u}-u=-V(\varTheta)$. The latter along
with condition $c(\varTheta)=p(Q,\varTheta)$
$P\upharpoonright\sigma(\varTheta)$–a.s. yields
$\beta\in\mathcal{F}^{ruin,\ell}_{P,\varTheta}$. Condition (7) follows by
Lemma 5.3, while (8) follows by ($RPM_{\xi}$) and (7).

Ad (ii): Since
$(\beta,\xi)\in{\mathcal{F}}^{\ell}_{P,\varTheta}\times\mathcal{R}^{\ast,\ell}_{+}(D)$,
it follows by Theorem 4.4 that there exists a unique pair
$(\rho,Q)\in\mathfrak{M}_{+}(D)\times{\mathcal{M}}^{\ast,\ell}_{S,{\bf
Exp}(\rho(\varTheta))}$ determined by ($RPM_{\xi}$) and satisfying condition
($\ast$ ‣ 1), and such that $Q$ is a martingale measure for $R^{u}$ with
$R^{u}-u=-V(\varTheta)$. Thus, taking into account condition (6) we obtain
$c(\varTheta)=p(Q,\varTheta)$ $P\upharpoonright\sigma(\varTheta)$–a.s.; hence
$Q\in\mathcal{M}^{ruin,\ell}_{S,\mathbf{Exp}(\rho(\varTheta))}$. Conditions
(7) and (8) follow as in (i).

Ad (iii): In both cases (i) and (ii), according to Theorem 4.4, there exist an
essentially unique rcp $\\{Q_{\theta}\\}_{\theta\in D}$ of $Q$ over
$Q_{\varTheta}$ consistent with $\varTheta$ and a $P_{\varTheta}$-null set
$M_{\ast}\in\mathfrak{B}(D)$ satisfying for any $\theta\notin{M}_{\ast}$
conditions $Q_{\theta}\in{{\mathcal{M}}}^{\ell}_{S,{\bf Exp}(\rho(\theta))}$,
($\widetilde{\ast}$ ‣ 3) and ($RPM_{\theta}$), and such that $Q_{\theta}$ is a
martingale measure for the reserve process $r^{u}(\theta)$ with
$r^{u}(\theta)-u=-V(\theta)$. Because $c(\varTheta)=p(Q,\varTheta)$, there
exists a $P_{\varTheta}$-null set $L_{Q,1}\in\mathfrak{B}(D)$ such that
$c(\theta)=p(Q_{\theta})$ for any $\theta\notin{L}_{Q,1}$ by Remark 5.2(a);
hence $Q_{\theta}\in\mathcal{M}^{ruin,\ell}_{S,\mathbf{Exp}(\rho(\theta))}$
for any $\theta\notin\widetilde{M}_{\ast}:=M_{\ast}\cup{L}_{Q,1}$. Condition
(9) follows by (7) along with [18], Remark 3.4(b), while (10) is a consequence
of ($RPM_{\theta}$) and (9). ∎

##  6\. Premium Calculation Principles and Change of Measures

In this section, we discuss implications of our results to the pricing of
actuarial risks (premium calculation principles) in an arbitrage free market.
In this context, the financial pricing of insurance (FPI for short) approach
proposed by Delbaen & Haezendonck [2] plays a key role.

We first need to establish the validity of the following proposition, which
may serve as a useful tool for a financial interpretation of our results.

######  Proposition 6.1.

For any given probability space $(\varOmega,\varSigma,P)$ and any aggregate
claims process $S$ induced by a $P$–risk process $(N,X)$, assume that
$W_{1}\in{\mathcal{L}}^{1}(P)$ and that $P$ satisfies conditions (a1) and
(a2). The following statements are equivalent:

  1. (i)

the process $\\{S_{t}-t\cdot c(\varTheta)\\}_{t\in{\mathbb{R}}_{+}}$ is a
martingale in ${\mathcal{L}}^{\ell}(P)$;

  2. (ii)

$P\in{\mathcal{M}}^{\ast,\ell}_{S,{\bf Exp}(g(\varTheta))}$ and
$c(\varTheta)=g(\varTheta)\cdot\mathbb{E}_{P}[X_{1}]$
$P\upharpoonright\sigma(\varTheta)$–a.s., where $g(\varTheta)$ is as in
Proposition 4.2(ii).

###### Proof.

Implication (ii) $\Longrightarrow$ (i) follows immediately by Proposition 4.2.

Ad (i) $\Longrightarrow$ (ii): Fix on arbitrary $v\in{\mathbb{R}}_{+}$. We
split the rest of the proof into the following steps.

(a) There exists a $P_{\varTheta}$–null set
$\widetilde{K}\in{\mathfrak{B}}(D)$ such that $\\{S_{t}-t\cdot
c(\theta)\\}_{t\in{\mathbb{R}}_{+}}$ is a $P_{\theta}$–martingale in
${\mathcal{L}}^{\ell}(P_{\theta})$ for any $\theta\notin\widetilde{K}$.

Since $\\{S_{t}-t\cdot c(\varTheta)\\}_{t\in{\mathbb{R}}_{+}}$ is a
$P$–martingale in ${\mathcal{L}}^{\ell}(P)$, we get that
$\mathbb{E}_{P}[S_{v}\mid\varTheta]=v\cdot c(\varTheta)$
$P\upharpoonright\sigma(\varTheta)$–a.s., implying $\\{S_{t}-t\cdot
c(\varTheta)\\}_{t\in{\mathbb{R}}_{+}}=\\{S_{t}-t\cdot\mathbb{E}_{P}[S_{1}\mid\varTheta]\\}_{t\in{\mathbb{R}}_{+}}$.
By [11], Proposition 5.2, which remains true for any $d$–dimensional random
vector $\varTheta$ in the place of a real-valued random variable, there exists
a $P_{\varTheta}$–null set $K\in{\mathfrak{B}}(D)$ such that
$\\{S_{t}-t\cdot\mathbb{E}_{P_{\theta}}[S_{1}]\\}_{t\in{\mathbb{R}}_{+}}$ is a
$P_{\theta}$–martingale in ${\mathcal{L}}^{\ell}(P_{\theta})$ for any
$\theta\notin K$. Since $S_{v}-v\cdot c(\varTheta)\in{\mathcal{L}}^{\ell}(P)$,
we get $S_{v},c(\varTheta)\in{\mathcal{L}}^{\ell}(P)$; hence applying [9],
Lemma 3.5, there exists a $P_{\varTheta}$–null set $K_{1}\in{\mathfrak{B}}(D)$
such that $c(\theta)=\mathbb{E}_{P_{\theta}}[S_{1}]$ for any $\theta\notin
K_{1}$. Putting $\widetilde{K}:=K\cup K_{1}\in{\mathfrak{B}}(D)$ we obtain
(a).

(b) The process $S$ is a $P$–CMPP$(g(\varTheta),P_{X_{1}})$ and
$c(\varTheta)=g(\varTheta)\cdot\mathbb{E}_{P}[X_{1}]$
$P\upharpoonright\sigma(\varTheta)$–a.s..

Fix on arbitrary $\theta\notin\widetilde{K}$. Since by (a) the process
$\\{S_{t}-t\cdot c(\theta)\\}_{t\in{\mathbb{R}}_{+}}$ is a martingale in
${\mathcal{L}}^{\ell}(P_{\theta})$, for any $u\leq v$ we get

$\int_{A}(S_{v}-v\cdot c(\theta))\,dP_{\theta}=\int_{A}(S_{u}-u\cdot
c(\theta))\,dP_{\theta}\quad\text{for
any}\,\,A\in{\mathcal{F}}^{N,\varTheta}_{u}\subseteq{\mathcal{F}}_{u},$

where by
${\mathcal{F}}^{N,\varTheta}:=\\{{\mathcal{F}}_{t}^{N,\varTheta}\\}_{t\in{\mathbb{R}}_{+}}$
is denoted the natural filtration generated by $N$ and $\varTheta$. Because
$X_{1}\in{\mathcal{L}}^{1}(P)$ and $P$ satisfies conditions (a1) and (a2), we
may apply [12], Lemma 3.3, which remains valid for any $d$–dimensional random
vector $\varTheta$ in the place of a random variable, in order to obtain

$\int_{A}(N_{v}\cdot\mathbb{E}_{P_{\theta}}[X_{1}]-v\cdot
c(\theta))\,dP_{\theta}=\int_{A}(N_{u}\cdot\mathbb{E}_{P_{\theta}}[X_{1}]-u\cdot
c(\theta))\,dP_{\theta}\quad\text{for
any}\,\,A\in{\mathcal{F}}^{N,\varTheta}_{u},$

and due to condition (a2), we equivalently get

$\int_{A}\left(N_{v}-v\cdot\frac{c(\theta)}{\mathbb{E}_{P}[X_{1}]}\right)\,dP_{\theta}=\int_{A}\left(N_{u}-u\cdot\frac{c(\theta)}{\mathbb{E}_{P}[X_{1}]}\right)\,dP_{\theta}\quad\text{for
any}\,\,A\in{\mathcal{F}}^{N,\varTheta}_{u};$

hence
$\left\\{N_{t}-t\cdot\frac{c(\theta)}{\mathbb{E}_{P}[X_{1}]}\right\\}_{t\in{\mathbb{R}}_{+}}$
is a $(P_{\theta},{\mathcal{F}}^{N,\varTheta})$–martingale in
${\mathcal{L}}^{\ell}(P_{\theta})$, implying that it is also a
$(P_{\theta},{\mathcal{F}}^{N})$–martingale in
${\mathcal{L}}^{\ell}(P_{\theta})$. Applying now [17], Theorem 2.3.4, we
equivalently obtain that $N$ is a
$P_{\theta}$–PP$\left(\frac{c(\theta)}{\mathbb{E}_{P}[X_{1}]}\right)$; hence
$\mathbb{E}_{P_{\theta}}[W_{1}]=\frac{1}{\mathbb{E}_{P_{\theta}}[N_{1}]}$, or
equivalently
$c(\theta)=\frac{\mathbb{E}_{P}[X_{1}]}{\mathbb{E}_{P_{\theta}}[W_{1}]}$.
Because $W_{1}\in{\mathcal{L}}^{1}(P)$, we may apply [9], Lemma 3.5, in order
to get
$c(\varTheta)=\frac{\mathbb{E}_{P}[X_{1}]}{\mathbb{E}_{P}[W_{1}\mid\varTheta]}=g(\varTheta)\cdot\mathbb{E}_{P}[X_{1}]$
$P\upharpoonright\sigma(\varTheta)$–a.s., implying along with [9], Proposition
4.4, that $N$ is a $P$–MPP$(g(\varTheta))$. But since
$S_{v}\in{\mathcal{L}}^{\ell}(P)$ by (a), we get that
$\mathbb{V}ar_{P}[S_{v}]<\infty$, implying along with Wald identities (cf.
e.g. [17], Lemma 5.2.10) and [17], Lemma 4.2.5, that
$g(\varTheta),X_{1}\in{\mathcal{L}}^{2}(P)$; hence
$P\in{\mathcal{M}}^{\ast,\ell}_{S,{\bf Exp}(g(\varTheta))}$. ∎

For given $u>0$ consider the price process
$U^{u}_{\mathbb{T}}:=\\{U^{u}_{t}\\}_{t\in\mathbb{T}}$, defined by means of
$U^{u}_{t}:=R^{u}_{t}-u$ for any $t\in\mathbb{T}:=[0,T],\,\,T>0$, and assume
that its random behaviour is described by a given probability measure $P$ on
$\varSigma$.

By the Harrison-Kreps theory (see Harisson & Kreps [8]) it follows that there
exists an equivalent to $P$ martingale measure $Q$ for $U^{u}_{\mathbb{T}}$ if
and only if the (re-)insurance market is arbitrage free. The next step is the
selection of an appropriate measure $Q$. The $P$–CMRP assumption for the
aggregate claims process $S$ provides a more realistic model than that of
(mixed) Cramér-Lundberg model. Thus, the framework of the FPI approach
proposed by Delbaen & Haezendonck, should be modified in the following way.

More precisely, $P$ should be a member of ${\mathcal{M}}^{\ast,2}_{S,{\bf
K}(\varTheta)}$, while $Q$ should be selected in such a way that it satisfies
conditions (a1) and (a2), $W_{1}\in{\mathcal{L}}^{1}(Q)$ and it is a
martingale measure for $U^{u}_{\mathbb{T}}$. But the latter is equivalent to
the statement $c(\varTheta)=p(Q,\varTheta)$
$Q\upharpoonright\sigma(\varTheta)$–a.s. and $S$ is a CMPP under $Q$ by
Proposition 6.1. Therefore, we are confronted with the following problem: If
$P\in{\mathcal{M}}^{\ast,2}_{S,{\bf K}(\varTheta)}$ find a characterization of
all those progressively equivalent to $P$ probability measures $Q$ on the
domain of $P$ such that $S$ is converted into CMPP under $Q$ and
$c(\varTheta)=p(Q,\varTheta)$ $Q\upharpoonright\sigma(\varTheta)$–a.s..

A positive answer to this problem is given by Theorem 5.4, which is a
consequence of our main result Theorem 4.4. However, the existence of a
measure $Q$ as above, does not always provide a realistic pricing framework,
as $Q$ should give more weight to unfavourable events in a risk-averse
environment. Thus, the conditional premium densities should satisfy condition

(11) $p(P,\varTheta)<p(Q,\varTheta)<\infty\quad
P\upharpoonright\sigma(\varTheta)\text{--a.s.}.$

According to Lemma 5.3, condition (11) is a necessary one, if we want to avoid
a $P$–a.s. ruin for the reserve process.

Under the FPI framework, we say that a premium calculation principle (PCP for
short) is a probability measure $Q$ on
$\widetilde{\varSigma}:={\mathcal{F}}^{S}_{\infty}$ such that
$Q\in{\mathcal{M}}^{1}_{S,{\bf Exp}(\theta)}$ with $\theta\in\varUpsilon$, see
[2], Definition 3.1, for more details. Since the distribution $Q_{\varTheta}$
cannot be degenerate, the probability measure
$Q\in{\mathcal{M}}^{\ast,1}_{S,{\bf Exp}(\rho(\varTheta))}$ fails to be a PCP.
Nevertheless, by virtue of Theorem 3.5 there exists an essentially unique rcp
$\\{Q_{\theta}\\}_{\theta\in D}$ of $Q$ over $Q_{\varTheta}$ consistent with
$\varTheta$ and a $P_{\varTheta}$–null set
$\widetilde{L}_{\ast\ast}\in{\mathfrak{B}}(D)$, such that
$Q_{\theta}\in{\mathcal{M}}^{1}_{S,{\bf Exp}(\rho(\theta))}$ for any
$\theta\notin\widetilde{L}_{\ast\ast}$, i.e. each probability measure
$Q_{\theta}$ is a PCP. Thus, it seems natural to call every probability
measure $Q\in{\mathcal{M}}^{\ast,1}_{S,{\bf Exp}(\rho(\varTheta))}$ a mixed
PCP.

######  Remarks 6.2.

(a) If $Q$ is a mixed PCP satisfying condition (6), then for any
$\theta\notin\widetilde{M}_{\ast}$ the corresponding PCP satisfies condition
$p(Q_{\theta})=\frac{\mathbb{E}_{P_{\theta}}[X_{1}\cdot
e^{\beta(X_{1},\theta)}]}{\mathbb{E}_{P_{\theta}}[W_{1}]}.$

The proof follows by arguments similar to those used in Remark 5.2(a).

(b) If $Q$ is a mixed PCP, then the following statements are equivalent:

  1. (i)

$p(P,\varTheta)<p(Q,\varTheta)<\infty\quad
P\upharpoonright\sigma(\varTheta)$–a.s.;

  2. (ii)

there exists a $P_{\varTheta}$–null set $M_{\ast\ast}\in{\mathfrak{B}}(D)$,
containing the $P_{\varTheta}$–null set $\widetilde{M}_{\ast}$, such that
$p(P_{\theta})<p(Q_{\theta})<\infty$ for any $\theta\notin M_{\ast\ast}$.

In fact, due to Remark 5.2(a)(ii) and [9], Lemma 3.5, condition
$p(Q,\varTheta)<\infty$ $P\upharpoonright\sigma(\varTheta)$–a.s. is equivalent
to the fact that there exist $P_{\varTheta}$–null sets
$L_{Q,1},M_{Q}\in{\mathfrak{B}}(D)$ such that
$p(Q,\theta)=p(Q_{\theta})<\infty$ for any $\theta\notin L_{Q,1}\cup M_{Q}$.
Putting $\widehat{M}_{\ast}:=\widetilde{M}_{\ast}\cup L_{P,1}\cup
M_{Q}\in{\mathfrak{B}}(D)$ and applying again Remark 5.2(a)(ii), for any
$B\in{\mathfrak{B}}(D)$ we get

$\displaystyle p(P,\varTheta)\leq p(Q,\varTheta)\quad
P\upharpoonright\sigma(\varTheta)$
$\displaystyle\Longleftrightarrow\int_{\varTheta^{-1}[B]}p(P,\varTheta)\,dP\leq\int_{\varTheta^{-1}[B]}p(Q,\varTheta)\,dP$
$\displaystyle\Longleftrightarrow\int_{B\cap\widehat{M}^{c}_{\ast}}p(P_{\theta})\,P_{\varTheta}(d\theta)\leq\int_{B\cap\widehat{M}^{c}_{\ast}}p(Q_{\theta})\,P_{\varTheta}(d\theta),$

or equivalently there exists a $P_{\varTheta}$–null set
$M_{\ast\ast}\in{\mathfrak{B}}(D)$, containing $\widehat{M}_{\ast}$, such that
$p(P_{\theta})\leq p(Q_{\theta})$ for any $\theta\notin M_{\ast\ast}$.

If (ii) holds, assume if possible that $p(P,\varTheta)=p(Q,\varTheta)$
$P\upharpoonright\sigma(\varTheta)$–a.s.. It then follows as above that
$\int_{B\cap\widehat{M}^{c}_{\ast}}p(P_{\theta})\,P_{\varTheta}(d\theta)=\int_{B\cap\widehat{M}^{c}_{\ast}}p(Q_{\theta})\,P_{\varTheta}(d\theta)$
for any $B\in{\mathfrak{B}}(D)$, implying that $p(P_{\theta})=p(Q_{\theta})$
for $P_{\varTheta}$–a.a. $\theta\in D$, a contradiction; hence (ii)
$\Longrightarrow$ (i). In the same way we get the inverse implication.

Condition (i) of Remark 6.2(b), does not in general imply condition $p(P)\leq
p(Q)$, see Remark 6.7. In the next proposition we find sufficient conditions
for such an implication to hold. To prove it, we need first to establish the
validity of the following lemma.

######  Lemma 6.3.

If $Z\in{\mathcal{L}}^{1}(P)$ is a real–valued positive random variable on
$(\varOmega,\varSigma)$ and $f,g$ two monotonic functions of the same
monotonicity such that $f(Z),g(Z)\in{\mathcal{L}}^{1}(P)$, then

$\mathbb{E}_{P}[\chi_{A}\cdot f(Z)\cdot
g(Z)]\geq\frac{1}{P(A)}\cdot\mathbb{E}_{P}[\chi_{A}\cdot
f(Z)]\cdot\mathbb{E}_{P}[\chi_{A}\cdot g(Z)]\quad\text{for
any}\,\,A\in\sigma(Z).$

###### Proof.

In fact, first note that for any $A\in\sigma(Z)$ there exists a set
$B\in{\mathfrak{B}}(\varUpsilon)$ such that $A:=Z^{-1}(B)$. The same
monotonicity for the functions $f$ and $g$ implies that
$\left(f(z_{1})-f(z_{2})\right)\cdot\left(g(z_{1})-g(z_{2})\right)\geq
0\quad\text{for any}\,\,\,z_{1},z_{2}\in B$; hence

$\displaystyle 0$
$\displaystyle\leq\int_{B}\int_{B}\left(f(z_{1})-f(z_{2})\right)\cdot\left(g(z_{1})-g(z_{2})\right)\,P_{Z}(dz_{1})\,P_{Z}(dz_{2})$
$\displaystyle=2\cdot P_{Z}(B)\cdot\int_{B}f(z_{1})\cdot
g(z_{1})\,P_{Z}(dz_{1})-2\cdot\int_{B}\int_{B}f(z_{1})\cdot
g(z_{2})\,P_{Z}(dz_{1})\,P_{Z}(dz_{2})$ $\displaystyle=2\cdot
P_{Z}(B)\cdot\int_{B}f(z_{1})\cdot
g(z_{1})\,P_{Z}(dz_{1})-2\cdot\left(\int_{B}f(z_{1})\,P_{Z}(dz_{1})\right)\left(\int_{B}g(z_{2})\,P_{Z}(dz_{2})\right).$

Thus the claimed result follows.∎

Remark. If the functions $f$, $g$ appearing in Lemma 6.3 have different
monotonicity, then the inequality continues to apply but in the opposite
direction.

######  Proposition 6.4.

Let $D:=\varUpsilon$ and assume that
$p(P_{\bullet})\in{\mathcal{L}}^{1}(P_{\varTheta})$ and that
$p(Q_{\bullet}),\frac{p(Q_{\bullet})}{\xi(\bullet)}\in{\mathcal{L}}^{1}(Q_{\varTheta})$.
If $p(Q_{\bullet})$ and $\xi(\bullet)$ are monotonic functions of $\theta$
with the same monotonicity and $p(P_{\theta})\leq p(Q_{\theta})$ for any
$\theta\notin M_{\ast\ast}$, then condition $p(P)\leq p(Q)$ holds true.

###### Proof.

Since $p(Q_{\bullet})$ and $\xi(\bullet)$ are monotonic functions of $\theta$
with the same monotonicity and $p(P_{\theta})\leq p(Q_{\theta})$ for any
$\theta\notin M_{\ast\ast}$, by Remark 5.2(b) we get

$\displaystyle p(P)$
$\displaystyle=\int_{M^{c}_{\ast\ast}}p(P_{\theta})\,P_{\varTheta}(d\theta)\leq\int_{M^{c}_{\ast\ast}}p(Q_{\theta})\,P_{\varTheta}(d\theta)=\int_{M^{c}_{\ast\ast}}\frac{p(Q_{\theta})}{\xi(\theta)}\,Q_{\varTheta}(d\theta)$
$\displaystyle\leq\left(\int_{M^{c}_{\ast\ast}}p(Q_{\theta})\,Q_{\varTheta}(d\theta)\right)\cdot\left(\int_{M^{c}_{\ast\ast}}\frac{1}{\xi(\theta)}\,Q_{\varTheta}(d\theta)\right)=\int_{M^{c}_{\ast\ast}}p(Q_{\theta})\,Q_{\varTheta}(d\theta),$

where the second inequality follows by Lemma 6.3 along with the Remark
following it, and the last equality holds since
$\mathbb{E}_{Q}[\frac{1}{\xi(\varTheta)}]=1$; hence $p(P)\leq p(Q)$,
completing in this way the whole proof.∎

In the next Examples 6.5 to 6.8, applying Theorems 3.5 and 4.6, we show how to
construct mixed PCPs $Q$ satisfying conditions

$p(P,\varTheta)<p(Q,\varTheta)<\infty\quad
P\upharpoonright\sigma(\varTheta)\text{--a.s.}\quad\text{ and }\quad
p(P)<p(Q)<\infty,$

and such that for any $T>0$ the process $V_{\mathbb{T}}(\varTheta)$ has the
property of (NFLVR).

######  Example 6.5.

Take $D:=(1,\infty)^{2}$, and let $\varTheta=(\varTheta_{1},\varTheta_{2})$ be
a $D$-valued random vector on $\varOmega$ with
$\varTheta_{1},\varTheta_{2}\in{\mathcal{L}}^{1}(P)$. Moreover, assume that
$P\in{\mathcal{M}}^{\ast,2}_{S,{\bf K}(\varTheta)}$ with

${\bf K}(\varTheta):=\frac{1}{2}\cdot{\bf
Exp}(1/\varTheta_{1})+\frac{1}{2}\cdot{\bf Exp}(1/\varTheta_{2}),$

and $P_{X_{1}}={\bf Ga}(\zeta,2)$ for $\zeta\in\varUpsilon$.

Consider the real-valued function $\beta:=\gamma+\alpha$ on $\varUpsilon\times
D$, with $\gamma(x):=\ln\frac{\mathbb{E}_{P}[X_{1}]}{2c}-\ln
x+\frac{2(c-1)}{c\mathbb{E}_{P}[X_{1}]}\cdot x$ for any $x\in\varUpsilon$ with
$c>2$ a real constant, and $\alpha(\theta):=0$ for any $\theta\in D$. Applying
standard computations, we obtain that $\mathbb{E}_{P}[e^{\gamma(X_{1})}]=1$
and $\mathbb{E}_{P}[X^{2}_{1}\cdot
e^{\gamma(X_{1})}]=\frac{2c^{2}}{\zeta}<\infty$, implying
$\beta\in{\mathcal{F}}^{2}_{P,\varTheta}$.

Let $\xi\in\mathfrak{M}_{+}(D)$ defined by means of
$\xi(\theta):=\xi(\theta_{1},\theta_{2}):=1$ for any $\theta\in D$. Clearly
$\mathbb{E}_{P}[\xi(\varTheta)]=1$, implying $\xi\in\mathcal{R}_{+}(D)$, and
since

$\mathbb{E}_{P}\left[\left(\frac{2}{\varTheta_{1}+\varTheta_{2}}\right)^{2}\cdot\xi(\varTheta)\right]<\infty,$

we deduce that $\xi\in\mathcal{R}^{\ast,2}(D)$.

Since
$(\beta,\xi)\in{\mathcal{F}}^{2}_{P,\varTheta}\times\mathcal{R}^{\ast,2}(D)$,
applying Theorem 3.5 we obtain a unique pair
$(\rho,Q)\in\mathfrak{M}_{+}(D)\times{\mathcal{M}}^{*,2}_{S,{\bf
Exp}(\rho(\varTheta))}$ determined by ($RPM_{\xi}$) and satisfying conditions
($\ast$ ‣ 1), an essentially unique rcp $\\{Q_{\theta}\\}_{\theta\in D}$ of
$Q$ over $Q_{\varTheta}$ consistent with $\varTheta$ and a
$P_{\varTheta}$–null set $\widetilde{L}_{\ast\ast}\in{\mathfrak{B}}(D)$ such
that for any $\theta\notin\widetilde{L}_{\ast\ast}$ conditions
$Q_{\theta}\in{{\mathcal{M}}}^{2}_{S,{\bf Exp}(\rho(\theta))}$,
($\widetilde{\ast}$ ‣ 3) and ($RPM_{\theta}$) hold true. It then follows by
condition ($\ast$ ‣ 1) that
$\rho(\varTheta)=\frac{2}{\varTheta_{1}+\varTheta_{2}}$
$P\upharpoonright\sigma(\varTheta)$–a.s., while condition ($RPM_{\xi}$) yields

$Q_{X_{1}}(B)=\mathbb{E}_{P}[\chi_{X^{-1}_{1}[B]}\cdot
e^{\gamma(X_{1})}]=\int_{B}\frac{\zeta}{c}\cdot e^{-\frac{\zeta}{c}\cdot
x}\,\lambda(dx)\quad\text{for any}\,\,B\in{\mathfrak{B}}(\varUpsilon)$

and

$Q_{\varTheta}(\widetilde{B})=\mathbb{E}_{P}[\chi_{\varTheta^{-1}[\widetilde{B}]}\cdot\xi(\varTheta)]=P_{\varTheta}(\widetilde{B})\quad\text{for
any}\,\,\widetilde{B}\in{\mathfrak{B}}(D);$

hence for any $\theta\notin\widetilde{L}_{\ast\ast}$ the probability measure
$Q_{\theta}$ is a PCP satisfying condition

$p(P_{\theta})=\frac{4}{\zeta\cdot(\theta_{1}+\theta_{2})}<\frac{2\cdot
c}{\zeta\cdot(\theta_{1}+\theta_{2})}=p(Q_{\theta}),$

where the inequality follows since $c>2$. The latter, along with Remark 5.2(b)
and standard computations yields
$p(P)=\int_{D}p(P_{\theta})\,P_{\varTheta}(d\theta)<\int_{D}p(Q_{\theta})\,Q_{\varTheta}(d\theta)=p(Q)<\infty$.

Fix on arbitrary $\theta\notin M_{\ast}$. According to Theorem 4.4 the
processes $\\{V_{t}(\varTheta)\\}_{t\in{\mathbb{R}}_{+}}$ and
$\\{V_{t}(\theta)\\}_{t\in{\mathbb{R}}_{+}}$ with

$V_{t}(\varTheta)=S_{t}-t\cdot\frac{2}{\varTheta_{1}+\varTheta_{2}}\cdot\frac{\mathbb{E}_{P}[X_{1}]}{2c}\cdot\mathbb{E}_{P}\left[e^{\frac{2\cdot(c-1)}{c\cdot\mathbb{E}_{P}[X_{1}]}\cdot
X_{1}}\right]=S_{t}-\frac{2\cdot
c}{\zeta\cdot(\varTheta_{1}+\varTheta_{2})}\quad
P\upharpoonright\sigma(\varTheta)$

and

$V_{t}(\theta)=S_{t}-t\cdot\frac{2}{\theta_{1}+\theta_{2}}\cdot\frac{\mathbb{E}_{P_{\theta}}[X_{1}]}{2c}\cdot\mathbb{E}_{P_{\theta}}\left[e^{\frac{2\cdot(c-1)}{c\cdot\mathbb{E}_{P_{\theta}}[X_{1}]}\cdot
X_{1}}\right]=S_{t}-\frac{2\cdot c}{\zeta\cdot(\theta_{1}+\theta_{2})}$

for any $t\geq 0$ are martingales in ${\mathcal{L}}^{2}(Q)$ and
${\mathcal{L}}^{2}(Q_{\theta})$, respectively. In particular, for any $T>0$,
Theorem 4.6, asserts that both processes $V_{\mathbb{T}}(\varTheta)$ and
$V_{\mathbb{T}}(\theta)$ satisfy condition (NFLVR).

######  Example 6.6.

Take $D:=\varUpsilon$ and let $\varTheta$ be a positive random variable.
Assume that $P\in{\mathcal{M}}^{\ast,2}_{S,{\bf Ga}(\varTheta,k)}$ with
$k\in\varUpsilon$, such that $P_{X_{1}}={\bf Exp}(\eta)$ with
$\eta\in\varUpsilon$ and $P_{\varTheta}={\bf Ga}(b_{1},a)$ with
$(b_{1},a)\in\varUpsilon^{2}$.

Consider the real-valued function $\beta:=\gamma+\alpha$ on $\varUpsilon\times
D$, with $\gamma(x):=\ln(1-c\cdot\mathbb{E}_{P}[X_{1}])+c\cdot x$ for any
$x\in\varUpsilon$ with $c<\eta$ a positive constant, and
$\alpha(\theta):=\ln\frac{\theta}{d}\cdot\mathbb{E}_{P_{\theta}}[W_{1}]$ for
any $\theta\in D$, where $d<k$ is a positive constant. It can be easily seen
that $\mathbb{E}_{P}[e^{\gamma(X_{1})}]=1$ and $\mathbb{E}_{P}[X^{2}_{1}\cdot
e^{\gamma(X_{1})}]=\frac{2}{(\eta-c)^{2}}<\infty$, implying
$\beta\in{\mathcal{F}}^{2}_{P,\varTheta}$.

Let $\xi\in\mathfrak{M}_{+}(\varUpsilon)$ defined by means of
$\xi(\theta):=\left(\frac{b_{2}}{b_{1}}\right)^{a}\cdot
e^{-(b_{2}-b_{1})\cdot\theta}$ for any $\theta\in\varUpsilon$, where
$b_{2}\in\varUpsilon$ is constant such that $b_{2}<b_{1}$. Clearly
$\mathbb{E}_{P}[\xi(\varTheta)]=1$, implying that
$\xi\in\mathcal{R}_{+}(\varUpsilon)$. Applying standard computations we get

$\mathbb{E}_{P}\left[\xi(\varTheta)\cdot\left(\frac{e^{\ln(\frac{\varTheta}{d}\cdot\mathbb{E}_{P}[W_{1}\mid\varTheta])}}{\mathbb{E}_{P}[W_{1}\mid\varTheta]}\right)^{2}\right]=\frac{\mathbb{E}_{P}[\xi(\varTheta)\cdot\varTheta^{2}]}{d^{2}}<\infty,$

implying that $\xi\in\mathcal{R}^{\ast,2}(D)$.

Since
$(\beta,\xi)\in{\mathcal{F}}^{2}_{P,\varTheta}\times\mathcal{R}^{\ast,2}(D)$,
applying Theorem 3.5 we obtain a unique pair
$(\rho,Q)\in\mathfrak{M}_{+}(D)\times{\mathcal{M}}^{*,2}_{S,{\bf
Exp}(\rho(\varTheta))}$ determined by ($RPM_{\xi}$) and satisfying conditions
($\ast$ ‣ 1), an essentially unique rcp $\\{Q_{\theta}\\}_{\theta\in D}$ of
$Q$ over $Q_{\varTheta}$ consistent with $\varTheta$ and a
$P_{\varTheta}$–null set $\widetilde{L}_{\ast\ast}\in{\mathfrak{B}}(D)$ such
that for any $\theta\notin\widetilde{L}_{\ast\ast}$ conditions
$Q_{\theta}\in{{\mathcal{M}}}^{2}_{S,{\bf Exp}(\rho(\theta))}$,
($\widetilde{\ast}$ ‣ 3) and ($RPM_{\theta}$) hold true. It then follows by
condition ($\ast$ ‣ 1) that $\rho(\varTheta)=\frac{\varTheta}{d}$
$P\upharpoonright\sigma(\varTheta)$–a.s., while condition ($RPM_{\xi}$) yields

$Q_{X_{1}}(B)=\mathbb{E}_{P}[\chi_{X^{-1}_{1}[B]}\cdot
e^{\gamma(X_{1})}]=\int_{B}(\eta-c)\cdot e^{-(\eta-c)\cdot
x}\,\lambda(dx)\quad\text{for any}\,\,B\in{\mathfrak{B}}(\varUpsilon)$

and

$Q_{\varTheta}(\widetilde{B})=\mathbb{E}_{P}[\chi_{\varTheta^{-1}[\widetilde{B}]}\cdot\xi(\varTheta)]=\int_{\widetilde{B}}\frac{b_{2}^{a}\cdot\theta^{a-1}}{\Gamma(a)}\cdot
e^{-b_{2}\cdot\theta}\,\lambda(d\theta)\quad\text{for
any}\,\,\widetilde{B}\in{\mathfrak{B}}(D);$

hence for any $\theta\notin\widetilde{L}_{\ast\ast}$ the probability measure
$Q_{\theta}$ is a PCP satisfying condition
$p(P_{\theta})=\frac{\theta}{k\cdot\eta}<\frac{\theta}{d\cdot(\eta-c)}=p(Q_{\theta})$.
Thus, applying Remark 5.2(b) together with by standard computations we obtain

$p(P)=\int_{D}p(P_{\theta})\,P_{\varTheta}(d\theta)=\frac{a}{b_{1}\cdot
k\cdot\eta}\quad\text{ and }\quad
p(Q)=\int_{D}p(Q_{\theta})\,Q_{\varTheta}(d\theta)=\frac{a}{b_{2}\cdot
d\cdot(\eta-c)},$

implying $p(P)<p(Q)$.

Fix on arbitrary $\theta\notin M_{\ast}$. According to Theorem 4.4 the
processes $\\{V_{t}(\varTheta)\\}_{t\in{\mathbb{R}}_{+}}$ and
$\\{V_{t}(\theta)\\}_{t\in{\mathbb{R}}_{+}}$ with

$V_{t}(\varTheta)=S_{t}-t\cdot\frac{\varTheta}{d}\cdot(1-c\cdot\mathbb{E}_{P}[X_{1}])\cdot\mathbb{E}_{P}\left[X_{1}\cdot
e^{c\cdot X_{1}}\right]=S_{t}-\frac{\varTheta}{d\cdot(\eta-c)}\quad
P\upharpoonright\sigma(\varTheta)$

and

$V_{t}(\theta)=S_{t}-t\cdot\frac{\theta}{d}\cdot(1-c\cdot\mathbb{E}_{P_{\theta}}[X_{1}])\cdot\mathbb{E}_{P_{\theta}}\left[X_{1}\cdot
e^{c\cdot X_{1}}\right]=S_{t}-\frac{\theta}{d\cdot(\eta-c)}$

for any $t\geq 0$ are martingales in ${\mathcal{L}}^{2}(Q)$ and
${\mathcal{L}}^{2}(Q_{\theta})$, respectively. In particular, for any $T>0$,
Theorem 4.6, asserts that both processes $V_{\mathbb{T}}(\varTheta)$ and
$V_{\mathbb{T}}(\theta)$ satisfy condition (NFLVR).

######  Remark 6.7.

In the situation of Example 6.6, assume that $c=0$. If we choose the constant
$b_{2}$ in the interval $\left(\frac{b_{1}\cdot k}{d},\infty\right)$, we get
that even though the corresponding measures of the rcp
$\\{Q_{\theta}\\}_{\theta\in D}$ satisfy condition
$p(P_{\theta})<p(Q_{\theta})$ for any $\theta\notin\widetilde{L}_{\ast\ast}$,
the inequality $p(P)<p(Q)$ fails.

In our last example we rediscover Wang’s risk-adjusted premium principle (see
[20] for the definition and its properties).

######  Example 6.8.

Take $D:=\varUpsilon$ and let $\varTheta$ be a positive real-valued random
variable. Moreover, assume that $P\in{\mathcal{M}}^{\ast,2}_{S,{\bf
Exp}(\varTheta)}$, such that
$P_{X_{1}}\ll\lambda\upharpoonright{\mathfrak{B}}([0,1])$ and denote by
$\widebar{F}_{X_{1}}(x):=P_{X_{1}}\left((x,\infty)\right)$ for any
$x\in\varUpsilon$ the corresponding survival function of the random variable
$X_{1}$. Finally assume that
$\int_{0}^{\infty}x\cdot\left(\widebar{F}_{X_{1}}(x)\right)^{\frac{1}{c}}\,\lambda(dx)<\infty$,
where $c>1$ is a real constant. Recall that the risk adjusted premium for a
risk $X_{1}$ is defined by means of

$\pi_{c}(X_{1}):=\int_{0}^{\infty}\left(\widebar{F}_{X_{1}}(x)\right)^{\frac{1}{c}}\,\lambda(dx)\quad\text{
for any }\,\,c\geq 1.$

(see [20], Definition 2).

Consider the real-valued function $\beta:=\gamma+\alpha$ on $\varUpsilon\times
D$, with $\gamma(x):=-\ln c+(\frac{1}{c}-1)\cdot\ln\widebar{F}_{X_{1}}(x)$ for
any $x\in\varUpsilon$ and $\alpha(\theta):=0$ for any $\theta\in D$. Applying
standard computations, we get that $\mathbb{E}_{P}[e^{\gamma(X_{1})}]=1$ and
$\mathbb{E}_{P}[X^{2}_{1}\cdot
e^{\gamma(X_{1})}]=2\cdot\int_{0}^{\infty}x\cdot\left(\widebar{F}_{X_{1}}(x)\right)^{\frac{1}{c}}\,\lambda(dx)<\infty$,
implying $\beta\in{\mathcal{F}}^{2}_{P,\varTheta}$.

For any $r>0$ such that $\mathbb{E}_{P}[e^{r\cdot\varTheta}]<\infty$, define
the function $\xi\in\mathfrak{M}_{+}(D)$ by means of
$\xi(\theta):=\frac{e^{r\cdot\theta}}{\mathbb{E}_{P}[e^{r\cdot\varTheta}]}$
for any $\theta\in D$. Clearly $\mathbb{E}_{P}[\xi(\varTheta)]=1$, implying
that $\xi\in\mathcal{R}_{+}(D)$. But since $P\in{\mathcal{M}}^{\ast,2}_{S,{\bf
Exp}(\varTheta)}$ we get $\mathbb{E}_{P}[\varTheta^{2}]<\infty$, implying that
$\left(M_{\varTheta}(r)\right)^{\prime\prime}<\infty$ for all $r$ in a
neighbourhood of $0$, where by $M_{\varTheta}$ is denoted the moment
generating function of $\varTheta$; hence

$\mathbb{E}_{P}[\varTheta^{2}\cdot\xi(\varTheta)]=\frac{\mathbb{E}_{P}[\varTheta^{2}\cdot
e^{r\cdot\varTheta}]}{\mathbb{E}_{P}[e^{r\cdot\varTheta}]}=\frac{\left(M_{\varTheta}(r)\right)^{\prime\prime}}{M_{\varTheta}(r)}<\infty,$

implying $\xi\in\mathcal{R}^{\ast,2}_{+}(D)$.

Since
$(\beta,\xi)\in{\mathcal{F}}^{2}_{P,\varTheta}\times\mathcal{R}^{\ast,2}_{+}(D)$,
we may apply Theorem 3.5 in order to get a unique pair
$(\rho,Q)\in\mathfrak{M}_{+}(D)\times{\mathcal{M}}^{\ast,2}_{S,{\bf
Exp}(\rho(\varTheta))}$ determined by ($RPM_{\xi}$) and satisfying conditions
($\ast$ ‣ 1), an essentially unique rcp $\\{Q_{\theta}\\}_{\theta\in D}$ of
$Q$ over $Q_{\varTheta}$ consistent with $\varTheta$ and a
$P_{\varTheta}$–null set $\widetilde{L}_{\ast\ast}\in{\mathfrak{B}}(D)$
satisfying for any $\theta\notin\widetilde{L}_{\ast\ast}$ conditions
$Q_{\theta}\in{{\mathcal{M}}}^{2}_{S,{\bf Exp}(\rho(\theta))}$,
($\widetilde{\ast}$ ‣ 3) and ($RPM_{\theta}$). It then follows by condition
($\ast$ ‣ 1) that $\rho(\varTheta)=\varTheta$
$P\upharpoonright\sigma(\varTheta)$–a.s., while condition ($RPM_{\xi}$) yields
that

$Q_{X_{1}}(B_{1})=\mathbb{E}_{P}[\chi_{X^{-1}_{1}[B_{1}]}\cdot
e^{\gamma(X_{1})}]=\int_{B_{1}}\frac{1}{c}\cdot\left(\widebar{F}(x)\right)^{\frac{1}{c}-1}\,P_{X_{1}}(dx)\quad\text{for
any}\,\,B_{1}\in{\mathfrak{B}}(\varUpsilon)$

and

$Q_{\varTheta}(B_{2})=\mathbb{E}_{P}[\chi_{\varTheta^{-1}[B_{2}]}\cdot\xi(\varTheta)]=\int_{B_{2}}\frac{e^{r\cdot\theta}}{\mathbb{E}_{P}[e^{r\cdot\varTheta}]}\,P_{\varTheta}(d\theta)\quad\text{for
any}\,\,B_{2}\in{\mathfrak{B}}(D);$

hence for any $\theta\notin\widetilde{L}_{\ast\ast}$ the corresponding measure
$Q_{\theta}$ is a PCP satisfying condition

(12)
$p(P_{\theta})=\theta\cdot\int_{0}^{\infty}\widebar{F}(x)\,\lambda(dx)<\theta\cdot\int_{0}^{\infty}\left(\widebar{F}(x)\right)^{\frac{1}{c}}\,\lambda(dx)=\theta\cdot\pi_{c}(X_{1})=p(Q_{\theta}),$

where the inequality follows since $c>1$. Since
$Q\in{\mathcal{M}}^{\ast,2}_{S,{\bf Exp}(\varTheta)}$, we obtain

$\int_{D}\frac{p(Q_{\theta})}{\xi(\theta)}\,Q_{\varTheta}(d\theta)=\int_{D}\theta\cdot
e^{-r\cdot\theta}\cdot\mathbb{E}_{P}[e^{r\cdot\varTheta}]\,Q_{\varTheta}(d\theta)\cdot\pi_{c}(X_{1})<\infty,$

while conditions $P,Q\in{\mathcal{M}}^{\ast,2}_{S,{\bf Exp}(\varTheta)}$ along
with Remark 5.2(b) yield $p(P),p(Q)<\infty$.

Because $\xi(\theta)$ and $p(Q_{\theta})$ have the same monotonicity as
functions of $\theta$ and condition (12) is valid, we may apply Proposition
6.4 to get $p(P)<p(Q)<\infty$.

Fix on arbitrary $\theta\notin M_{\ast}$. According to Theorem 4.4 the
processes $\\{V_{t}(\varTheta)\\}_{t\in{\mathbb{R}}_{+}}$ and
$\\{V_{t}(\theta)\\}_{t\in{\mathbb{R}}_{+}}$ with
$V_{t}(\varTheta)=S_{t}-t\cdot\varTheta\cdot\pi_{c}(X_{1})\quad
P\upharpoonright\sigma(\varTheta)\text{--a.s.}$ and
$V_{t}(\theta)=S_{t}-t\cdot\theta\cdot\pi_{c}(X_{1})$ for any $t\geq 0$ are
martingales in ${\mathcal{L}}^{2}(Q)$ and ${\mathcal{L}}^{2}(Q_{\theta})$,
respectively. In particular, for any $T>0$, Theorem 4.6, asserts that both
processes $V_{\mathbb{T}}(\varTheta)$ and $V_{\mathbb{T}}(\theta)$ satisfy
condition (NFLVR).

## References

  * [1] Cohn, D. L. (2013) Measure Theory, 2nd ed. Birkhäuser Advanced Texts.
  * [2] Delbaen, F. and Haezendock, J. (1989) A martingale approach to premium calculation principles in an arbitrage free market. Insur. Math. Econ., 8, 269–-277.
  * [3] Delbaen, F. and Schachermayer, W. (2006) The Mathematics of Arbitrage, Springer-Verlag Berlin Heidelberg.
  * [4] Embrechts, P. (2000) Actuarial versus Financial Pricing of Insurance. The Journal of Risk Finance 1, 17–26.
  * [5] Embrechts, P. and Meister, S. (1995) Pricing Insurance Derivatives: The Case of CAT-Futures. Securitization of Insurance Risk, 1995 Bowles Symposium. SOA Monograph M-FI97-1, 15–26.
  * [6] Grandell, J. (1997) Mixed Poisson Processes, Chapman & Hall.
  * [7] Gut, A. (2009) Stopped Random Walks: Limit Theorems and Applications, 2nd ed. Springer-Verlag New York.
  * [8] Harisson, J. M. and Kreps, D. M. (1979) Martingales and arbitrage in multiperiod securities markets, J. Econ. Theory, 20, 381–408.
  * [9] Lyberopoulos, D. P. and Macheras, N. D. (2012) Some characterizations of mixed Poisson processes, Sankhyā A, 74, 57–79.
  * [10] Lyberopoulos, D. P. and Macheras, N. D. (2014) Some characterizations of mixed renewal processes, arXiv: 1205.4441.
  * [11] Lyberopoulos, D. P. and Macheras, N. D. (2019) A characterization of martingale-equivalent compound mixed Poisson processes, arXiv:1905.07629.
  * [12] Lyberopoulos, D. P. and Macheras, N. D. (2020) Some martingale characterizations of compound mixed Poisson processes, arXiv: 2004.07835
  * [13] Macheras, N.D. and Tzaninis, S. M. (2018) Some characterizations for Markov processes as mixed renewal processes, Math. Slovaca., 68, 1477–-1494.
  * [14] Macheras, N. D. and Tzaninis, S. M. (2020) A characterization of equivalent martingale measures in a renewal risk model with applications to premium calculation principles, Mod. Stoch.: Theory Appl., 7, 43–60.
  * [15] Pachl, J. K. (1978) Disintegration and compact measures, Math. Scand. 43, 157-168.
  * [16] Schmidli, H. (2017) Risk Theory. Springer-Verlag.
  * [17] Schmidt, K.D. (1996) Lectures on Risk Theory, B.G. Teubner, Stuttgart.
  * [18] Tzaninis, S. M. and Macheras, N. D. (2020) A characterization of progressively equivalent probability measures preserving the structure of a compound mixed renewal process, arXiv:2007.05289.
  * [19] von Weizsäcker, H. and Winkler, G. (1990) Stochastic Integrals: An Introduction, Vieweg+Teubner Verlag.
  * [20] Wang, S. (1995) Insurance pricing and increased limits ratemaking by proportional hazards transforms. Insur. Math. Econ., 17, 43–54.