# Multiple-prior valuation of cash flows subject to capital requirements

Hampus<EMAIL_ADDRESS>Stockholm University, Filip
Lindskog222corresponding author<EMAIL_ADDRESS>Stockholm University,
Julie<EMAIL_ADDRESS>Aarhus University

###### Abstract

We study market-consistent valuation of liability cash flows motivated by
current regulatory frameworks for the insurance industry. Building on the
theory on multiple-prior optimal stopping we propose a valuation functional
with sound economic properties that applies to any liability cash flow.
Whereas a replicable cash flow is assigned the market value of the replicating
portfolio, a cash flow that is not fully replicable is assigned a value which
is the sum of the market value of a replicating portfolio and a positive
margin. The margin is a direct consequence of considering a hypothetical
transfer of the liability cash flow from an insurance company to an empty
corporate entity set up with the sole purpose to manage the liability run-off,
subject to repeated capital requirements, and considering the valuation of
this entity from the owner’s perspective taking model uncertainty into
account. Aiming for applicability, we consider a detailed insurance
application and explain how the optimisation problems over sets of probability
measures can be cast as simpler optimisation problems over parameter sets
corresponding to parameterised density processes appearing in applications.

##  1 Introduction

We consider the valuation of an aggregate insurance liability cash flow in
run-off. The valuation approach is a direct consequence of considering a
hypothetical transfer of the liability cash flow from an insurance company to
an empty corporate entity set up with the sole purpose to manage the liability
run-off. The owner of this entity needs to make sure at any time, in order to
continue ownership of the entity, to pay claims and also to provide buffer
capital according to the externally imposed solvency capital requirement (e.g.
by a regulatory framework such as Solvency II). The owner accepts ownership
given a suitable initial compensation from the original insurance company
wanting to transfer its liabilities. This compensation determine the value of
the liability cash flow. However, the owner has the right to, at any time, any
surplus exceeding what is required to manage the liability run-off and meet
solvency capital requirements. Therefore, the amount of compensation that the
owner finds acceptable depends of the owner’s view of such surplus.

The setting and the valuation approach we consider are similar to those
considered in Engsner et al. [13]. An essential difference here is that we
acknowledge that an agent who assigns a value to possible future dividends and
capital injections from managing a run-off of a liability may consider a
valuation functional depending on a set of pricing measures, in the incomplete
market setting, rather than a single one. The agent is uncertain about which
pricing measure to use and may change view depending on new information.
Although this appears to be a modest difference it on the one hand lead to
significant mathematical challenges and on the other hand it means that the
conservative valuation functional, corresponding to expected discounted values
according to the worst pricing measure, can be applied to a wide range of
liability cash flows rather than having to be chosen in order to match a
specific type of liability cash flow. In order to make this statement clear we
may think of cash flows from life insurance. If the agent benefits from
survival of policyholders, then a conservative valuation from the agents
perspective corresponds to choosing a pricing measure $\mathbb{Q}$ that
assigns higher probability to the occurrences of deaths compared to
$\mathbb{P}$. However, if the agent instead benefits from deaths of
policyholders, then such a $\mathbb{Q}$ no longer corresponds to conservative
valuation.

Insurance liability cash flows may be partly defined in terms of financial
asset prices, specific interest rates or inflations indices. For liability
cash flows where this is not the case, the cash flows may show significant
correlation with market prices. Therefore, any insurance liability valuation
methodology must be such not to introduce arbitrage opportunities and must
consider replicating portfolios that hedge the financial component of a
liability cash flow, whenever that is relevant. Consequently, there is a
significant literature on market-consistent insurance valuation covering
single-period, multiple-period and continuous-time valuation problems with
varying assumptions on the financial market forming the basis for designing
replicating portfolios of varying degrees of sophistication. We refer (in
chronological order) to Grosen and Jørgensen [16], Malamud et al. [19],
Wüthrich et al. [27], Möhr [20], Tsanakas et al. [26], Wüthrich and Merz [28],
Pelsser and Stadje [21], Engsner et al. [14], Delong et al. [9], Barigou and
Dhaene [3], Engsner et al. [13], and references therein.

A common theme in the literature on market-consistent insurance valuation is
that the value assigned to a liability cash flow can be expressed as the sum
of a market price of a replicating portfolio and a value assigned to the
replication error (notice that a substantial replication error is a common
feature of insurance liabilities). The liability values in this paper are also
of this kind. Rebalancing times of a dynamic replicating portfolio means that
the replication error has to be reassessed over time and taking this into
consideration leads to the notion of time-consistent valuation. Similarly,
repeated capital requirements lead to capital costs that are not known at the
initial valuation time and taking such costs into account appropriately also
require time-consistent valuation. Time consistency is a key concept in the
literature on dynamic risk measurement. We refer (in chronological order) to
Riedel [22], Detlefsen and Scandolo [11], Rosazza Gianin [25], Cheridito et
al. [5], Artzner et al. [1], Bion-Nadal [4], Cheridito and Kupper [6],
Cheridito and Kupper [7], and references therein.

In Artzner et al. [2] and Deelstra et al. [8] it is argued that diversifiable
insurance risk should only be assigned a value corresponding to the
$\mathbb{P}$-expectation of such risk since the law of large numbers applies
if the insurance company may form arbitrarily large portfolios. In our setting
this argument is not valid since the corporate entity to which the insurance
company’s aggregate liability is transferred is a separate entity (referred to
as reference undertaking in Solvency II) that may not be merged with other
corporate entities. In that sense the entity to which the liabilities are
transferred may be seen as a special purpose vehicle. Although this entity
benefits from diversification when capital requirements are computed, it can
not diversify the liability further.

Optimal stopping with multiple priors for agents assessing risk in terms of
dynamic convex risk measures is analyzed in Cheridito et al. [5]. Similar
problems are analyzed in Engelage [12], where the framework of optimal
stopping with multiple priors in [23] is extended to so-called dynamic
variational preferences. From an applied perspective: whereas all
priors/probability measures in a given set of priors are treated as equally
likely in the framework in Riedel [23], introducing (dynamic) penalty terms as
in Cheridito et al. [5] and Engelage [12] means that the optimizing agent may
assign different (dynamic) weights to the priors in the optimization problem.
Optimal stopping is a key element in our approach to valuation since the owner
of the entity managing the run-off of the liability, just as shareholders in
general, has limited liability. At any time, taking the value of assets and
future liability cash flows into account, if a capital injection is needed to
meet capital requirements, the owner may choose between making a capital
injection of not. Without such a capital injection, ownership is terminated
and the the remaining assets are transferred to policyholders. Therefore, the
rational owner determines optimal stopping times.

Although we assume that the replicating portfolio is chosen to ensure that the
valuation of the liability cash flow is market consistent, we do not discuss
market consistency in detail since this was treated in detail in Engsner et
al. [13] and the material on market consistency in Engsner et al. [13] applies
without any modification also in the present paper. However, we emphasise that
we advocate choosing a replicating portfolio in agreement with what
recommended by EIOPA in [10, Article 38] in Article 38(h) on the Reference
Undertaking: ”the assets are selected in such a way that they minimise the
Solvency Capital Requirement for market risk that the reference undertaking is
exposed to”. The reference undertaking in Solvency II is similar in spirit to
the corporate entity managing the run-off of the liability in the present
paper.

The paper is organised as follows. Section 2 presents the valuation framework.
Basic assumptions, notation and terminology are introduced in Subsection 2.1.
Subsection 2.2 introduces the agents involved and explains how capital
requirements and limited liability are key ingredients in the valuation
philosophy that originates from the idea of a hypothetical transfer of an
insurance company’s liabilities to an empty corporate entity. Definitions and
results are presented in Subsection 2.2 for very general capital requirements.
Subsection 2.3 then specialises by considering capital requirements given in
terms of conditional monetary risk measures, in line with current regulatory
frameworks. Section 3 presents a general construction of a parametric set of
priors that cover natural choices for applications and shows that the set of
priors satisfies the properties making it suitable for optimal stopping with
multiple priors. Section 4 considers a specific insurance application that
illustrates the use of the valuation framework and the results presented.

##  2 The valuation framework

###  2.1 Preliminaries

We consider time periods $1,\dots,T$, corresponding time points $0,1,\dots,T$,
and a filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$,
where $\mathbb{F}=(\mathcal{F}_{t})_{t=0}^{T}$ with
$\\{\emptyset,\Omega\\}=\mathcal{F}_{0}\subseteq\dots\subseteq\mathcal{F}_{T}=\mathcal{F}$,
and $\mathbb{P}$ denotes the real-world measure. For $p\in[1,\infty)$, we
write $L^{p}(\mathcal{F}_{t},\mathbb{P})$ for the normed linear space of
$\mathcal{F}_{t}$-measurable random variables $X$ with norm
$\mathbb{E}^{\mathbb{P}}[|X|^{p}]^{1/p}$. We write
$L^{\infty}(\mathcal{F}_{t},\mathbb{P})$ for the normed linear space of
$\mathcal{F}_{t}$-measurable essentially bounded random variables. Equalities
and inequalities between random variables should be interpreted in the
$\mathbb{P}$-almost sure sense. A stopping time is a function
$\tau:\Omega\to\\{0,1,\dots,T\\}\cup\\{+\infty\\}$ such that
$\\{\tau=t\\}\in\mathcal{F}_{t}$ for $t=0,1,\dots,T$.

For two probability measures $\mathbb{Q}^{(1)},\mathbb{Q}^{(2)}$ equivalent to
$\mathbb{P}$ and a stopping time $\tau\leq T$, the probability measure
$\mathbb{Q}^{(3)}(A):=\mathbb{E}^{\mathbb{Q}^{(1)}}[\mathbb{Q}^{(2)}(A\mid\mathcal{F}_{\tau})]$,
$A\in\mathcal{F}_{T}$, is called the pasting of $\mathbb{Q}^{(1)}$ and
$\mathbb{Q}^{(2)}$ in $\tau$. It is often convenient to express the pasting
$\mathbb{Q}^{(3)}$ of $\mathbb{Q}^{(1)},\mathbb{Q}^{(2)}$ in $\tau$ in terms
of the density processes $D^{(1)},D^{(2)}$ with respect to $\mathbb{P}$,

$\displaystyle
D^{(1)}_{t}=\frac{d\mathbb{Q}^{(1)}}{d\mathbb{P}}\Big{|}_{\mathcal{F}_{t}},\quad
D^{(2)}_{t}=\frac{d\mathbb{Q}^{(2)}}{d\mathbb{P}}\Big{|}_{\mathcal{F}_{t}}.$

The density process $D^{(3)}$ given by

$\displaystyle
D^{(3)}_{t}=\mathbb{I}\\{t\leq\tau\\}D^{(1)}_{t}+\mathbb{I}\\{t>\tau\\}\frac{D^{(1)}_{\tau}D^{(2)}_{t}}{D^{(2)}_{\tau}}$

corresponds to the pasting $\mathbb{Q}^{(3)}$ of
$\mathbb{Q}^{(1)},\mathbb{Q}^{(2)}$ in $\tau$. Equivalently, we can write

$D^{(3)}_{t}=\prod_{s=1}^{t}\bigg{(}\mathbb{I}\\{s\leq\tau\\}\frac{D^{(1)}_{s}}{D^{(1)}_{s-1}}+\mathbb{I}\\{s>\tau\\}\frac{D^{(2)}_{s}}{D^{(2)}_{s-1}}\bigg{)}.$

A set $\mathcal{Q}$ of probability measures equivalent to $\mathbb{P}$ is
called stable under pasting if for any
$\mathbb{Q}^{(1)},\mathbb{Q}^{(2)}\in\mathcal{Q}$ and any stopping time
$\tau\leq T$, the pasting $\mathbb{Q}^{(3)}$ of
$\mathbb{Q}^{(1)},\mathbb{Q}^{(2)}$ in $\tau$ is an element in $\mathcal{Q}$.
We call such a set stable under pasting. Such sets are also referred to as
m-stable, time consistent or rectangular in the related literature.

We assume the existence of a financial market containing assets for which
$\mathbb{F}$-adapted price processes $(S^{0}_{t})_{t=0}^{T}$ and
$(S^{i}_{t})_{t=0}^{T}$, $i=1,\dots,d$, are available. $(S^{0}_{t})_{t=0}^{T}$
is the price process of a (predictable) locally riskless bond. The price
processes correspond to traded assets for which reliable price quotes are
available. We take the price process of the locally riskless bond as numéraire
process and in what follows all financial values are discounted by this
numéraire. This saves us from having to explicitly take interest rates
processes into account and makes the mathematical expressions less involved.
We will also allow for $\mathbb{F}$-adapted cash flows that depend on
insurance events independent of the filtration generated by the traded assets.
In particular, we consider an incomplete market setting. We assume that the
set $\mathcal{P}$ of equivalent martingale measures (for each
$\mathbb{Q}\in\mathcal{P}$, $\mathbb{Q}$ is equivalent to $\mathbb{P}$ and the
$(S^{0}_{t})_{t=0}^{T}$-discounted price processes are
$\mathbb{Q}$-martingales) is non-empty. By Proposition 6.43 in Föllmer and
Schied [15] the set $\mathcal{P}$ is stable under pasting. We will consider a
non-empty subset $\mathcal{Q}\subset\mathcal{P}$. We refer to
$\mathbb{Q}\in\mathcal{Q}$ as a market risk neutral probability measure. We
use the conventions $\sum_{l=k}^{k-1}:=0$ and $\inf\emptyset:=+\infty$ for
sums over an empty index set and the infimum of an empty set. We use the
notation $(x)^{+}:=\max(0,x)$.

###  2.2 Valuation with general capital requirements

We consider an insurance company with an aggregate insurance liability
corresponding to a liability cash flow given by the $\mathbb{F}$-adapted
stochastic process $X^{o}=(X^{o}_{t})_{t=1}^{T}$. Regulation forces the
insurance company to comply with externally imposed capital requirements. The
requirements put restrictions on the asset portfolio of the insurance company.
A subset of the assets forms a replicating portfolio with $\mathbb{F}$-adapted
cash flow $X^{r}=(X^{r}_{t})_{t=1}^{T}$ intended to, to some extent, offset
the liability cash flow. Depending on the degree of replicability of the
liability cash flow, the replicating portfolio could be anything from simply a
position in the numéraire asset to a portfolio that is rebalanced dynamically
according to some strategy. $X:=X^{o}-X^{r}$ is the residual liability cash
flow. We will, in accordance with current solvency regulation (Möhr [20] and
prescribed by EIOPA, see [10, Article 38]) define the value of the liability
cash flow $X^{o}$ by considering a hypothetical transfer of the liability and
the replicating portfolio to a separate entity referred to as a reference
undertaking. The reference undertaking has initially neither assets nor
liabilities and its sole purpose is to manage the run-off of the liability.
The benefit of ownership is the right to receive certain dividends/surplus,
defined below, until either the run-off of the liability cash flow is complete
or until letting the reference undertaking default on its obligations to the
policyholders. The term default means termination of ownership of the
reference undertaking. The precise details are as follows,

  * •

At time $t=0$: The liabilities corresponding to the cash flow $X^{o}$, the
replicating portfolio corresponding to the cash flow $X^{r}$ and an amount
$R_{0}$ in the numéraire are supposed to be transferred from the insurance
company to the reference undertaking, where $R_{0}$ is the amount making the
reference undertaking precisely meet the externally imposed capital
requirement. In return, an agent aspiring ownership of the reference
undertaking must first pay the original insurance company an amount $C_{0}$
corresponding to the value of receiving future dividends resulting from
managing the run-off of the liability. In case there are several agents
aspiring ownership, the one offering the highest amount $C_{0}$ wins the
ownership.

_In summary: the new owner of the reference undertaking receives compensation
$V_{0}:=R_{0}-C_{0}$ from the original insurance company as compensation for
accepting to receive the liabilities and replicating portfolio and agreeing to
manage the liability run-off._

  * •

At time $t=0$: By paying the amount $C_{0}$ to the original insurance company,
the owner receives full ownership of the reference undertaking. However, the
cash-flow $X^{r}$ of the replicating portfolio (possibly defined in terms of a
dynamic strategy) cannot be modified by the owner, for instance in order to
boost dividend payments in a way that may not be in the interest of
policyholders.

  * •

At time $t=1$: The owner has the option to either default on its obligations
to the policyholders or not to default.

The decision to default means to give up ownership and transfer $R_{0}$ and
the replicating portfolio to the policyholders. The owner neither receives any
dividend payment nor incurs any loss upon a decision to default.

If $T>1$ and given the decision not to default, a new amount $R_{1}$ in the
numéraire asset is needed to make the reference undertaking precisely meet the
externally imposed capital requirement. If
$R_{0}-R_{1}-X^{o}_{1}+X^{r}_{1}\geq 0$, then the positive surplus
$R_{0}-R_{1}-X^{o}_{1}+X^{r}_{1}\geq 0$ is paid to the owner and $X^{o}_{1}$,
which the policyholders are entitled to, is paid to the policyholders. If
$R_{0}-R_{1}-X^{o}_{1}+X^{r}_{1}<0$, then the owner faces a deficit that must
be offset by injecting $-R_{0}+R_{1}+X^{o}_{1}-X^{r}_{1}>0$. Also in this case
$X^{o}_{1}$ is paid to the policyholders.

If $T=1$, then the above description of cash flows to policyholders and owner
applies upon setting $R_{1}=0$.

  * •

At time $t\in\\{2,\dots,T\\}$: If the owner has not defaulted on its
obligations, then the situation is completely analogous to that at time $t=1$
described above.

From the above follows that the owner of the reference undertaking has to
decide on a decision rule defining under which circumstances default occurs.
The default time is a stopping time $\tau\in\mathcal{S}_{1,T+1}$, where
$\mathcal{S}_{t,T+1}$ denotes the set of $\mathbb{F}$ stopping times taking
values in $\\{t,\dots,T+1\\}$. The event $\\{\tau=T+1\\}$ is to be interpreted
as a complete liability run-off without default at any time. Formally,
$\mathcal{S}_{t,T+1}:=\\{\tau:\tau\text{ is a stopping time with }\tau\geq
t\\}\wedge(T+1)$.

The cumulative cash flow to the owner can be written as

$\displaystyle\sum_{t=1}^{\tau-1}(R_{t-1}-R_{t}-X_{t}),\quad
X_{t}:=X^{o}_{t}-X^{r}_{t}.$ (1)

For ease of notation, define the payoff process $(H_{t})_{t=1}^{T}$ by

$\displaystyle H_{1}:=0,\quad
H_{t}:=\sum_{s=1}^{t-1}(R_{s-1}-R_{s}-X_{s})\quad\text{for }t>1.$ (2)

Note that this payoff process is predictable. The conservative value of the
cash flow (1) is

$\displaystyle\inf_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{0}^{\mathbb{Q}}[H_{\tau}].$
(3)

We assume that the owner of the reference undertaking chooses a default time
$\tau$ maximizing the value (3). Consequently, the value at time $0$ of the
reference undertaking is

$\displaystyle\sup_{\tau\in\mathcal{S}_{1,T+1}}\inf_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{0}^{\mathbb{Q}}[H_{\tau}].$
(4)

For $t\in\\{1,\dots,T\\}$, the value of the reference undertaking at time $t$,
given no default at times $\leq t$, is given by the completely analogous
expression upon replacing $\sup$ and $\inf$ in (4) by the essential supremum
$\operatorname*{ess\,sup}$ and essential infimum $\operatorname*{ess\,inf}$
(see Appendix A.5 in Föllmer and Schied [15] for details) and conditioning on
$\mathcal{F}_{t}$ rather than $\mathcal{F}_{0}$. Notice that since no cash
flows occur at times $>T$, the value of the reference undertaking is zero at
time $T$. The value of the reference undertaking can thus be identified as the
value of an American type derivative. Details on arbitrage-free pricing of
American derivatives can be found in Section 6.3 in [15].

Since we are considering sets $\mathcal{Q}$ of probability measures we need
the cash flows to be suitably integrable with respect to all
$\mathbb{Q}\in\mathcal{Q}$. The following notion of uniform integrability,
from Riedel [23], will be used. The process $(H_{t})_{t=1}^{T}$ in (2) is
bounded by a $\mathcal{Q}$-uniformly integrable random variable in the sense
that there exists $Z\geq 0$ such that

$\displaystyle\sup_{t\in\\{1,\dots,T\\}}|H_{t}|\leq Z\text{ and
}\lim_{K\to\infty}\sup_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}[Z\mathbb{I}_{\\{Z\geq
K\\}}]=0.$ (5)

We now define the value of the reference undertaking, corresponding to what an
external party would pay to become owner of the entity managing the run-off of
the liability, and also the value of the residual liability. The sum of the
latter and the market price of the replicating portfolio is the value of the
original liability to policyholders and therefore is a theoretical aggregate
premium.

######  Definition 1.

Let $\mathcal{Q}$ be a set of market risk neutral probability measures.
Consider sequences $(X_{t})_{t=1}^{T}$ and $(R_{t})_{t=0}^{T}$ with $X_{t}\in
L^{1}(\mathcal{F}_{t},\mathbb{Q})$ for $t\in\\{1,\dots,T\\}$ for every
$\mathbb{Q}\in\mathcal{Q}$, $R_{T}=0$ and $R_{t}\in
L^{1}(\mathcal{F}_{t},\mathbb{Q})$ for $t\in\\{0,\dots,T-1\\}$ for every
$\mathbb{Q}\in\mathcal{Q}$. Define $C_{T}:=0$ and, for
$t\in\\{0,\dots,T-1\\}$,

$\displaystyle
C_{t}:=\operatorname*{ess\,sup}_{\tau\in\mathcal{S}_{t+1,T+1}}\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}\bigg{[}\sum_{s=t+1}^{\tau-1}(R_{s-1}-R_{s}-X_{s})\bigg{]}.$
(6)

$C_{t}$ is _the value of the reference undertaking at time $t$_ given no
default at times $\leq t$. $V_{t}:=R_{t}-C_{t}$ is _the value of the residual
liability at time $t$_ given no default at times $\leq t$.

Notice that

$\displaystyle V_{t}$ $\displaystyle:=R_{t}-C_{t}$
$\displaystyle=R_{t}-\operatorname*{ess\,sup}_{\tau\in\mathcal{S}_{t+1,T+1}}\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}\bigg{[}\sum_{s=t+1}^{\tau-1}(R_{s-1}-R_{s}-X_{s})\bigg{]}$
$\displaystyle=\operatorname*{ess\,inf}_{\tau\in\mathcal{S}_{t+1,T+1}}\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}\bigg{[}R_{t}-\sum_{s=t+1}^{\tau-1}(R_{s-1}-R_{s}-X_{s})\bigg{]}$
$\displaystyle=\operatorname*{ess\,inf}_{\tau\in\mathcal{S}_{t+1,T+1}}\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}\bigg{[}\sum_{s=t+1}^{\tau-1}X_{s}+R_{\tau-1}\bigg{]}$
$\displaystyle\leq\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}\bigg{[}\sum_{s=t+1}^{T}X_{s}\bigg{]}=:\overline{V}_{t}.$

The general upper bound

$\displaystyle\overline{V}_{0}:=\sup_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{0}^{\mathbb{Q}}\bigg{[}\sum_{s=1}^{T}X_{s}\bigg{]}\geq
V_{0}$ (7)

does neither depend on the filtration nor on the capital requirements, and is
typically much easier to compute than $V_{0}$. Therefore, this upper bound
provides a useful conservative estimate of $V_{0}$. This statement is
illustrated in the numerical example in Section 4. Notice that in general

$\displaystyle V_{t}$
$\displaystyle=\operatorname*{ess\,inf}_{\tau\in\mathcal{S}_{t+1,T+1}}\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}\bigg{[}\sum_{s=t+1}^{\tau-1}X_{s}+R_{\tau-1}\bigg{]}$
$\displaystyle\geq\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\operatorname*{ess\,inf}_{\tau\in\mathcal{S}_{t+1,T+1}}\mathbb{E}_{t}^{\mathbb{Q}}\bigg{[}\sum_{s=t+1}^{\tau-1}X_{s}+R_{\tau-1}\bigg{]}=:\underline{V}_{t}.$
(8)

In particular, the general lower bound

$\displaystyle\underline{V}_{0}:=\sup_{\mathbb{Q}\in\mathcal{Q}}\inf_{\tau\in\mathcal{S}_{1,T+1}}\mathbb{E}_{0}^{\mathbb{Q}}\bigg{[}\sum_{s=1}^{\tau-1}X_{s}+R_{\tau-1}\bigg{]}\leq
V_{0}$ (9)

may be attractive since it is typically easier to compute than $V_{0}$, see
Section 4 for an illustration. Computing $\underline{V}_{0}$ means solving a
standard optimal stopping problem for each $\mathbb{Q}\in\mathcal{Q}$ followed
by finding the maximum of the obtained values $V_{0}^{\mathbb{Q}}$.

Notice that the value $L_{0}$ of the original liability cash flow $X^{o}$
follows directly from the procedure for transferring the liabilities and
replicating portfolio to an external party (the new owner of the reference
undertaking) accepting the transfer: $L_{0}$ equals the sum of the market
value of the replicating portfolio and the value $V_{0}$ of the residual
liability:

$L_{0}=\mathbb{E}_{0}^{\mathbb{Q}}\bigg{[}\sum_{s=1}^{T}X^{r}_{s}\bigg{]}+V_{0},$

where $\mathbb{Q}$ is any market risk neutral probability measure making the
expectation equal the market value of the replicating portfolio. For details
on the market consistency of the value $L_{0}$ we refer to the material on
market consistency in Engsner et al. [13].

We intend to build on the theory of multiple prior optimal stopping in Riedel
[23] where four assumptions on a set $\mathcal{Q}$ of probability measures are
imposed in order for key results to hold. These assumptions are
$\mathcal{Q}$-uniform integrability together with properties (i)-(iii) of the
following definition.

######  Definition 2.

A set $\mathcal{Q}$ of probability measures is suitable for multiple prior
optimal stopping if the following properties hold. (i) Each
$\mathbb{Q}\in\mathcal{Q}$ is equivalent to $\mathbb{P}$; (ii) $\mathcal{Q}$
is stable under pasting; (iii) For each $t\in\\{0,\dots,T\\}$,

$\bigg{\\{}\frac{d\mathbb{Q}}{d\mathbb{P}}\bigg{|}_{\mathcal{F}_{t}}:\mathbb{Q}\in\mathcal{Q}\bigg{\\}}$

is weakly compact in $L^{1}(\mathcal{F}_{T},\mathbb{P})$.

######  Remark 1.

If $\mathcal{Q}$ satisfies the properties (i)-(iii) in Definition 2, then it
follows from Theorem 2 in Riedel [23] that the lower bound $\underline{V}_{0}$
in (9) equals $V_{0}$. This holds since for such $\mathcal{Q}$ the inequality
in (8) is in fact a minimax identity. Notice also that for an arbitrary
$\mathbb{Q}$ equivalent to $\mathbb{P}$, $\\{\mathbb{Q}\\}$ satisfies
properties (i)-(iii) in Definition 2.

As a basis for applying the theory to be presented, we will later in Section 3
explicitly construct a useful set $\mathcal{Q}$ satisfying the properties in
Definition 2 and present a detailed numerical example in Section 4.

We are now ready to state a key result which shows that $(C_{t},V_{t})$
defined in terms of a multiple prior optimal stopping problem may equivalently
be defined as the solution to a backward recursion.

######  Theorem 1.

Let $\mathcal{Q}$ be a set of probability measures satisfying properties
(i)-(iii) of Definition 2. Consider sequences $(X^{o}_{t})_{t=1}^{T}$,
$(X^{r}_{t})_{t=1}^{T}$, $(R_{t})_{t=0}^{T}$ with $X^{o}_{t},X^{r}_{t}\in
L^{1}(\mathcal{F}_{t},\mathbb{Q})$ for $t\in\\{1,\dots,T\\}$ for every
$\mathbb{Q}\in\mathcal{Q}$, $R_{T}=0$ and $R_{t}\in
L^{1}(\mathcal{F}_{t},\mathbb{Q})$ for $t\in\\{0,\dots,T-1\\}$ for every
$\mathbb{Q}\in\mathcal{Q}$. Set $X_{t}:=X^{o}_{t}-X^{r}_{t}$ and assume that
$(H_{t})_{t=1}^{T}$ in (2) is bounded by a $\mathcal{Q}$-uniformly integrable
random variable. Then

(i) If the sequences $(C_{t})_{t=0}^{T}$ and $(V_{t})_{t=0}^{T}$ are given by
Definition 1, then

$\displaystyle C_{t}$
$\displaystyle=\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[(R_{t}-X_{t+1}-V_{t+1})^{+}],\quad
C_{T}=0,$ (10) $\displaystyle V_{t}$
$\displaystyle=\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\Big{(}R_{t}-\mathbb{E}_{t}^{\mathbb{Q}}[(R_{t}-X_{t+1}-V_{t+1})^{+}]\Big{)},\quad
V_{T}=0.$ (11)

(ii) The stopping times $(\tau^{*}_{t})_{t=0}^{T-1}$ given by

$\displaystyle\tau^{*}_{t}=\inf\\{s\in\\{t+1,\dots,T\\}:R_{s-1}-X_{s}-V_{s}<0\\}\wedge(T+1)$

are optimal in (6).

(iii) If the sequences $(C_{t})_{t=0}^{T}$ and $(V_{t})_{t=0}^{T}$ are given
by (10) and (11), then, for $t\in\\{0,\dots,T-1\\}$, $C_{t}$ is given by (6)
and $V_{t}=R_{t}-C_{t}$.

######  Remark 2.

Stability under pasting of $\mathcal{Q}$ is a necessary requirement in Theorem
1. However, we show later in Theorem 6 that instead of the weak compactness
property (iii) in Definition 2, which is assumed in Theorem 1, it is
sufficient to verify weak relative compactness together with some natural
additional properties. Notice that a bounded and uniformly integrable subset
of $L^{1}(\mathcal{F}_{t},\mathbb{P})$ is weakly relatively compact in
$L^{1}(\mathcal{F}_{t},\mathbb{P})$ (Theorem A.70 in [15]). Without weak
compactness we can however not guarantee that there exists a
$\mathbb{Q}^{*}\in\mathcal{Q}$ which solves the optimization problems (10) and
(11).

###### Proof of Theorem 1.

We will first consider the problem

$\displaystyle\operatorname*{ess\,sup}_{\tau\in\mathcal{S}_{t,T+1}}\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[H_{\tau}].$
(12)

We define the multiple prior Snell envelope of $H$ with respect to
$\mathcal{Q}$ as in [23] by

$\displaystyle U^{\mathcal{Q}}_{T+1}:=H_{T+1},\quad
U^{\mathcal{Q}}_{t}:=\max\big{\\{}H_{t},\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[U^{\mathcal{Q}}_{t+1}]\big{\\}}\quad\text{for
}t\leq T.$ (13)

We know from Theorem 1 in [23] that

$\displaystyle
U^{\mathcal{Q}}_{t}=\operatorname*{ess\,sup}_{\tau\in\mathcal{S}_{t,T+1}}\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[H_{\tau}]$
(14)

and that $\tau^{*}_{t}:=\inf\\{s\geq t:U^{\mathcal{Q}}_{s}=H_{s}\\}$ is an
optimal stopping time that solves (12). Define $\widetilde{U}^{\mathcal{Q}}$
by

$\displaystyle\widetilde{U}^{\mathcal{Q}}_{t}:=\operatorname*{ess\,sup}_{\tau\in\mathcal{S}_{t+1,T+1}}\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[H_{\tau}].$

We claim that the relation
$\widetilde{U}^{\mathcal{Q}}_{t}=\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[U^{\mathcal{Q}}_{t+1}]$
holds. Indeed, from (14),

$\displaystyle
U^{\mathcal{Q}}_{t}=\max\big{\\{}H_{t},\operatorname*{ess\,sup}_{\tau\in\mathcal{S}_{t+1,T+1}}\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[H_{\tau}]\big{\\}}=\max\big{\\{}H_{t},\widetilde{U}^{\mathcal{Q}}_{t}\big{\\}}.$

Therefore, from (13), we have the relation

$\displaystyle\max\big{\\{}H_{t},\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[U^{\mathcal{Q}}_{t+1}]\big{\\}}=\max\big{\\{}H_{t},\widetilde{U}^{\mathcal{Q}}_{t}\big{\\}}.$

Since this holds for arbitrary adapted $H$, the claim is proved and gives

$\displaystyle C_{t}$
$\displaystyle=\widetilde{U}^{\mathcal{Q}}_{t}-H_{t+1}=\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[U^{\mathcal{Q}}_{t+1}]-H_{t+1}$
$\displaystyle=\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[\max\\{H_{t+1},\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t+1}^{\mathbb{Q}}[U^{\mathcal{Q}}_{t+2}]\\}-H_{t+1}]$
$\displaystyle=\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[\max\\{0,\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t+1}^{\mathbb{Q}}[U^{\mathcal{Q}}_{t+2}]-H_{t+1}\\}]$
$\displaystyle=\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[\max\\{0,C_{t+1}+H_{t+2}-H_{t+1}\\}]$
$\displaystyle=\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[(R_{t}-X_{t+1}-V_{t+1})^{+}].$

Hence, we have shown (10) from which (11) is an immediate consequence. This
concludes the proof of statement (i). ∎

###  2.3 Valuation with capital requirements by conditional monetary risk
measures

We now consider the valuation problem in the setting where the capital
requirements are given in terms of conditional monetary risk measures.

######  Definition 3.

For $p\in[0,\infty]$ and $t\in\\{0,\dots,T-1\\}$, a conditional monetary risk
measure is a mapping $\rho_{t}:L^{p}(\mathcal{F}_{t+1},\mathbb{P})\to
L^{p}(\mathcal{F}_{t},\mathbb{P})$ satisfying

$\displaystyle\textrm{if }\lambda\in L^{p}(\mathcal{F}_{t},\mathbb{P})\textrm{
and }Y\in L^{p}(\mathcal{F}_{t+1},\mathbb{P}),\textrm{ then
}\rho_{t}(Y+\lambda)=\rho_{t}(Y)-\lambda,$ (15) $\displaystyle\textrm{if
}Y,\widetilde{Y}\in L^{p}(\mathcal{F}_{t+1},\mathbb{P})\textrm{ and
}Y\leq\widetilde{Y},\textrm{ then }\rho_{t}(Y)\geq\rho_{t}(\widetilde{Y}),$
(16) $\displaystyle\rho_{t}(0)=0.$ (17)

A sequence $(\rho_{t})_{t=0}^{T-1}$ of conditional monetary risk measures is
called a dynamic monetary risk measure.

The natural conditional monetary risk measures corresponding to current
regulatory frameworks are defined in terms of conditional quantile functions.
For integer $t\geq 0$, $x\in\mathbb{R}$, $u\in(0,1)$ and
$\mathcal{F}_{t+1}$-measurable $Z$, let

$\displaystyle F_{t,-Z}(x)$ $\displaystyle:=\mathbb{P}_{t}(-Z\leq x),$
$\displaystyle F_{t,-Z}^{-1}(1-u)$
$\displaystyle:=\operatorname*{ess\,inf}\\{m\in
L^{0}(\mathcal{F}_{t},\mathbb{P}):F_{t,-Z}(m)\geq 1-u\\}$

and define the conditional versions of value-at-risk and average value-at-risk
as

$\displaystyle\operatorname{V@R}_{t,u}(Z):=F_{t,-Z}^{-1}(1-u),\quad\operatorname{AV@R}_{t,u}(Z):=\frac{1}{u}\int_{0}^{u}\operatorname{V@R}_{t,v}(Z)dv.$

Both $\operatorname{V@R}_{t,u}$ and $\operatorname{AV@R}_{t,u}$ are
conditional monetary risk measures in the sense of Definition 3 for $p\geq 1$.
Given conditional monetary risk measures
$\rho_{t}:L^{1}(\mathcal{F}_{t+1},\mathbb{P})\to
L^{1}(\mathcal{F}_{t},\mathbb{P})$ we consider here

$\displaystyle R_{t}:=\rho_{t}(-X_{t+1}-V_{t+1}),\quad R_{T}:=0.$ (18)

Notice that if $R_{t+1}$ is given and $C_{t+1}$ is given by Definition 1, then
also $V_{t+1}:=R_{t+1}-C_{t+1}$ is given and therefore $R_{t}$ is well defined
by setting $R_{t}:=\rho_{t}(-X_{t+1}+V_{t+1})$. Moreover, we may write

$\displaystyle V_{t}:=\varphi_{t}(X_{t+1}+V_{t+1}),\quad V_{T}:=0,$ (19)

where

$\displaystyle\varphi_{t}(Y):=\rho_{t}(-Y)-\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[(\rho_{t}(-Y)-Y)^{+}].$

######  Theorem 2.

Let $\rho_{t}:L^{1}(\mathcal{F}_{t+1},\mathbb{P})\to
L^{1}(\mathcal{F}_{t},\mathbb{P})$ be a conditional monetary risk measure in
the sense of Definition 3 and let
$\varphi_{t}:L^{1}(\mathcal{F}_{t+1},\mathbb{P})\to
L^{1}(\mathcal{F}_{t},\mathbb{P})$ be given by (19). Then

$\displaystyle\textrm{if }\lambda\in L^{p}(\mathcal{F}_{t},\mathbb{P})\textrm{
and }Y\in L^{p}(\mathcal{F}_{t+1},\mathbb{P}),\textrm{ then
}\varphi_{t}(Y+\lambda)=\varphi_{t}(Y)+\lambda,$ (20) $\displaystyle\textrm{if
}Y,\widetilde{Y}\in L^{p}(\mathcal{F}_{t+1},\mathbb{P})\textrm{ and
}Y\leq\widetilde{Y},\textrm{ then
}\varphi_{t}(Y)\leq\varphi_{t}(\widetilde{Y}),$ (21)
$\displaystyle\varphi_{t}(0)=0.$ (22)

Moreover, if $\widetilde{\rho}_{t}:L^{1}(\mathcal{F}_{t+1},\mathbb{P})\to
L^{1}(\mathcal{F}_{t},\mathbb{P})$ is a conditional monetary risk measure in
the sense of Definition 3 such that $\rho_{t}\leq\widetilde{\rho}_{t}$, then

$\displaystyle\widetilde{\varphi}_{t}(Y):=\widetilde{\rho}_{t}(-Y)-\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[(\widetilde{\rho}_{t}(-Y)-Y)^{+}]$

satisfies $\varphi_{t}\leq\widetilde{\varphi}_{t}$.

###### Proof of Theorem 2.

The properties (20), (21) and (22) follow immediately by arguments similar to
those in the proof of Proposition 1 in Engsner et al. [14]. The final property
follows immediately from the inequality

$(\widetilde{R}-Y)^{+}\leq(R-Y)^{+}+\widetilde{R}-R\quad\text{for
}\widetilde{R}\geq R.$

∎

Theorem 2 has consequences that should be seen as necessary requirements of
any sound valuation method. If $X_{1}+\dots+X_{T}=c$ for some constant $c$,
then the corresponding value $V_{0}=c$. If we consider two residual liability
cash flows $(X_{t})_{t=1}^{T}$ and $(\widetilde{X}_{t})_{t=1}^{T}$ such that
$X_{t}\leq\widetilde{X}_{t}$ for every $t$, then the corresponding values
satisfy $V_{0}\leq\widetilde{V}_{0}$. Similarly, if the sequence of
conditional monetary risk measures $(\rho_{t})_{t=0}^{T-1}$ are replaced by a
more prudent choice $(\widetilde{\rho}_{t})_{t=0}^{T-1}$ such that
$\rho_{t}\leq\widetilde{\rho}_{t}$ for every $t$, then the corresponding
values satisfy $V_{0}\leq\widetilde{V}_{0}$.

Let $(V^{S}_{t})_{t=0}^{T}$ be given by $V^{S}_{t}=\sum_{u=1}^{t}X_{u}+V_{t}$,
where $V_{T}=0$,

$V_{t}=R_{t}-\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}_{t}[(R_{t}-X_{t+1}-V_{t+1})_{+}],\quad
R_{t}=\rho_{t}(-X_{t+1}-V_{t+1}),$

where $\rho_{t}$ is a suitable conditional monetary risk measure such as
$\operatorname{V@R}_{t}$ or $\operatorname{AV@R}_{t}$. It is reasonable to
require that $V^{S}$ is a $\mathbb{P}$-supermartingale which is equivalent to
$V_{t}\geq\mathbb{E}^{\mathbb{P}}_{t}[X_{t+1}+V_{t+1}]$ which implies
$V_{t}\geq\mathbb{E}^{\mathbb{P}}_{t}[X_{t+1}+\dots+X_{T}]$. In particular,
the $\mathbb{P}$-supermartingale property guarantees the existence of a
nonnegative “risk margin”
$V_{t}-\mathbb{E}^{\mathbb{P}}_{t}[X_{t+1}+\dots+X_{T}]\geq 0$.

######  Theorem 3.

Let $X_{t}\in L^{1}(\mathcal{F}_{t},\mathbb{P})$ for $t=1,\dots,T$. Let
$L^{1}(\mathcal{F}_{t+1},\mathbb{P})\ni Y_{t+1}\mapsto\rho_{t}(-Y_{t+1})\in
L^{1}(\mathcal{F}_{t},\mathbb{P})$, for $t=0,\dots,T-1$, be a conditional
monetary risk measure such that

$\displaystyle\mathbb{E}^{\mathbb{P}}_{t}[\rho_{t}(-Y_{t+1})-Y_{t+1}]>0\quad\text{or}\quad\mathbb{P}_{t}(\rho_{t}(-Y_{t+1})-Y_{t+1}=0)=1.$
(23)

Then there exists a set $\mathcal{Q}$ of probability measures such that
$(V^{S}_{t})_{t=0}^{T}$ is a $\mathbb{P}$-supermartingale.

###### Proof of Theorem 3.

Notice that the supermartingale requirement is equivalent to

$\displaystyle\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}_{t}[(R_{t}-X_{t+1}-V_{t+1})_{+}]\leq\mathbb{E}^{\mathbb{P}}_{t}[R_{t}-X_{t+1}-V_{t+1}]$
(24)

It is sufficient to find some $\mathbb{Q}$ such that the statement holds for
$\mathcal{Q}=\\{\mathbb{Q}\\}$. We construct this $\mathbb{Q}$ by defining a
suitable $\mathbb{P}$-martingale $(D_{t})_{t=0}^{T}$ corresponding to the
change of measure from $\mathbb{P}$ to $\mathbb{Q}$.

Let $W_{t+1}:=R_{t}-X_{t+1}-V_{t+1}$, let
$G_{t}(x):=\mathbb{P}_{t}(W_{t+1}\leq x)$ denote the
$\mathcal{F}_{t}$-conditional distribution function of $W_{t+1}$, and let
$p_{t}:=G_{t}(0)$. Let $(D_{t})_{t=0}^{T}$, with $D_{0}=1$, be a
$\mathbb{P}$-martingale satisfying

$\displaystyle\frac{D_{t+1}}{D_{t}}=\left\\{\begin{array}[]{ll}1&\text{if
}p_{t}\in\\{0,1\\},\\\
\exp\big{(}\lambda_{t}\Phi^{-1}(U_{t+1})-\lambda_{t}^{2}/2\big{)}&\text{if
}p_{t}\in(0,1),\end{array}\right.$

where $U_{t+1}$ is independent of $\mathcal{F}_{t}$ and uniformly distributed
on $(0,1)$ and, conditional of $\mathcal{F}_{t}$, $U_{t+1}$ and $W_{t+1}$ are
countermonotone. Let $\lambda_{t}$ be some $\mathcal{F}_{t}$-measurable random
variable satisfying

$\displaystyle\exp\big{(}\lambda_{t}\Phi^{-1}(1-p_{t})-\lambda_{t}^{2}/2\big{)}\mathbb{E}^{\mathbb{P}}_{t}\big{[}W_{t+1}^{+}\big{]}\leq\mathbb{E}^{\mathbb{P}}_{t}\big{[}W_{t+1}\big{]}\quad\text{on
}\\{p_{t}\in(0,1)\\}.$

By construction,

$\displaystyle\mathbb{E}^{\mathbb{P}}_{t}\bigg{[}\frac{D_{t+1}}{D_{t}}W_{t+1}^{+}\bigg{]}=\mathbb{E}^{\mathbb{P}}_{t}\big{[}W_{t+1}\big{]}\quad\text{on
}\\{p_{t}\in\\{0,1\\}\\}.$

Moreover, on $\\{p_{t}\in(0,1)\\}$,

$\displaystyle\mathbb{E}^{\mathbb{P}}_{t}\bigg{[}\frac{D_{t+1}}{D_{t}}W_{t+1}^{+}\bigg{]}$
$\displaystyle=\int_{0}^{1-p_{t}}\exp\big{(}\lambda_{t}\Phi^{-1}(u)-\lambda_{t}^{2}/2\big{)}G_{t}^{-1}(1-u)du$
$\displaystyle\leq\exp\big{(}\lambda_{t}\Phi^{-1}(1-p_{t})-\lambda_{t}^{2}/2\big{)}\int_{0}^{1-p_{t}}G_{t}^{-1}(1-u)du$
$\displaystyle=\exp\big{(}\lambda_{t}\Phi^{-1}(1-p_{t})-\lambda_{t}^{2}/2\big{)}\mathbb{E}^{\mathbb{P}}_{t}\big{[}W_{t+1}^{+}\big{]}$
$\displaystyle\leq\mathbb{E}^{\mathbb{P}}_{t}\big{[}W_{t+1}\big{]}.$

∎

Property (23) in Theorem 3 is satisfied by $\operatorname{AV@R}_{t,u}$ which
is an example of so-called strictly expectation bounded risk measures, see
Definition 5 and Example 3 in Rockafellar et al. [24].

The following lemma is useful for constructing a bounding
$\mathcal{Q}$-uniformly integrable random variable.

######  Lemma 1.

For any $\mathcal{Q}$-uniformly integrable $Z\geq 0$,
$\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[Z]$
is a $\mathcal{Q}$-uniformly integrable random variable.

###### Proof of Lemma 1.

We need to show that

$\lim_{K\to\infty}\sup_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}\Big{[}\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[Z]\mathbb{I}_{\\{\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[Z]\geq
K\\}}\Big{]}=0.$

If we set
$X=Z\mathbb{I}_{\\{\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[Z]\geq
K\\}}$, then $X$ is $\mathcal{Q}$-uniformly integrable since it is of the form
$Z\mathbb{I}_{A}$. Hence by the law of iterated expectations for
$\mathcal{Q}$-uniformly integrable random variables (Lemma 1 in [23]),

$\displaystyle\sup_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}\Big{[}\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[Z]\mathbb{I}_{\\{\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[Z]\geq
K\\}}\Big{]}$
$\displaystyle=\sup_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}\Big{[}\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[X]\Big{]}$
$\displaystyle=\sup_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}[X].$

Notice that $\sup_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}[Z]<\infty$
since for any $r>0$,

$\sup_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}[Z]\leq
r+\sup_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}[|Z|\mathbb{I}_{|Z|>r}]$

and, due to $\mathcal{Q}$-uniformly integrability of $Z$, we may choose $r$ to
make the second term on the right-hand side sufficiently small. Since

$\sup_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}\Big{[}\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[Z]\Big{]}=\sup_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}[Z]<\infty,$

the events
$A_{n}=\\{\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}_{t}^{\mathbb{Q}}[Z]\geq
n\\}$ satisfy $\sup_{\mathbb{Q}\in\mathcal{Q}}\mathbb{Q}(A_{n})\to 0$ as
$n\to\infty$. For any $\mathbb{Q}\in\mathcal{Q}$ and $r_{n}>0$,

$\displaystyle\sup_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}\Big{[}Z\mathbb{I}_{A_{n}}\Big{]}\leq
r_{n}\sup_{\mathbb{Q}\in\mathcal{Q}}\mathbb{Q}(A_{n})+\sup_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}\Big{[}|Z|\mathbb{I}_{\\{|Z|>r_{n}\\}}\Big{]}.$
(25)

Consider a sequence $(r_{n})_{n=1}^{\infty}$ such that $r_{n}\to\infty$ and
$r_{n}\sup_{\mathbb{Q}\in\mathcal{Q}}\mathbb{Q}(A_{n})\to 0$ as $n\to\infty$.
Applying this sequence to (25), taking the supremum over $\mathcal{Q}$ and
letting $n\to\infty$ proves the statement of the lemma. ∎

The following theorem says that if the conditional monetary risk measures
$\rho_{t}$ defining $R_{t}$ in (18) satisfy natural and verifiable bounds,
then statements in Theorem 1 hold also in this setting.

######  Theorem 4.

Let $\mathcal{Q}$ be a set of probability measures satisfying properties
(i)-(iii) of Definition 2. Consider sequences $(X^{o}_{t})_{t=1}^{T}$,
$(X^{r}_{t})_{t=1}^{T}$, with $X^{o}_{t},X^{r}_{t}\in
L^{1}(\mathcal{F}_{t},\mathbb{Q})$ for $t\in\\{1,\dots,T\\}$ for every
$\mathbb{Q}\in\mathcal{Q}$, $R_{T}=0$ and $R_{t}\in
L^{1}(\mathcal{F}_{t},\mathbb{Q})$ for $t\in\\{0,\dots,T-1\\}$ for every
$\mathbb{Q}\in\mathcal{Q}$. Let $X_{t}:=X^{o}_{t}-X^{r}_{t}$ and let
$(R_{t})_{t=0}^{T}$ be defined by (18). Assume that $\sum_{t=1}^{T}|X_{t}|$ is
$\mathcal{Q}$-uniformly integrable. If the conditional monetary risk measures
$\rho_{t}$ in (18) satisfy either

$\displaystyle|\rho_{t}(Z)|\leq
K_{\rho}\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}_{t}[|Z|]\text{
for some }K_{\rho}\in(1,\infty)$ (26)

or

$\displaystyle\mathbb{P}\in\mathcal{Q}\text{ and }|\rho_{t}(Z)|\leq
K_{\rho}\mathbb{E}^{\mathbb{P}}_{t}[|Z|]\text{ for some
}K_{\rho}\in(1,\infty),$ (27)

then $(H_{t})_{t=0}^{T}$ defined in (2) satisfies that
$\operatorname*{ess\,sup}_{t=1,\dots T}H_{t}$ is bounded by a
$\mathcal{Q}$-uniformly integrable random variable. In particular, the
statements in Theorem 1 hold.

###### Proof of Theorem 4.

Set

$\displaystyle S_{T}:=0,\quad
S_{t}:=\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}_{t}\bigg{[}\sum_{u=t+1}^{T}|X_{u}|\bigg{]}\text{
for }t=0,1,\dots,T-1.$

By Lemma 1 all variables $S_{t}$ are $\mathcal{Q}$-uniformly integrable. We
will show by induction that, for all $t$, there exist constants
$K_{V,t},K_{R,t}\in(1,\infty)$ such that

$\displaystyle|V_{t}|\leq K_{V,t}S_{t},\quad|R_{t}|\leq K_{R,t}S_{t}$ (28)

from which the statement of the theorem follows. Note that (28) trivially
holds for $t=T$. In order to show the induction step, assume that (28) holds
with $t$ replaced by $t+1$. If (27) holds, then

$\displaystyle|R_{t}|$ $\displaystyle=|\rho_{t}(-X_{t+1}-V_{t+1})|$
$\displaystyle\leq K_{\rho}\mathbb{E}^{\mathbb{P}}_{t}[|X_{t+1}|+|V_{t+1}|]$
$\displaystyle\leq
K_{\rho}\mathbb{E}^{\mathbb{P}}_{t}[|X_{t+1}|+K^{V}_{t+1}S_{t+1}]$
$\displaystyle\leq
K_{\rho}K_{V,t+1}\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}_{t}[|X_{t+1}|+S_{t+1}]$
$\displaystyle=K_{\rho}K_{V,t+1}S_{t},$

where the law of iterated expectations for $\mathcal{Q}$-uniformly integrable
random variables (Lemma 1 in [23]) was used in the last step. If (26) holds,
then similarly

$\displaystyle|R_{t}|$ $\displaystyle=|\rho_{t}(-X_{t+1}-V_{t+1})|$
$\displaystyle\leq K_{\rho}\mathbb{E}^{\mathbb{Q}}_{t}[|X_{t+1}|+|V_{t+1}|]$
$\displaystyle\leq
K_{\rho}\mathbb{E}^{\mathbb{Q}}_{t}[|X_{t+1}|+K_{V,t+1}S_{t+1}]$
$\displaystyle\leq
K_{\rho}K_{V,t+1}\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}_{t}[|X_{t+1}|+S_{t+1}]$
$\displaystyle=K_{\rho}K_{V,t+1}S_{t}.$

We also note that

$\displaystyle C_{t}$
$\displaystyle=\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}_{t}[(R_{t}-X_{t+1}-V_{t+1})^{+}]$
$\displaystyle\leq|R_{t}|+\operatorname*{ess\,sup}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}_{t}[|X_{t+1}|+|V_{t+1}|]$
$\displaystyle\leq(K_{\rho}+1)K_{V,t+1}S_{t}$

which implies $|V_{t}|\leq|R_{t}|+C_{t}\leq(2K_{\rho}+1)K_{V,t+1}S_{t}$. We
have proved that (28) holds, i.e. the induction step. By the induction
principle (28) holds for all $t$ and the proof is complete. ∎

##  3 Construction of sets of probability measures for multiple prior optimal
stopping

Our aim here is to define a useful set $\mathcal{Q}$ of parametric probability
measures that enables the analysis of a wide range of models and that provide
solutions to the multiple-prior optimization problem (6). In particular, the
set $\mathcal{Q}$ constructed below will imply that optimization over
$\mathcal{Q}$ can be reduced to optimization over the set of parameters, see
Theorem 6 for the precise statement.

We will define a useful set of probability measures, satisfying all the
requirements for applying key results on multiple prior optimal stopping, by
defining the corresponding set of density processes
$(D_{\lambda,t})_{t=0}^{T}$ of the form

$\displaystyle D_{\lambda,0}:=1,\quad
D_{\lambda,t}:=\prod_{s=1}^{t}\int_{\Theta}f_{s}(\theta)\lambda_{s}(\mathrm{d}\theta)\text{
for }t\in\\{1,\dots,T\\},$

where $\Theta$ is a set of parameters and $(f_{s})_{s=1}^{T}$ and
$(\lambda_{s})_{s=1}^{T}$ are defined below.

On $(\Omega,\mathcal{F}_{T})$, a probability measure $\mathbb{Q}$ absolutely
continuous with respect to $\mathbb{P}$ corresponds to a Radon-Nikodym
derivative $D_{T}\in L^{1}(\mathcal{F}_{T},\mathbb{P})$ and together with the
filtration $(\mathcal{F}_{t})_{t=0}^{T}$ give rise to the density process
$(D_{t})_{t=0}^{T}$ given by $D_{t}=\mathbb{E}^{\mathbb{P}}_{t}[D_{T}]$.
Similarly, a set $\mathcal{Q}$ of probability measures, absolutely continuous
with respect to $\mathbb{P}$, corresponds to the set $\mathcal{D}_{T}\subset
L^{1}(\mathcal{F}_{T},\mathbb{P})$ of Radon-Nikodym derivatives. Write
$\overline{\mathcal{D}}_{T}$ for the $L^{1}$ closure of $\mathcal{D}_{T}$ and
let $\overline{\mathcal{Q}}$ be the set of probability measures corresponding
to the Radon-Nikodym derivatives $\overline{\mathcal{D}}_{T}$. For two
probability measures $\mathbb{Q}^{(1)},\mathbb{Q}^{(2)}$ with Radon-Nikodym
derivatives $D^{(1)}_{T},D^{(2)}_{T}$ the Radon-Nikodym derivative of the
pasting of $\mathbb{Q}^{(1)},\mathbb{Q}^{(2)}$ in $\tau$ is

$D^{(1)}_{\tau}\frac{D^{(2)}_{T}}{D^{(2)}_{\tau}}.$

The following result is both of independent interest and will be relevant for
constructing stable sets of probability measures, depending on a parameter,
that are useful for multiple prior optimal stopping problems.

######  Theorem 5.

Given $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t=0}^{T},\mathbb{P})$, let
$\mathcal{Q}$ be a set of probability measures equivalent to $\mathbb{P}$ that
is convex and stable under pasting. Let $\mathcal{D}_{T}$ be the corresponding
set of Radon-Nikodym derivatives and let $\overline{\mathcal{D}}_{T}$ be the
$L^{1}$ closure of $\mathcal{D}_{T}$. Let
$\mathcal{D}_{t}:=\\{D_{t}=\mathbb{E}^{\mathbb{P}}_{t}[D_{T}]:D_{T}\in\mathcal{D}_{T}\\}$
and
$\overline{\mathcal{D}}_{t}:=\\{D_{t}=\mathbb{E}^{\mathbb{P}}_{t}[D_{T}]:D_{T}\in\overline{\mathcal{D}}_{T}\\}$.
Then

  * (i)

The set $\overline{\mathcal{Q}}$ corresponding to $\overline{\mathcal{D}}_{T}$
is convex and stable under pasting.

  * (ii)

For each $t$, $\overline{\mathcal{D}}_{t}$ is convex and closed in
$L^{1}(\mathcal{F}_{T},\mathbb{P})$.

  * (iii)

If $\mathcal{D}_{T}$ is $\mathbb{P}$-uniformly integrable, then for each $t$,
$\mathcal{D}_{t}$ is weakly relatively compact in
$L^{1}(\mathcal{F}_{T},\mathbb{P})$ and $\overline{\mathcal{D}}_{t}$ is weakly
compact in $L^{1}(\mathcal{F}_{T},\mathbb{P})$.

  * (iv)

If $\mathcal{D}_{T}$ is $\mathbb{P}$-uniformly integrable, then, for any
$\mathcal{F}_{t+1}$-measurable $\mathcal{Q}$-uniformly integrable random
variable $Y_{t+1}$,

$\displaystyle\operatorname*{ess\,inf}_{\mathbb{Q}\in\overline{\mathcal{Q}}}\mathbb{E}^{\mathbb{Q}}_{t}\big{[}Y_{t+1}\big{]}=\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}_{t}\big{[}Y_{t+1}\big{]}$

and similarly with $\operatorname*{ess\,inf}$ replaced by
$\operatorname*{ess\,sup}$.

######  Remark 3.

Since $\mathbb{E}^{\mathbb{P}}[D]=1$ for $D\in\mathcal{D}_{T}$ (and similarly
for $\overline{\mathcal{D}}_{T}$), by Lemma 4.10 in Kallenberg [17],
$\mathcal{D}_{T}$ is uniformly integrable if

$\lim_{\mathbb{P}(A)\to 0}\sup_{\mathbb{Q}\in\mathcal{Q}}\mathbb{Q}(A)=0$

(and similarly for $\overline{\mathcal{D}}_{T}$).

###### Proof of Theorem 5.

For both statements $(i)$ and $(ii)$, it is straightforward to verify that
convexity holds so we only prove the remaining claims.

To prove $(i)$, consider any stopping time $\tau$ and
$D^{(1)}_{T},D^{(2)}_{T}\in\overline{\mathcal{D}}_{T}$. Take
$(D^{(1)}_{n,T})_{n\geq 1},(D^{(2)}_{n,T})_{n\geq 1}\subset\mathcal{D}_{T}$
such that $D^{(1)}_{n,T}\to D^{(1)}_{T}$ and $D^{(2)}_{n,T}\to D^{(2)}_{T}$ in
$L^{1}$. Since $D^{(1)}_{n,T},D^{(2)}_{n,T}\in\mathcal{D}_{T}$ and
$\mathcal{Q}$ is stable under pasting, the Radon-Nikodym derivative of the
pasting of $D^{(1)}_{n,T},D^{(2)}_{n,T}\in\mathcal{D}_{T}$ in $\tau$ is also
an element in $\mathcal{D}_{T}$. Therefore, statement (i) is proved if we show
that there exists a subsequence $(n_{i})$ such that

$\displaystyle
D^{(1)}_{n_{i},\tau}\frac{D^{(2)}_{n_{i},T}}{D^{(2)}_{n_{i},\tau}}\to
D^{(1)}_{\tau}\frac{D^{(2)}_{T}}{D^{(2)}_{\tau}}\text{ in }L^{1}.$ (29)

Since, for $k=1,2$,

$\displaystyle\mathbb{E}^{\mathbb{P}}[|D^{(k)}_{n,T}-D^{(k)}_{T}|]$
$\displaystyle=\mathbb{E}^{\mathbb{P}}[\mathbb{E}^{\mathbb{P}}_{\tau}[|D^{(k)}_{n,T}-D^{(k)}_{T}|]]$
$\displaystyle\geq\mathbb{E}^{\mathbb{P}}[|\mathbb{E}^{\mathbb{P}}_{\tau}[D^{(k)}_{n,T}-D^{(k)}_{T}]|]$
$\displaystyle=\mathbb{E}^{\mathbb{P}}[|D^{(k)}_{n,\tau}-D^{(k)}_{\tau}|],$

we see that $D^{(k)}_{n,\tau}\to D^{(k)}_{\tau}$ in $L^{1}$. Since convergence
in $L^{1}$ implies convergence in probability which in turn implies a.s.
convergence along some subsequence $(n_{i})$, we have

$D^{(1)}_{n_{i},\tau}\frac{D^{(2)}_{n_{i},T}}{D^{(2)}_{n_{i},\tau}}\to
D^{(1)}_{\tau}\frac{D^{(2)}_{T}}{D^{(2)}_{\tau}}\text{ a.s.},$

where we used the fact that $D^{(2)}_{n_{i},\tau}$ and $D^{(2)}_{\tau}$ are
strictly positive a.s. Since the terms of the sequence on the left-hand side
are positive and all have expected values equal to $1$, this sequence is
uniformly integrable. Therefore, by Proposition 4.12 in [17], the a.s.
convergence can be replaced by convergence in $L^{1}$, i.e. (29) holds.

We now prove $(ii)$. Consider an $L^{1}$ convergent sequence $(D_{n,t})_{n\geq
1}\subset\overline{\mathcal{D}}_{t}$ with limit $D_{t}$. We will prove that
$D_{t}\in\overline{\mathcal{D}}_{t}$. Take an arbitrary
$D^{\prime}_{T}\in\overline{\mathcal{D}}_{T}$ and let
$D^{\prime}_{t}:=\mathbb{E}^{\mathbb{P}}_{t}[D^{\prime}_{T}]$. Set

$\displaystyle D_{n,T}:=D_{n,t}\frac{D^{\prime}_{T}}{D^{\prime}_{t}},\quad
D_{T}:=D_{t}\frac{D^{\prime}_{T}}{D^{\prime}_{t}}.$

By construction $(D_{n,T})_{n\geq 1}\subset\overline{\mathcal{D}}_{T}$ and
$D_{n,T}\to D_{T}$ in $L^{1}$. Hence, $D_{T}\in\overline{\mathcal{D}}_{T}$.
Since $D_{t}=\mathbb{E}^{\mathbb{P}}[D_{T}]\in\overline{\mathcal{D}}_{t}$ and
$D_{n,t}\to D_{t}$ in $L^{1}$ the proof of $(ii)$ is complete.

We now prove $(iii)$. From $(i)$ and $(ii)$ follow that
$\overline{\mathcal{Q}}$ is convex and stable under pasting and, for each $t$,
$\overline{\mathcal{D}}_{t}$ is a convex and closed subset of
$L^{1}(\mathcal{F}_{T},\mathbb{P})$. A convex and closed subset of
$L^{1}(\mathcal{F}_{T},\mathbb{P})$ is weakly closed (Theorem A.63 in [15]). A
bounded and uniformly integrable subset of $L^{1}(\mathcal{F}_{T},\mathbb{P})$
is weakly relatively compact (Theorem A.70 in [15]). Each $\mathcal{D}_{t}$ is
a bounded subset of $L^{1}(\mathcal{F}_{T},\mathbb{P})$:

$\sup_{D_{t}\in\mathcal{D}_{t}}\|D_{t}\|_{L^{1}}=\sup_{D_{t}\in\mathcal{D}_{t}}\mathbb{E}^{\mathbb{P}}[D_{t}]=1.$

Hence, weak relative compactness of $\mathcal{D}_{t}$ follows if
$\mathcal{D}_{t}$ is uniformly integrable. Similarly for
$\overline{\mathcal{D}}_{t}$. Moreover, $\mathcal{D}_{t}$ is uniformly
integrable if $\mathcal{D}_{T}$ is uniformly integrable, as the following
argument shows. By Lemma 4.10 in [17], since
$\sup_{D\in\mathcal{D}_{T}}\mathbb{E}^{\mathbb{P}}[D]=1$, $\mathcal{D}_{T}$ is
$\mathbb{P}$-uniformly integrable if and only if $\lim_{\mathbb{P}(A)\to
0}\sup_{D\in\mathcal{D}_{T}}\mathbb{E}^{\mathbb{P}}[D;A]=0$. If the latter
holds, then in particular
$\lim_{r\to\infty}\sup_{D\in\mathcal{D}_{T}}\mathbb{E}^{\mathbb{P}}[D;\mathbb{E}^{\mathbb{P}}_{t}[D]>r]=0$
which is equivalent to

$\lim_{r\to\infty}\sup_{D\in\mathcal{D}_{T}}\mathbb{E}^{\mathbb{P}}[\mathbb{E}^{\mathbb{P}}_{t}[D];\mathbb{E}^{\mathbb{P}}_{t}[D]>r]=0.$

Hence, $\mathcal{D}_{t}$ is $\mathbb{P}$-uniformly integrable if
$\mathcal{D}_{T}$ is $\mathbb{P}$-uniformly integrable. By the same argument,
$\overline{\mathcal{D}}_{t}$ is uniformly integrable if
$\overline{\mathcal{D}}_{T}$ is uniformly integrable. However,
$\overline{\mathcal{D}}_{T}$ is uniformly integrable since it is the closure
of a uniformly integrable set in $L^{1}$. Hence, $\overline{\mathcal{D}}_{t}$
is weakly compact in $L^{1}(\mathcal{F}_{T},\mathbb{P})$ for every $t$. The
proof of $(iii)$ is complete.

It remains to prove $(iv)$. Notice first that

$\displaystyle\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^{\mathbb{Q}}_{t}[Y_{t+1}]=\operatorname*{ess\,inf}_{D\in\mathcal{D}_{T}}\frac{1}{D_{t}}\mathbb{E}^{\mathbb{P}}_{t}[DY_{t+1}]=\operatorname*{ess\,inf}_{D\in\mathcal{D}_{T}}\frac{1}{D_{t}}\mathbb{E}^{\mathbb{P}}_{t}[D_{t+1}Y_{t+1}].$

Take $D^{*}\in\overline{\mathcal{D}}_{T}$ and $(D_{n})\subset\mathcal{D}_{T}$
with $D_{n}\to D^{*}$ in $L^{1}$. Therefore,
$D_{n}\stackrel{{\scriptstyle\mathbb{P}}}{{\to}}D^{*}$ which implies
$D_{n}Y_{t+1}\stackrel{{\scriptstyle\mathbb{P}}}{{\to}}D^{*}Y_{t+1}$.
Moreover, $D_{n,t}\stackrel{{\scriptstyle\mathbb{P}}}{{\to}}D^{*}_{t}$ since
$D_{n,t}=\mathbb{E}^{\mathbb{P}}_{t}[D_{n}]$,
$D^{*}_{t}=\mathbb{E}^{\mathbb{P}}_{t}[D^{*}]$ and

$\displaystyle\mathbb{E}^{\mathbb{P}}[|\mathbb{E}^{\mathbb{P}}_{t}[D_{n}]-\mathbb{E}^{\mathbb{P}}_{t}[D^{*}]|]\leq\mathbb{E}^{\mathbb{P}}[\mathbb{E}^{\mathbb{P}}_{t}[|D_{n}-D^{*}|]]=\mathbb{E}^{\mathbb{P}}[|D_{n}-D^{*}|]\to
0$

and convergence in $L^{1}$ implies convergence in probability. Since $Y_{t+1}$
is $\mathcal{Q}$-uniformly integrable, $\\{DY_{t+1}:D\in\mathcal{D}_{T}\\}$ is
$\mathbb{P}$-uniformly integrable. Therefore,
$D_{n}Y_{t+1}\stackrel{{\scriptstyle\mathbb{P}}}{{\to}}D^{*}Y_{t+1}$ implies
$D_{n}Y_{t+1}\to D^{*}Y_{t+1}$ in $L^{1}$. This further implies that
$\mathbb{E}^{\mathbb{P}}_{t}[D_{n}Y_{t+1}]\to\mathbb{E}^{\mathbb{P}}_{t}[D^{*}Y_{t+1}]$
in $L^{1}$ since

$\displaystyle\mathbb{E}^{\mathbb{P}}[|\mathbb{E}^{\mathbb{P}}_{t}[D_{n}Y_{t+1}]-\mathbb{E}^{\mathbb{P}}_{t}[D^{*}Y_{t+1}]|]$
$\displaystyle\leq\mathbb{E}^{\mathbb{P}}[\mathbb{E}^{\mathbb{P}}_{t}[|D_{n}Y_{t+1}-D^{*}Y_{t+1}|]]$
$\displaystyle=\mathbb{E}^{\mathbb{P}}[|D_{n}Y_{t+1}-D^{*}Y_{t+1}|]\to 0.$

In particular,

$\displaystyle\frac{1}{D_{n,t}}\mathbb{E}^{\mathbb{P}}_{t}[D_{n}Y_{t+1}]\stackrel{{\scriptstyle\mathbb{P}}}{{\to}}\frac{1}{D^{*}_{t}}\mathbb{E}^{\mathbb{P}}_{t}[D^{*}Y_{t+1}]$

which implies that there exists a subsequence $(n_{i})$ such that

$\displaystyle\frac{1}{D_{n_{i},t}}\mathbb{E}^{\mathbb{P}}_{t}[D_{n_{i}}Y_{t+1}]\stackrel{{\scriptstyle\text{a.s.}}}{{\to}}\frac{1}{D^{*}_{t}}\mathbb{E}^{\mathbb{P}}_{t}[D^{*}Y_{t+1}].$

Therefore, for any $D^{*}\in\overline{\mathcal{D}}_{T}$,

$\displaystyle\frac{1}{D^{*}_{t}}\mathbb{E}^{\mathbb{P}}_{t}[D^{*}Y_{t+1}]\geq\operatorname*{ess\,inf}_{D\in\mathcal{D}_{T}}\frac{1}{D_{t}}\mathbb{E}^{\mathbb{P}}_{t}[DY_{t+1}].$

The same argument shows the corresponding identity for
$\operatorname*{ess\,sup}$. The proof is complete. ∎

Consider a parameter set $\Theta$ which is taken to be a subset of a complete
and separable metric space. For each $t\in\\{1,\dots,T\\}$, let $f_{t}\geq 0$
be a measurable function on $\Omega\times\Theta$ such that $\omega\mapsto
f_{t}(\omega,\theta)$ is $\mathcal{F}_{t}$-measurable for each
$\theta\in\Theta$. It is assumed that the $\mathcal{F}_{t}$-measurable random
variables $f_{t}(\theta)$ satisfy

$\displaystyle\operatorname*{ess\,inf}_{\theta\in\Theta}f_{t}(\theta)>0\;\mathbb{P}\text{-a.s.~{}for
}t\in\\{1,\dots,T\\},$ (30)
$\displaystyle\mathbb{E}^{\mathbb{P}}_{t-1}[f_{t}(\theta)]=1\text{ for
}(t,\theta)\in\\{1,\dots,T\\}\times\Theta.$ (31)

For each $t\in\\{1,\dots,T\\}$, let $\lambda_{t}$ be an
$\mathcal{F}_{t-1}$-measurable random element in the space
$\mathcal{P}(\Theta)$ of probability measures on $\Theta$ equipped with the
topology of weak convergence. Let $\Lambda_{t}$ be the set of all such random
probability measures. For $\lambda_{t}\in\Lambda_{t}$ for all $t$, let

$\displaystyle
D_{\lambda,T}:=\prod_{t=1}^{T}\int_{\Theta}f_{t}(\theta)\lambda_{t}(\mathrm{d}\theta),\quad
D_{\lambda,t}:=\mathbb{E}^{\mathbb{P}}_{t}[D_{\lambda,T}]\text{ for }t<T.$
(32)

Notice that, due to properties (30) and (31), $(D_{\lambda,t})_{t=0}^{T}$ is a
positive $\mathbb{P}$-martingale with
$\mathbb{E}^{\mathbb{P}}[D_{\lambda,t}]=1$. Let

$\displaystyle\mathcal{D}_{f,T}:=\bigg{\\{}\prod_{t=1}^{T}f_{t}(\theta):\theta\in\Theta\bigg{\\}},\quad\widetilde{\mathcal{D}}_{f,T}:=\Big{\\{}D_{\lambda,T}:\lambda_{t}\in\Lambda_{t}\text{
for all }t\Big{\\}}$

and let $\overline{\mathcal{D}}_{f,T}$ be the $L^{1}$-closure of
$\mathcal{D}_{f,T}$. For $t=0,1,\dots,T-1$, let

$\displaystyle\mathcal{D}_{f,t}:=\Big{\\{}D_{t}=\mathbb{E}^{\mathbb{P}}_{t}[D_{T}]:D_{T}\in\mathcal{D}_{f,T}\Big{\\}}$

and let $\widetilde{\mathcal{D}}_{f,t}$ and $\overline{\mathcal{D}}_{f,t}$ be
defined analogously.

######  Definition 4.

Denote by
$\mathcal{Q}_{\Theta},\widetilde{\mathcal{Q}}_{\Theta},\overline{\mathcal{Q}}_{\Theta}$
the sets of probability measures corresponding to the sets
$\mathcal{D}_{f,T},\widetilde{\mathcal{D}}_{f,T},\overline{\mathcal{D}}_{f,T}$
of Radon-Nikodym derivatives with respect to $\mathbb{P}$.

Notice that $\mathcal{Q}_{\Theta}$ corresponds to only considering measures
$\lambda_{t}(\cdot)=1_{\\{\theta\\}}(\cdot)$ in (32). We will show in Theorem
6 that $\overline{\mathcal{Q}}_{\Theta}$ has the properties assumed in Theorem
1. We also show that Theorem 1 holds also for
$\widetilde{\mathcal{Q}}_{\Theta}$.

######  Theorem 6.

Consider the sets
$\mathcal{Q}_{\Theta},\widetilde{\mathcal{Q}}_{\Theta},\overline{\mathcal{Q}}_{\Theta}$
and
$\mathcal{D}_{f,T},\widetilde{\mathcal{D}}_{f,T},\overline{\mathcal{D}}_{f,T}$
in Definition 4.

  * (i)

The sets $\widetilde{\mathcal{Q}}_{\Theta}$ and
$\overline{\mathcal{Q}}_{\Theta}$ are convex and stable under pasting.

  * (ii)

For every $t\in\\{1,\dots,T\\}$, $\overline{\mathcal{D}}_{f,t}$ is closed in
$L^{1}(\mathcal{F}_{T},\mathbb{P})$.

  * (iii)

If $\widetilde{\mathcal{D}}_{f,T}$ is $\mathbb{P}$-uniformly integrable, then
$\widetilde{\mathcal{D}}_{f,t}$ is weakly relatively compact in
$L^{1}(\mathcal{F}_{T},\mathbb{P})$ and $\overline{\mathcal{D}}_{f,t}$ is
weakly compact in $L^{1}(\mathcal{F}_{T},\mathbb{P})$ for every
$t\in\\{0,\dots,T\\}$.

  * (iv)

If $\widetilde{\mathcal{D}}_{f,T}$ is $\mathbb{P}$-uniformly integrable, then,
for any $\mathcal{F}_{t+1}$-measurable
$\widetilde{\mathcal{Q}}_{\Theta}$-uniformly integrable random variable
$Y_{t+1}$,

$\displaystyle\operatorname*{ess\,inf}_{\mathbb{Q}\in\overline{\mathcal{Q}}_{\Theta}}\mathbb{E}^{\mathbb{Q}}_{t}\big{[}Y_{t+1}\big{]}$
$\displaystyle=\operatorname*{ess\,inf}_{\mathbb{Q}\in\widetilde{\mathcal{Q}}_{\Theta}}\mathbb{E}^{\mathbb{Q}}_{t}\big{[}Y_{t+1}\big{]}$
$\displaystyle=\operatorname*{ess\,inf}_{\mathbb{Q}\in\mathcal{Q}_{\Theta}}\mathbb{E}^{\mathbb{Q}}_{t}\big{[}Y_{t+1}\big{]}$
$\displaystyle=\operatorname*{ess\,inf}_{\theta\in\Theta}\mathbb{E}^{\mathbb{P}}_{t}\big{[}Y_{t+1}f_{t+1}(\theta)\big{]}$

and similarly with $\operatorname*{ess\,inf}$ replaced by
$\operatorname*{ess\,sup}$.

###### Proof of Theorem 6.

We first prove $(i)$. We first prove convexity and stability under pasting for
the set of probability measures with Radon-Nikodym derivatives
$D_{T}\in\widetilde{\mathcal{D}}_{f,T}$ with respect to $\mathbb{P}$. We first
prove convexity. Note that for a density process $(D_{t})_{t=0}^{T}$ with
$D_{T}\in\widetilde{\mathcal{D}}_{f,T}$,

$\displaystyle\frac{D_{t+1}}{D_{t}}=\frac{\prod_{s=1}^{t+1}\int_{\Theta}f_{s}(\theta)\lambda_{s}(\mathrm{d}\theta)}{\prod_{s=1}^{t}\int_{\Theta}f_{s}(\theta)\lambda_{s}(\mathrm{d}\theta)}=\int_{\Theta}f_{t+1}(\theta)\lambda_{t+1}(\mathrm{d}\theta).$

Consider density processes $D^{(1)},D^{(2)}$ with
$D^{(1)}_{T},D^{(2)}_{T}\in\widetilde{\mathcal{D}}_{f,T}$, let $c\in(0,1)$ and
set $D^{(3)}:=cD^{(1)}+(1-c)D^{(2)}$. Then

$\displaystyle\frac{D^{(3)}_{t+1}}{D^{(3)}_{t}}$
$\displaystyle=\frac{1}{D^{(3)}_{t}}\Big{(}cD^{(1)}_{t}\int_{\Theta}f_{t+1}(\theta)\lambda^{(1)}_{t+1}(\mathrm{d}\theta)+(1-c)D^{(2)}_{t}\int_{\Theta}f_{t+1}(\theta)\lambda^{(2)}_{t+1}(\mathrm{d}\theta)\Big{)}$
$\displaystyle=\int_{\Theta}f_{t+1}(\theta)\bigg{(}\frac{1}{D^{(3)}_{t}}\Big{(}cD^{(1)}_{t}\lambda^{(1)}_{t+1}+(1-c)D^{(2)}_{t}\lambda^{(2)}_{t+1}\Big{)}(\mathrm{d}\theta)\bigg{)}$

and the convexity property follows since

$\displaystyle\frac{1}{D^{(3)}_{t}}\Big{(}cD^{(1)}_{t}\lambda^{(1)}_{t+1}+(1-c)D^{(2)}_{t}\lambda^{(2)}_{t+1}\Big{)}\in\Lambda_{t+1}.$

We now prove stability under pasting. Consider density processes
$D^{(1)},D^{(2)}$ with
$D^{(1)}_{T},D^{(2)}_{T}\in\widetilde{\mathcal{D}}_{f,T}$, and let $\tau$ be a
stopping time. Then

$\displaystyle D^{(3)}_{t}$
$\displaystyle:=\prod_{s=1}^{t}\bigg{(}\mathbb{I}\\{s\leq\tau\\}\frac{D^{(1)}_{s}}{D^{(1)}_{s-1}}+\mathbb{I}\\{s>\tau\\}\frac{D^{(2)}_{s}}{D^{(2)}_{s-1}}\bigg{)}$
$\displaystyle=\prod_{s=1}^{t}\int_{\Theta}f_{s}(\theta)\Big{(}\mathbb{I}\\{s\leq\tau\\}\lambda^{(1)}_{s}+\mathbb{I}\\{s>\tau\\}\lambda^{(2)}_{s}\Big{)}(\mathrm{d}\theta)$

and since $\mathbb{I}\\{s\leq\tau\\},\mathbb{I}\\{s>\tau\\}$ are
$\mathcal{F}_{s-1}$-measurable,

$\displaystyle\mathbb{I}\\{s\leq\tau\\}\lambda^{(1)}_{s}+\mathbb{I}\\{s>\tau\\}\lambda^{(2)}_{s}\in\Lambda_{s},$

which proves stability under pasting. Theorem 5$(i)$ completes the proof of
$(i)$.

Notice that $(ii)$ follows immediately from $(i)$ together with Theorem
5$(ii)$. Similarly, $(iii)$ follows immediately from $(i)$ together with
Theorem 5$(iii)$.

It remains to prove $(iv)$. The two last identities in $(iv)$ follow from the
definition of $\mathcal{Q}_{\Theta}$ and $\widetilde{\mathcal{Q}}_{\Theta}$:

$\displaystyle\operatorname*{ess\,inf}_{\mathbb{Q}\in\widetilde{\mathcal{Q}}_{\Theta}}\mathbb{E}^{\mathbb{Q}}_{t}\big{[}Y_{t+1}\big{]}$
$\displaystyle=\operatorname*{ess\,inf}_{\lambda_{t+1}\in\Lambda_{t+1}}\mathbb{E}^{\mathbb{P}}_{t}\bigg{[}Y_{t+1}\int_{\Theta}f_{t+1}(\theta)\lambda_{t+1}(\mathrm{d}\theta)\bigg{]}$
$\displaystyle=\operatorname*{ess\,inf}_{\theta\in\Theta}\mathbb{E}^{\mathbb{P}}_{t}\big{[}Y_{t+1}f_{t+1}\big{(}\theta\big{)}\big{]}.$

The first identity follows from Theorem 5$(iv)$. The proof is complete. ∎

##  4 Gaussian example

In this section we consider an application illustrating the general theory
presented up to this point. We consider a setting where the liability cash
flow is Gaussian, both under $\mathbb{P}$ and under any
$\mathbb{Q}_{\theta}\in\mathcal{Q}_{\Theta}$. We consider two cases:

  * Case 1

In this case we assume that $\mathcal{Q}=\mathcal{Q}_{\Theta}$. Recall that
$\mathcal{Q}_{\Theta}$ is not stable under pasting: this decision maker does
not consider probability measures corresponding to switching between
probability measures in $\mathcal{Q}_{\Theta}$ depending on information
revealed over time.

  * Case 2

In this case we assume that $\mathcal{Q}=\widetilde{\mathcal{Q}}_{\Theta}$.
This decision maker exhibits a behaviour that is time consistent. Notice that
$\widetilde{\mathcal{Q}}_{\Theta}$ is considerably larger than
$\mathcal{Q}_{\Theta}\subset\widetilde{\mathcal{Q}}_{\Theta}$.

The liability cash flow is assumed to be fully nonhedgeable by financial
assets and consequently we take $X^{r}=0$ which means that $X=X^{o}$. In order
to make the illustration clear, we choose $T=2$. Let
$C_{i,k}:=C_{i,k}^{\text{orig}}/v_{i}$ denote the exposure adjusted cumulative
amount of payments to policyholders for accident year $i$, where $v_{i}$ is a
known exposure measure for accident year $i$. The evolution of the exposure
adjusted cumulative amounts is assumed to satisfy

$\displaystyle
C_{i,1}=\beta_{0}^{\mathbb{P}}+\frac{\sigma_{0}^{\mathbb{P}}}{\sqrt{v_{i}}}\varepsilon_{i,1},\quad
C_{i,2}=\beta_{1}^{\mathbb{P}}C_{i,1}+\frac{\sigma_{1}^{\mathbb{P}}}{\sqrt{v_{i}}}\varepsilon_{i,2},$

where all $\varepsilon_{i,k}$ are independent and $N(0,1)$ with respect to
$\mathbb{P}$. Suppose that we are uncertain about the parameter values and
want to consider probability measures $\mathbb{Q}_{\theta}$,
$\theta=(\beta_{0},\sigma_{0},\beta_{1},\sigma_{1})$, such that

$\displaystyle
C_{i,1}=\beta_{0}+\frac{\sigma_{0}}{\sqrt{v_{i}}}\varepsilon_{i,1}^{\theta},\quad
C_{i,2}=\beta_{1}C_{i,1}+\frac{\sigma_{1}}{\sqrt{v_{i}}}\varepsilon_{i,2}^{\theta},$

where all $\varepsilon_{i,k}^{\theta}$ are independent and $N(0,1)$ with
respect to $\mathbb{Q}_{\theta}$. We choose a parameter set
$\Theta\subset(0,\infty)^{4}$ that describes the uncertainty about the
parameter values.

Suppose that $C_{i,k}$ with $i+k\leq 0$ are observed at time $0$ and that
$C_{i,k}$ with $i+k=t$, $t=1,2$, are observed at times $t>0$ and therefore
contain cash flows that are part of the outstanding liability to the
policyholders. The (incremental) liability cash flow $X=(X_{1},X_{2})$ is
given by

$\displaystyle
X=\bigg{(}v_{-1}(C_{-1,2}-C_{-1,1})+v_{0}C_{0,1},v_{0}(C_{0,2}-C_{0,1})\bigg{)}.$

Notice that $C_{-1,1}$ is here considered to be a known constant. Direct
computations give

$\displaystyle\mathbb{E}^{\mathbb{P}}[X_{1}+X_{2}]$
$\displaystyle=v_{-1}(\beta_{1}^{\mathbb{P}}-1)C_{-1,1}+v_{0}\beta_{0}^{\mathbb{P}}\beta_{1}^{\mathbb{P}},$
$\displaystyle\mathbb{E}^{\mathbb{Q}_{\theta}}[X_{1}+X_{2}]$
$\displaystyle=v_{-1}(\beta_{1}-1)C_{-1,1}+v_{0}\beta_{0}\beta_{1}.$

The filtration is given by the $\sigma$-algebras
$\mathcal{F}_{0}=\\{\emptyset,\Omega\\}$,
$\mathcal{F}_{1}=\sigma(\varepsilon_{-1,2},\varepsilon_{0,1})$ and
$\mathcal{F}_{2}=\sigma(\varepsilon_{0,2})\vee\mathcal{F}_{1}$. In order to
have the correct evolution with respect to $\mathbb{Q}_{\theta}$ of the
cumulative amounts it is seen that we must require that

$\displaystyle\mathbb{Q}_{\theta}(\varepsilon_{-1,2}\in\cdot\mid\mathcal{F}_{0})\sim
N(\mu_{-1,2},\sigma_{-1,2}^{2}),\quad\mu_{-1,2}=\frac{\beta_{1}-\beta_{1}^{\mathbb{P}}}{\sigma_{1}^{\mathbb{P}}/\sqrt{v_{-1}}}C_{-1,1},\,\sigma_{-1,2}=\frac{\sigma_{1}}{\sigma_{1}^{\mathbb{P}}},$
$\displaystyle\mathbb{Q}_{\theta}(\varepsilon_{0,1}\in\cdot\mid\mathcal{F}_{0})\sim
N(\mu_{0,1},\sigma_{0,1}^{2}),\quad\mu_{0,1}=\frac{\beta_{0}-\beta_{0}^{\mathbb{P}}}{\sigma_{0}^{\mathbb{P}}/\sqrt{v_{0}}},\,\sigma_{0,1}=\frac{\sigma_{0}}{\sigma_{0}^{\mathbb{P}}},$
$\displaystyle\mathbb{Q}_{\theta}(\varepsilon_{0,2}\in\cdot\mid\mathcal{F}_{1})\sim
N(\mu_{0,2},\sigma_{0,2}^{2}),\quad\mu_{0,2}=\frac{\beta_{1}-\beta_{1}^{\mathbb{P}}}{\sigma_{1}^{\mathbb{P}}/\sqrt{v_{0}}}C_{0,1},\,\sigma_{0,2}=\frac{\sigma_{1}}{\sigma_{1}^{\mathbb{P}}}.$

This corresponds to, in the setting of Section 3, choosing

$\displaystyle f_{1}(\theta)$
$\displaystyle=\frac{\varphi(\varepsilon_{-1,2};\mu_{-1,2},\sigma_{-1,2}^{2})\varphi(\varepsilon_{0,1};\mu_{0,1},\sigma_{0,1}^{2})}{\varphi(\varepsilon_{-1,2};0,1)\varphi(\varepsilon_{0,1};0,1)},$
(33) $\displaystyle f_{2}(\theta)$
$\displaystyle=\frac{\varphi(\varepsilon_{0,2};\mu_{0,2},\sigma_{0,2}^{2})}{\varphi(\varepsilon_{0,2};0,1)},$

where $\varphi(x;\mu,\sigma^{2})$ denotes the density function of
$N(\mu,\sigma^{2})$. By Remark 3, the set $\widetilde{\mathcal{D}}_{f,2}$ is
$\mathbb{P}$-uniformly integrable if

$\lim_{\mathbb{P}(A)\to
0}\sup_{\mathbb{Q}\in\widetilde{\mathcal{Q}}_{\Theta}}\mathbb{Q}(A)=0$

which holds here since $\sigma_{k}^{\mathbb{P}}/\sigma_{k}$ and
$|\beta_{k}-\beta_{k}^{\mathbb{P}}|$ both take values in bounded intervals
bounded away from $0$. The sets $A\in\mathcal{F}_{2}$ are of type
$\\{(\varepsilon_{-1,2},\varepsilon_{0,1},\varepsilon_{0,2})\in B\\}$ for
measurable sets $B\subset\mathbb{R}^{3}$ such that
$\mathbb{P}((\varepsilon_{-1,2},\varepsilon_{0,1},\varepsilon_{0,2})\in B)\to
0$. Therefore, it follows from Theorem 6 that the set
$\widetilde{\mathcal{Q}}_{\Theta}$ in Definition 4 satisfies the requirements
for multiple prior optimal stopping. In particular, Theorem 1 holds with
$\mathcal{Q}=\widetilde{\mathcal{Q}}_{\Theta}$.

$\Theta$ can be chosen to reflect parameter uncertainty. To illustrate how
such a choice may be implemented, consider the regression estimators from
Lindholm et al.[18] based on data from accident years $i=i_{0},\dots,-1$:

$\displaystyle\widehat{\beta_{0}^{\mathbb{P}}}$
$\displaystyle=\frac{\sum_{i=i_{0}}^{-1}v_{i}C_{i,1}}{\sum_{i=i_{0}}^{-1}v_{i}},\quad\widehat{(\sigma_{0}^{\mathbb{P}})^{2}}=\frac{1}{-i_{0}-1}\sum_{i=i_{0}}^{-1}v_{i}(C_{i,1}-\widehat{\beta_{0}^{\mathbb{P}}})^{2},$
$\displaystyle\widehat{\beta_{1}^{\mathbb{P}}}$
$\displaystyle=\frac{\sum_{i=i_{0}}^{-2}v_{i}C_{i,1}C_{i,2}}{\sum_{i=i_{0}}^{-2}v_{i}C_{i,1}^{2}},\quad\widehat{(\sigma_{1}^{\mathbb{P}})^{2}}=\frac{1}{-i_{0}-2}\sum_{i=i_{0}}^{-2}v_{i}(C_{i,2}-\widehat{\beta_{1}^{\mathbb{P}}}C_{i,1})^{2}.$

Here $i_{0}$ denotes the index of the first accident year observed. These
estimators are unbiased and uncorrelated. We now proceed with a numerical
illustration, with parameter values
$(\beta_{0}^{\mathbb{P}},\sigma_{0}^{\mathbb{P}},\beta_{1}^{\mathbb{P}},\sigma_{1}^{\mathbb{P}})=(2/3,1/5,3/2,1/5)$,
$i_{0}=-10$, and $v_{i}=1$ for $i=-10,\dots,0$. Based on these parameter
values and a large number $n$ of simulated independent standard normal
$\varepsilon_{i,j}$, leading to $n$ iid copies of
$C_{-10,1},\dots,C_{-1,1},C_{-10,2},\dots,C_{-2,2}$, we estimate
$(\beta_{0}^{\mathbb{P}},\sigma_{0}^{\mathbb{P}},\beta_{1}^{\mathbb{P}},\sigma_{1}^{\mathbb{P}})$
$n$ times. Figure 1 presents scatter plots, which suggests that the iid
vectors of estimators are approximately $N_{4}(\mu,\Sigma)$-distributed, where
$\mu$ and $\Sigma$ are the sample mean and sample covariance matrix. We can
therefore shape an approximative confidence region with confidence level $p$
of the parameter values by the squared Mahalanobis distance as

$\displaystyle\Theta$
$\displaystyle=\Big{\\{}z\in\mathbb{R}^{4}:(z-\mu)^{{\mathrm{T}}}\Sigma^{-1}(z-\mu)\leq
F_{\chi^{2}(4)}^{-1}(p)\Big{\\}}$
$\displaystyle=\Big{\\{}\mu+rLs\in\mathbb{R}^{4}:r^{2}\leq
F_{\chi^{2}(4)}^{-1}(p),\,s\in\mathbb{S}^{3}\Big{\\}},$

where $L$ is the (lower triangular) Cholesky decomposition of
$LL^{{\mathrm{T}}}=\Sigma$, $F_{\chi^{2}(4)}$ is the distribution function of
the $\chi^{2}(4)$ and $\mathbb{S}^{3}$ is the unit sphere in $\mathbb{R}^{4}$.
For the evaluation at time 1, only $(\beta_{1},\sigma_{1})$ needs to be
considered, leading to a set
$\Theta_{\beta_{1},\sigma_{1}}\subset\mathbb{R}^{2}$ satisfying that

$\\{(0,0,\beta_{1},\sigma_{1}):(\beta_{1},\sigma_{1})\in\Theta_{\beta_{1},\sigma_{1}}\\}$

is the orthogonal projection of $\Theta$ onto the $(\beta_{1},\sigma_{1})$
coordinate plane: $\beta_{0}=\sigma_{0}=0$. Explicitly,

$\displaystyle\Theta_{\beta_{1},\sigma_{1}}$
$\displaystyle=\Big{\\{}z\in\mathbb{R}^{2}:(z-\mu_{\beta_{1},\sigma_{1}})^{{\mathrm{T}}}\Sigma_{\beta_{1},\sigma_{1}}^{-1}(z-\mu_{\beta_{1},\sigma_{1}})\leq
F_{\chi^{2}(4)}^{-1}(p)\Big{\\}}$
$\displaystyle=\Big{\\{}\mu_{\beta_{1},\sigma_{1}}+rL_{\beta_{1},\sigma_{1}}s\in\mathbb{R}^{2}:r^{2}\leq
F_{\chi^{2}(4)}^{-1}(p),\,s\in\mathbb{S}^{1}\Big{\\}},$

where $\mathbb{S}^{1}$ is the unit sphere in $\mathbb{R}^{2}$,
$\mu_{\beta_{1},\sigma_{1}}$ is the subvector of the last two entries of $\mu$
and $L_{\beta_{1},\sigma_{1}}$ is the Cholesky decomposition of the submatrix
$\Sigma_{\beta_{1},\sigma_{1}}$ of $\Sigma$. Similarly, to compute the upper
bound of $V_{0}$ in (34), only $(\beta_{0},\beta_{1})$ need to be considered,
leading to a similar set $\Theta_{\beta_{0},\beta_{1}}\subset\mathbb{R}^{2}$.

The left plot in Figure 1 shows a scatted plot of $1000$ iid estimates of
$(\beta_{0}^{\mathbb{P}},\beta_{1}^{\mathbb{P}})$ together with boundaries
$\partial\Theta_{\beta_{0},\beta_{1}}$ for $p=0.1$ (blue) and for $p=0.9$
(red). The right plot in Figure 1 shows a scatted plot of $1000$ iid estimates
of $(\beta_{1}^{\mathbb{P}},\sigma_{1}^{\mathbb{P}})$ together with boundaries
$\partial\Theta_{\beta_{1},\sigma_{1}}$ for $p=0.1$ (blue) and for $p=0.9$
(red).

Figure 1: Scatter plots of $1000$ iid estimates of
$(\beta_{0}^{\mathbb{P}},\beta_{1}^{\mathbb{P}})$ (left) and of
$(\beta_{1}^{\mathbb{P}},\sigma_{1}^{\mathbb{P}})$ (right), together with
boundaries of the parameter regions $\Theta_{\beta_{0},\beta_{1}}$ (left) and
$\Theta_{\beta_{1},\sigma_{1}}$ (right) for $p=0.1$ (blue) and for $p=0.9$
(red).

Let $\rho_{0},\rho_{1}$ be conditional monetary risk measures defined in terms
of conditional quantiles with respect to $\mathbb{P}$, such as, for $t=0,1$,
$\rho_{t}=\operatorname{V@R}_{t,p}$ or $\rho_{t}=\operatorname{AV@R}_{t,p}$.
In both cases, $c:=\rho_{0}(e^{\mathbb{P}}_{1})=\rho_{1}(e^{\mathbb{P}}_{2})$
is a constant for an $\mathcal{F}_{t+1}$-measurable $e^{\mathbb{P}}_{t}\sim
N(0,1)$ and independent of $\mathcal{F}_{t}$ with respect to $\mathbb{P}$.
Then

$\displaystyle R_{1}$
$\displaystyle=\rho_{1}(-X_{2})=\rho_{1}(-\mathbb{E}^{\mathbb{P}}_{1}[X_{2}]+\operatorname{Var}^{\mathbb{P}}_{1}(X_{2})^{1/2}e^{\mathbb{P}}_{2})=\mathbb{E}^{\mathbb{P}}_{1}[X_{2}]+\operatorname{Var}^{\mathbb{P}}_{1}(X_{2})^{1/2}c$
$\displaystyle=v_{0}(\beta_{1}^{\mathbb{P}}-1)C_{0,1}+\sqrt{v_{0}}\sigma_{1}^{\mathbb{P}}c.$

###  4.1 Case 1: computing upper and lower bounds for $V_{0}$

In this case $\mathcal{Q}=\mathcal{Q}_{\Theta}$ does not satisfy the
conditions of Theorem 1 and therefore we can not compute $V_{0}$ by backward
recursion. However, upper and lower bounds for $V_{0}$ are easily computed.
From (7) we have the upper bound

$\displaystyle V_{0}$
$\displaystyle\leq\sup_{\mathbb{Q}\in\mathcal{Q}_{\Theta}}\mathbb{E}^{\mathbb{Q}}[X_{1}+X_{2}]$
$\displaystyle=\sup\Big{\\{}v_{-1}(\beta_{1}-1)C_{-1,1}+v_{0}\beta_{0}\beta_{1}:(\beta_{0},\sigma_{0},\beta_{1},\sigma_{1})\in\Theta\Big{\\}}=:\overline{V}_{0}.$
(34)

From (9) we have the lower bound

$\displaystyle V_{0}$
$\displaystyle\geq\sup_{\mathbb{Q}\in\mathcal{Q}_{\Theta}}\inf_{\tau\in\mathcal{S}_{1,T+1}}\mathbb{E}_{0}^{\mathbb{Q}}\bigg{[}\sum_{s=1}^{\tau-1}X_{s}+R_{\tau-1}\bigg{]}=:\underline{V}_{0}.$

In the setting of Section 3, for each $\theta\in\Theta$, with
$V_{T}^{\theta}=R_{T}^{\theta}=0$, we solve the backward recursion

$\displaystyle R_{t}^{\theta}$
$\displaystyle=\rho_{t}(-X_{t+1}-V_{t+1}^{\theta}),$ $\displaystyle
V_{t}^{\theta}$
$\displaystyle=R_{t}^{\theta}-\mathbb{E}^{\mathbb{Q}_{\theta}}_{t}[(R_{t}^{\theta}-X_{t+1}-V_{t+1}^{\theta})^{+}],$

and then compute

$\displaystyle\underline{V}_{0}=\sup_{\theta\in\Theta}V_{0}^{\theta}.$

Notice that $V_{t}^{\theta},R_{t}^{\theta}$ corresponds to the quantities
$V_{t},R_{t}$ in the special case $\mathcal{Q}=\\{\mathbb{Q}_{\theta}\\}$.
Computing $\underline{V}_{0}$ is simpler than computing $V_{0}$ since the
former involves just one optimisation over the parameter set $\Theta$ rather
than $T$ nested optimisations for the latter.

We now demonstrate how $\underline{V}_{0}$ is computed in the current Gaussian
setting. As shown above
$R_{1}^{\theta}=v_{0}(\beta_{1}^{\mathbb{P}}-1)C_{0,1}+\sqrt{v_{0}}\sigma_{1}^{\mathbb{P}}c$
(which does not depend on $\theta$) and

$\displaystyle
C_{1}^{\theta}=\mathbb{E}^{\mathbb{Q}_{\theta}}_{1}\big{[}\big{(}\rho_{1}(-X_{2})-X_{2}\big{)}^{+}\big{]}=\mathbb{E}^{\mathbb{Q}_{\theta}}_{1}[\big{(}a(\theta,C_{0,1})-b(\theta)e^{\theta}_{2}\big{)}^{+}],$

where $e^{\theta}_{2}\sim N(0,1)$ with respect to $\mathbb{Q}_{\theta}$ and
independent of $\mathcal{F}_{1}$, and

$\displaystyle a(\theta,C_{0,1})$
$\displaystyle=-\mathbb{E}^{\mathbb{Q}_{\theta}}_{1}[X_{2}]+\mathbb{E}^{\mathbb{P}}_{1}[X_{2}]+\operatorname{Var}^{\mathbb{P}}_{1}(X_{2})^{1/2}\rho_{1}(e^{\mathbb{P}}_{2})$
$\displaystyle=v_{0}(\beta_{1}^{\mathbb{P}}-\beta_{1})C_{0,1}+\sqrt{v_{0}}\sigma_{1}^{\mathbb{P}}c,$
$\displaystyle b(\theta)$
$\displaystyle=\operatorname{Var}^{\mathbb{Q}_{\theta}}_{1}(X_{2})^{1/2}=\sqrt{v_{0}}\sigma_{1}.$

Straightforward calculations show that

$\displaystyle\mathbb{E}^{\mathbb{Q}_{\theta}}_{1}[\big{(}a(\theta,C_{0,1})-b(\theta)e^{\theta}_{2}\big{)}^{+}]$
$\displaystyle=a(\theta,C_{0,1})\Phi\bigg{(}\frac{a(\theta,C_{0,1})}{b(\theta)}\bigg{)}+b(\theta)\varphi\bigg{(}\frac{a(\theta,C_{0,1})}{b(\theta)}\bigg{)}$
$\displaystyle=:g(\theta,C_{0,1}).$

Consequently,

$\displaystyle X_{1}+V_{1}^{\theta}$
$\displaystyle=X_{1}+R_{1}^{\theta}-C_{1}^{\theta}$
$\displaystyle=v_{-1}(\beta_{1}-1)C_{-1,1}+\sqrt{v_{-1}}\sigma_{1}\varepsilon_{-1,2}^{\theta}+\sqrt{v_{0}}\sigma_{1}^{\mathbb{P}}c$
$\displaystyle\quad+\beta_{1}^{\mathbb{P}}\bigg{(}v_{0}\beta_{0}+\sqrt{v_{0}}\sigma_{1}\varepsilon_{0,1}^{\theta}\bigg{)}-g\bigg{(}\theta,v_{0}\beta_{0}+\sqrt{v_{0}}\sigma_{1}\varepsilon_{0,1}^{\theta}\bigg{)}$
$\displaystyle=v_{-1}(\beta_{1}^{\mathbb{P}}-1)C_{-1,1}+\sqrt{v_{-1}}\sigma_{1}^{\mathbb{P}}\varepsilon_{-1,2}+\sqrt{v_{0}}\sigma_{1}^{\mathbb{P}}c$
$\displaystyle\quad+\beta_{1}^{\mathbb{P}}\bigg{(}v_{0}\beta_{0}^{\mathbb{P}}+\sqrt{v_{0}}\sigma_{1}^{\mathbb{P}}\varepsilon_{0,1}\bigg{)}-g\bigg{(}\theta,v_{0}\beta_{0}^{\mathbb{P}}+\sqrt{v_{0}}\sigma_{1}^{\mathbb{P}}\varepsilon_{0,1}\bigg{)}$

from which $R_{0}^{\theta}=\rho_{0}(-X_{1}-V_{1}^{\theta})$ can be estimated
with arbitrary accuracy by simulating iid copies of $X_{1}+V_{1}^{\theta}$
with respect to $\mathbb{P}$ and computing the empirical estimate, and
$C_{0}^{\theta}=\mathbb{E}^{\mathbb{Q}_{\theta}}[(R_{0}^{\theta}-X_{1}-V_{1}^{\theta})^{+}]$
can be estimated similarly by simulating iid copies with respect to
$\mathbb{Q}_{\theta}$ and approximating the expectation by the empirical mean.
Finally,

$\displaystyle\underline{V}_{0}=\sup_{\theta\in\Theta}\big{(}R_{0}^{\theta}-C_{0}^{\theta}\big{)}=\sup_{\theta\in\partial\Theta}\big{(}R_{0}^{\theta}-C_{0}^{\theta}\big{)}.$

Table 1 shows numerical values for lower bounds $\underline{V}_{0}$ and for
upper bounds $\overline{V}_{0}$. These values are based on $v_{-1}=v_{0}=1$,
$C_{-1,1}=\beta_{0}^{\mathbb{P}}$, $\rho_{t}=\operatorname{V@R}_{t,q}$ with
$q=0.005,0.01,0.05,0.10$ and parameters sets $\Theta$ of varying size
corresponding to $r^{2}\leq F_{\chi^{2}(4)}^{-1}(p)$ with $p=0.1,0.5,0.9$. The
main message of Table 1 is that the intervals
$(\underline{V}_{0},\overline{V}_{0})$ are very narrow for $q$ small and
therefore the upper bound $\overline{V}_{0}$ is an accurate estimate of
$V_{0}$ when $q$ is small. Notice that the upper bound is both easily computed
and has attractive theoretical properties.

###  4.2 Case 2: computing $V_{0}$ and an upper bound for $V_{0}$

In this case $\mathcal{Q}=\widetilde{\mathcal{Q}}_{\Theta}$ and the general
lower bound $\underline{V}_{0}$ coincides with $V_{0}$ and therefore its
computation by backward recursion is somewhat involved. However, the upper
bound is still fairly straightforward to compute. Notice that the lower bound
computed for Case 1 is a lower bound for $V_{0}$ in the current Case 2 since
$\mathcal{Q}_{\Theta}\subset\widetilde{\mathcal{Q}}_{\Theta}$.

We begin by computing the upper bound using the law of iterated expectations,
extended to the multiple prior setting, and Theorem 6:

$\displaystyle\overline{V}_{0}$
$\displaystyle=\sup_{\mathbb{Q}\in\widetilde{\mathcal{Q}}_{\Theta}}\mathbb{E}^{\mathbb{Q}}_{0}[X_{1}+X_{2}]$
$\displaystyle=\sup_{\mathbb{Q}\in\widetilde{\mathcal{Q}}_{\Theta}}\mathbb{E}^{\mathbb{Q}}_{0}[X_{1}+\operatorname*{ess\,sup}_{\mathbb{Q}^{\prime}\in\widetilde{\mathcal{Q}}_{\Theta}}\mathbb{E}^{\mathbb{Q}^{\prime}}_{1}[X_{2}]]$
$\displaystyle=\sup_{\mathbb{Q}\in\mathcal{Q}_{\Theta}}\mathbb{E}^{\mathbb{Q}}_{0}[X_{1}+\operatorname*{ess\,sup}_{\mathbb{Q}^{\prime}\in\mathcal{Q}_{\Theta}}\mathbb{E}^{\mathbb{Q}^{\prime}}_{1}[X_{2}]].$

Notice that, with $\beta_{1,\min}>1$,

$\displaystyle\operatorname*{ess\,sup}_{\mathbb{Q}^{\prime}\in\mathcal{Q}_{\Theta}}\mathbb{E}^{\mathbb{Q}^{\prime}}_{1}[X_{2}]$
$\displaystyle=v_{0}(\beta_{1,\max}-1)C_{0,1}\mathbb{I}_{\\{C_{0,1}\geq
0\\}}+v_{0}(\beta_{1,\min}-1)C_{0,1}\mathbb{I}_{\\{C_{0,1}<0\\}},$

where
$\beta_{1,\max}:=\max\\{\beta_{1}:(\beta_{0},\sigma_{0},\beta_{1},\sigma_{1})\in\Theta\\}$
and similarly for $\beta_{1,\min}$. Therefore,

$\displaystyle\overline{V}_{0}=\sup_{(\beta_{0},\sigma_{0},\beta_{1},\sigma_{1})\in\Theta}\Big{(}$
$\displaystyle v_{-1}(\beta_{1}-1)C_{-1,1}+v_{0}\beta_{0}$
$\displaystyle+v_{0}(\beta_{1,\max}-1)\big{(}\beta_{0}+\sigma_{0}\Phi(-\beta_{0}/\sigma_{0})\big{)}$
$\displaystyle-
v_{0}(\beta_{1,\min}-1)\sigma_{0}\Phi(-\beta_{0}/\sigma_{0})\Big{)}$

$R_{1}$ is calculated explicitly as above. Computing $C_{1}$ means computing

$C_{1}=\operatorname*{ess\,sup}_{\theta_{1}\in\partial\Theta_{\beta_{1},\sigma_{1}}}g(\theta_{1},C_{0,1}),$

where, with some abuse of notation, we consider $g$ to be defined for
parameters $\theta_{1}\in\Theta_{\beta_{1},\sigma_{1}}$ rather than
$\theta\in\Theta$. In practice, this means determining a function
$h:\mathbb{R}\to\mathbb{R}$ such that
$h(C^{k}_{0,1})=\max_{\theta_{1}\in\partial\Theta_{\beta_{1},\sigma_{1}}}g(\theta_{1},C^{k}_{0,1})$
for suitably many simulated iid copies $C^{1}_{0,1},\dots,C^{n}_{0,1}$ of
$C_{0,1}$ and approximating $C_{1}\approx h(C_{0,1})$. Given the choice of
$h$, $R_{0}=\rho_{0}(-X_{1}-R_{1}+C_{1})$ is approximated by its empirical
estimate based on simulated iid copies with respect to $\mathbb{P}$ of

$\displaystyle
v_{-1}(\beta_{1}^{\mathbb{P}}-1)C_{-1,1}+\sqrt{v_{-1}}\sigma_{1}^{\mathbb{P}}\varepsilon_{-1,2}+\sqrt{v_{0}}\sigma_{1}^{\mathbb{P}}c$
$\displaystyle+\beta_{1}^{\mathbb{P}}\bigg{(}v_{0}\beta_{0}^{\mathbb{P}}+\sqrt{v_{0}}\sigma_{1}^{\mathbb{P}}\varepsilon_{0,1}\bigg{)}-h\bigg{(}v_{0}\beta_{0}^{\mathbb{P}}+\sqrt{v_{0}}\sigma_{1}^{\mathbb{P}}\varepsilon_{0,1}\bigg{)}$

Similarly, $C_{0}$ is approximated by, for each $\theta$ in a dense subset of
$\partial\Theta$, simulating iid copies with respect to $\mathbb{Q}_{\theta}$
of

$\displaystyle
v_{-1}(\beta_{1}-1)C_{-1,1}+\sqrt{v_{-1}}\sigma_{1}\varepsilon_{-1,2}^{\theta}+\sqrt{v_{0}}\sigma_{1}^{\mathbb{P}}c$
$\displaystyle+\beta_{1}^{\mathbb{P}}\bigg{(}v_{0}\beta_{0}+\sqrt{v_{0}}\sigma_{1}\varepsilon_{0,1}^{\theta}\bigg{)}-h\bigg{(}v_{0}\beta_{0}+\sqrt{v_{0}}\sigma_{1}\varepsilon_{0,1}^{\theta}\bigg{)},$

estimating $\mathbb{E}^{\mathbb{Q}_{\theta}}[(R_{0}-X_{1}-R_{1}+C_{1})^{+}]$
by the empirical mean, and computing the minimum of these expectations over
the $\theta$ values. Finally, $V_{0}$ is estimated by the difference of the
estimates of $R_{0}$ and $C_{0}$.

Table 1 shows numerical values for lower bounds $\underline{V}_{0}$ and for
upper bounds $\overline{V}_{0}$ with the same parameter values as those
considered for Case 1. Similarly to Case 1, the intervals
$(\underline{V}_{0},\overline{V}_{0})$ are very narrow for $q$ small and
therefore the upper bound $\overline{V}_{0}$ is an accurate estimate of
$V_{0}$ when $q$ is small.

$\begin{array}[]{|l|ccc|}\hline\cr\vrule\lx@intercol\hfil\text{Case
1}\hfil\lx@intercol\vrule\lx@intercol\\\ \hline\cr&p=0.1&p=0.5&p=0.9\\\
\hline\cr q=0.10&(1.452,1.491)&(1.562,1.624)&(1.686,1.787)\\\
q=0.05&(1.473,1.491)&(1.592,1.624)&(1.730,1.787)\\\
q=0.01&(1.490,1.491)&(1.618,1.624)&(1.772,1.787)\\\
q=0.005&(1.491,1.491)&(1.622,1.624)&(1.780,1.787)\\\
\hline\cr\vrule\lx@intercol\hfil\text{Case
2}\hfil\lx@intercol\vrule\lx@intercol\\\ \hline\cr&p=0.1&p=0.5&p=0.9\\\
\hline\cr q=0.10&(1.470,1.513)&(1.595,1.666)&(1.734,1.856)\\\
q=0.05&(1.491,1.513)&(1.628,1.666)&(1.786,1.856)\\\
q=0.01&(1.509,1.513)&(1.656,1.666)&(1.835,1.856)\\\
q=0.005&(1.511,1.513)&(1.661,1.666)&(1.845,1.856)\\\ \hline\cr\end{array}$
Table 1: Case 1 and Case 2: lower and upper bounds
$(\underline{V}_{0},\overline{V}_{0})$ rounded to three decimals, where the
size of the parameter uncertainty region is determined by $r^{2}\leq
F_{\chi^{2}(4)}^{-1}(p)$ and $\rho_{t}=\operatorname{V@R}_{t,q}$. Empirical
estimates were based on iid samples of size $10^{5}$.

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