# Approximations of multi-period liability values by simple formulas

Nils<EMAIL_ADDRESS>Department of Mathematics, Stockholm
University, Filip<EMAIL_ADDRESS>Department of Mathematics,
Stockholm University

###### Abstract

This paper is motivated by computational challenges arising in multi-period
valuation in insurance. Aggregate insurance liability cashflows typically
correspond to stochastic payments several years into the future. However,
insurance regulation requires that capital requirements are computed for a
one-year horizon, by considering cashflows during the year and end-of-year
liability values. This implies that liability values must be computed
recursively, backwards in time, starting from the year of the most distant
liability payments. Solving such backward recursions with paper and pen is
rarely possible, and numerical solutions give rise to major computational
challenges.

The aim of this paper is to provide explicit and easily computable expressions
for multi-period valuations that appear as limit objects for a sequence of
multi-period models that converge in terms of conditional weak convergence.
Such convergence appears naturally if we consider large insurance portfolios
such that the liability cashflows, appropriately centered and scaled, converge
weakly as the size of the portfolio tends to infinity.

Keywords: valuation, multi-period models, conditional weak convergence

##  1 Introduction and motivation

This paper is motivated by problems arising in multi-period valuation in
insurance, but the applicability of the results extends beyond the insurance
setting. Aggregate insurance liability cashflows typically correspond to
payments several years into the future. However, insurance regulation requires
that capital requirements are calculated for a one-year horizon, considering
cashflows during the year and end-of-year liability values. This implies that
liability values must be computed recursively, backwards in time, starting
from the year of the most distant liability payments. Solving such backward
recursions leads to major computational challenges. Consequently, simple and
explicit (standard) formulas with questionable conceptual soundness have
become the preferred alternative. With this paper we say that there are
alternatives to commonly encountered formulas that give rise to explicit
formulas that are easy to use in practice and that retain both economic
interpretability and the conceptual soundness of the original principles for
insurance valuation. Mathematically, the explicit formulas follow from
combining the original principles for insurance valuation with widely
applicable large portfolio asymptotics.

We begin by presenting the motivating insurance problem. Let
$(C^{n_{0}},\mathbb{F}^{n_{0}})$ be a stochastic processes in discrete time
with $(C^{n_{0}}_{t})_{t=1}^{T}$ denoting the discounted payments at times $t$
due to insurance claims, and
$\mathbb{F}^{n_{0}}=(\mathcal{F}^{n_{0}}_{t})_{t=0}^{T}$, with
$\mathcal{F}^{n_{0}}_{0}=\\{\Omega,\emptyset\\}$, denoting a filtration to
which $C^{n_{0}}$ is adapted. $C^{n_{0}}$ describes discounted claim payments
over time in a so-called runoff situation, when no new contracts are written.
The number $n_{0}$ is intended as a measure of volume or exposure, for
instance the number of contracts that may generate future claims payments. We
want to assign a value to this liability cashflow. In order to do this we
consider a sequence $(C^{n},\mathbb{F}^{n})_{n\geq 1}$ of such stochastic
processes, with $\|C^{n}\|\to\infty$ almost surely as $n\to\infty$, where
$\|\cdot\|$ denotes the Euclidean norm on $\mathbb{R}^{T}$. An example could
be $C^{n}$ given by

$\displaystyle C^{n}_{t}=\sum_{k=1}^{M_{n}}1_{\\{D_{k}=t\\}}Z_{k},\quad
t=1,\dots,T,$

where $M_{n}$ denotes the total number of claims payments and $M_{n}\to\infty$
almost surely as $n\to\infty$, $D_{k}$ denotes the time and $Z_{k}$ denotes
the size of the $k$th claims payment. For this example, the filtration could
be the one generated by the discounted cashflow $C^{n}$ or a larger filtration
also including information about the number of payments
$\sum_{k=1}^{M_{n}}1_{\\{D_{k}=t\\}}$ at each point in time, or more.

We assume nonrandom sequences with terms $a_{n}\in(0,\infty)$ and
$b_{n}\in\mathbb{R}^{T}$ such that there is convergence in distribution

$\displaystyle X^{n}=a_{n}^{-1}(C^{n}-b_{n})\stackrel{{\scriptstyle
d}}{{\to}}X\quad\text{as }n\to\infty.$

The value $V^{n}_{0}(C^{n})$ of the insurance liabilities in a multi-period
model is the result of applying a suitable functional to the pair
$(C^{n},\mathbb{F}^{n})$. Natural valuation functionals satisfy the property
$V^{n}_{0}(C^{n})=a_{n}V^{n}_{0}(X^{n})+\sum_{s=1}^{T}b_{n,s}$. Convergence
$X^{n}\stackrel{{\scriptstyle d}}{{\to}}X$ suggests $V^{n}_{0}(X^{n})\to
V_{0}(X)$ and therefore the approximation

$\displaystyle V^{n}_{0}(C^{n})\approx a_{n}V_{0}(X)+\sum_{s=1}^{T}b_{n,s}.$
(1)

However, the flow of information over time is an essential ingredient in
valuation and it is not true that convergence $X^{n}\stackrel{{\scriptstyle
d}}{{\to}}X$ implies the convergence $V^{n}_{0}(X^{n})\to V_{0}(X)$ (even if
$\mathbb{F}^{n}$ is taken to be the filtration generated by $C^{n}$).
Consequently, one of the main objectives of the present paper is to determine
sharp sufficient conditions for the convergence of $V^{n}_{0}(X^{n})$ to
$V_{0}(X)$ in terms of an appropriate mode of convergence of
$(X^{n},\mathbb{F}^{n})$ to $(X,\mathbb{F})$, similar to so-called extended
weak convergence introduced by Aldous in [2] and conditional weak convergence
studied by Sweeting in [13]. The importance of the approximation (1) is
because $V^{n}_{0}(C^{n})$ is typically very difficult to compute numerically
whereas $V_{0}(X)$ is easier to compute numerically and, more importantly, in
the case of a Gaussian limit model, is given by an explicit expression in
terms of conditional variances of components of $X$ (note that Gaussian
vectors have the rare feature that conditional variances of one component
given a subset of components are nonrandom).

The paper is organized as follows. Section 2 introduces notation and basic
properties of conditional distributions and risk measures. Section 3 presents
the main contents of the paper and, following a general presentation of the
mathematical setup, gives economic motivation of key quantities in Section 3.1
and presents the main results in Section 3.2. All proofs together with
auxiliary results are found in Section 4.

##  2 Preliminaries

$\mathbb{N}=\\{1,2,\dots\\}$, $\mathbb{R}$ denotes the real numbers and
$\mathbb{R}_{+}=[0,\infty)$. Whenever relevant, for $T\in\mathbb{N}$, a vector
$x\in\mathbb{R}^{T}$ is assumed to be column vector and its transpose
$x^{{\mathrm{T}}}$ a row vector. For
$x=(x_{1},\dots,x_{T})^{{\mathrm{T}}}\in\mathbb{R}^{T}$,
${\|x\|}^{2}=x^{{\mathrm{T}}}x$ and we let $x_{\leq
t}=(x_{1},\dots,x_{t})^{{\mathrm{T}}}$ and
$x_{>t}=(x_{t+1},\dots,x_{T})^{{\mathrm{T}}}$. For $d\in\mathbb{N}$ and a
Borel set $A\in\mathcal{B}(\mathbb{R}^{d})$, $\mathcal{P}(A)$ denotes the set
of probability measures on $A$. For $\mu\in\mathcal{P}(A)$,
$\operatorname{supp}(\mu)$ denotes its support. We consider a probability
space $(\Omega,\mathcal{F},\operatorname{P})$. For a $\sigma$-algebra
$\mathcal{F}_{t}\subset\mathcal{F}$, $L^{0}(\mathcal{F}_{t},\operatorname{P})$
denotes the vector space of all real-valued $\mathcal{F}_{t}$-measurable
random variables, and for $p\in(0,\infty)$,
$L^{p}(\mathcal{F}_{t},\operatorname{P})$ denotes the subset $\\{Y\in
L^{0}(\mathcal{F}_{t},\operatorname{P}):\operatorname{E}[|Y|^{p}]<\infty\\}$.

For an $\mathbb{R}^{d}$-valued random variable $Y$ we let $\mathcal{L}(Y)$
denote its distribution, i.e. the induced probability measure
$\operatorname{P}(Y\in\cdot)$ on $\mathbb{R}^{d}$. Given $\sigma$-algebras
$\mathcal{F}_{t}\subset\mathcal{F}$ and a random variable $Y$, a regular
conditional distribution $Q_{\mathcal{F}_{t},Y}$ is a version of
$\operatorname{P}(Y\in\cdot\mid\mathcal{F}_{t})$ which forms a probability
kernel from $(\Omega,\mathcal{F}_{t})$ to
$(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$. If $\mathcal{F}_{t}=\sigma(Z)$
for an $\mathbb{R}^{d^{\prime}}$-valued random variable $Z$, then
$Q_{\mathcal{F}_{t},Y}$ is an $\mathcal{F}_{t}$-measurable random measure
$\omega\mapsto\kappa(Z(\omega),\cdot)$ on $\mathbb{R}^{d}$, where $\kappa$ is
a probability kernel from
$(\mathbb{R}^{d^{\prime}},\mathcal{B}(\mathbb{R}^{d^{\prime}}))$ to
$(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$, see Theorem 6.3 in Kallenberg
[8]. The notation $\mathcal{L}(Y\mid Z=z)$ means $\kappa(z,\cdot)$. In
particular, $z\mapsto\mathcal{L}(Y\mid Z=z)$ is well defined on
$\operatorname{supp}(\mathcal{L}(Z))$.

We define the conditional $p$-quantile of $Y\in
L^{0}(\mathcal{F},\operatorname{P})$ given $\mathcal{F}_{t}$ as the random
variable $F_{Y\mid\mathcal{F}_{t}}^{-1}(p)$ given by

$\displaystyle\omega\mapsto\min\\{m\in\mathbb{R}:Q_{\mathcal{F}_{t},Y}(\omega,(-\infty,m])\geq
p\\}.$

If $Q_{\mathcal{F}_{t},Y}(\omega,\cdot)=\kappa(Z(\omega),\cdot)$, then
$F_{Y\mid Z=z}^{-1}(p)$ means $\min\\{m\in\mathbb{R}:\kappa(z,(-\infty,m])\geq
p\\}$ and is well defined on $\operatorname{supp}(\mathcal{L}(Z))$. Similar to
ordinary quantiles, the conditional quantiles have the property
$F_{aY+b\,\mid\mathcal{F}_{t}}^{-1}(p)=aF_{Y\mid\mathcal{F}_{t}}^{-1}(p)+b$ if
$a\in\mathbb{R}_{+}$ and $b\in L^{0}(\mathcal{F}_{t},\operatorname{P})$ (in
particular if $b\in\mathbb{R}$). Monetary conditional risk measures appear
naturally for multi-period valuations, see e.g. Chapter 11 in Föllmer and
Schied [6]. The risk measures Value-at-Risk and Average Value-at-Risk,
conditional on $\mathcal{F}_{t}$ and for $u\in(0,1)$, are given by

$\displaystyle\operatorname{V@R}_{u}(Y\mid\mathcal{F}_{t})$
$\displaystyle=F_{-Y\mid\mathcal{F}_{t}}^{-1}(1-u),$
$\displaystyle\operatorname{AV@R}_{u}(Y\mid\mathcal{F}_{t})$
$\displaystyle=\frac{1}{u}\int_{0}^{u}\operatorname{V@R}_{v}(Y\mid\mathcal{F}_{t})dv.$

In [6], $\operatorname{V@R}_{u}(Y\mid\mathcal{F}_{t})$ is defined as
$\operatorname{essinf}\\{m_{t}\in
L^{0}(\mathcal{F}_{t},\operatorname{P}):\operatorname{P}(Y+m_{t}<0\mid\mathcal{F}_{t})\leq
u\\}$ but the two definitions of
$\operatorname{V@R}_{u}(Y\mid\mathcal{F}_{t})$ are equal almost surely. For
$p\geq 1$, $\operatorname{V@R}_{u}(Y\mid\mathcal{F}_{t})\in
L^{p}(\mathcal{F}_{t},\operatorname{P})$ if $Y\in
L^{p}(\mathcal{F},\operatorname{P})$, and similarly for
$\operatorname{AV@R}_{u}(Y\mid\mathcal{F}_{t})$. Let
$\mathcal{P}([0,1])^{\prime}$ denote the subset of $\mathcal{P}([0,1])$
consisting of measures $\mu\in\mathcal{P}([0,1])$ that either have a bounded
density with respect to Lebesgue measure or satisfy
$\operatorname{supp}(\mu)\subset[a,b]$ for some $0<a<b<1$. Both
$\operatorname{V@R}_{u}(Y\mid\mathcal{F}_{t})$ and
$\operatorname{AV@R}_{u}(Y\mid\mathcal{F}_{t})$ can be expressed as

$\displaystyle\rho(Y\mid\mathcal{F}_{t})=\int_{0}^{1}F_{-Y\mid\mathcal{F}_{t}}^{-1}(p)\mu(dp),\quad\mu\in\mathcal{P}([0,1])^{\prime}.$
(2)

$\operatorname{V@R}_{u}(Y\mid\mathcal{F}_{t})$ corresponds to
$\mu(dp)=\delta_{1-u}(dp)$ (a unit point mass at $1-u$) and
$\operatorname{AV@R}_{u}(Y\mid\mathcal{F}_{t})$ corresponds to
$\mu(dp)=u^{-1}I\\{p\in[1-u,1]\\}(dp)$ (a bounded density). Conditional risk
measures of the form (2) satisfy
$\rho(aY+b\mid\mathcal{F}_{t})=a\rho(Y\mid\mathcal{F}_{t})-b$ if
$a\in\mathbb{R}_{+}$ and $b$ is $\mathcal{F}_{t}$-measurable, called positive
homogeneity and conditional cash additivity. Moreover, they are monotone:
$\widetilde{Y}\geq Y$ implies
$\rho(\widetilde{Y}\mid\mathcal{F}_{t})\leq\rho(Y\mid\mathcal{F}_{t})$.
Throughout the paper (in)equalities between random variables should be
interpreted in the almost sure sense.

##  3 Convergence of multi-period valuations

Fix $T\in\mathbb{N}$ and let $X,X^{1},X^{2},\dots$ be random vectors in
$\mathbb{R}^{T}$ and let $Y,Y^{1},Y^{2},\dots$ be random vectors in
$(\mathbb{R}^{d})^{T}$, $d\in\mathbb{N}$. Suppose that
$\mathcal{L}(X^{n},Y^{n})\stackrel{{\scriptstyle w}}{{\to}}\mathcal{L}(X,Y)$
as $n\to\infty$, where $\mathcal{L}(X,Y)$ is Gaussian. Set
$\mathcal{F}_{t}=\sigma((X,Y)_{\leq t})$,
$\mathcal{F}^{n}_{t}=\sigma((X^{n},Y^{n})_{\leq t})$. $(X,Y)$ and
$(X^{n},Y^{n})$ are adapted discrete-time stochastic processes with respect to
the filtrations $(\mathcal{F}_{t})_{t=0}^{T}$ and
$(\mathcal{F}^{n}_{t})_{t=0}^{T}$, respectively, where
$\mathcal{F}_{0}=\mathcal{F}^{n}_{0}=\\{\Omega,\emptyset\\}$. For each $n$,
$X^{n}=(X^{n}_{t})_{t=1}^{T}$ corresponds to a discounted incremental cashflow
in a multi-period model with $T$ periods and time points $0,1,\dots,T$. For
each $n$, $Y^{n}=(Y^{n}_{t})_{t=1}^{T}$ corresponds to a stochastic process
that provides additional information, additional to the information provided
by the process $X^{n}$. Taking $Y^{n}$ to be nonrandom means that no such
additional information is considered and that
$(\mathcal{F}^{n}_{t})_{t=0}^{T}$ is the natural filtration generated by
$X^{n}$. The discounted value at time $t$ of the cashflow occurring after time
$t$ is denoted by $V^{n}_{t}(X^{n})$. Since no cashflows occur after time $T$,
$V^{n}_{T}(X^{n})=0$. The process of values of the cashflow $X^{n}$ is
$(V^{n}_{t}(X^{n}))_{t=0}^{T}$. Both $X^{n}_{t}$ and $V^{n}_{t}(X^{n})$ are
$\mathcal{F}^{n}_{t}$-measurable. The processes $(V_{t}(X))_{t=0}^{T}$ and
$(V^{n}_{t}(X^{n}))_{t=0}^{T}$ are defined backward recursively by

$\displaystyle V_{T}(X)=0,\quad V_{t}(X)=\varphi_{t}(X_{t+1}+V_{t+1}(X)),\quad
t<T,$ (3) $\displaystyle V^{n}_{T}(X^{n})=0,\quad
V^{n}_{t}(X^{n})=\varphi^{n}_{t}(X^{n}_{t+1}+V^{n}_{t+1}(X^{n})),\quad t<T,$
(4)

where $\varphi_{t}:L^{1}(\mathcal{F}_{T})\to L^{1}(\mathcal{F}_{t})$ and
$\varphi^{n}_{t}:L^{1}(\mathcal{F}^{n}_{T})\to L^{1}(\mathcal{F}^{n}_{t})$ are
mappings that are made precise below. In particular, the mappings
$\varphi_{t}$ and $\varphi^{n}_{t}$ will satisfy

$\displaystyle\varphi_{t}(aY+\widetilde{Y})=a\varphi_{t}(Y)+\widetilde{Y},\quad\varphi^{n}_{t}(aY+\widetilde{Y})=a\varphi^{n}_{t}(Y)+\widetilde{Y}\quad\text{for
}a\in\mathbb{R}_{+},\widetilde{Y}\in L^{1}(\mathcal{F}_{t}).$ (5)

These properties are called positive homogeneity (the effect of multiplication
by a nonnegative scalar) and conditional cash additivity. As a consequence,
for $a\in\mathbb{R}_{+},b\in\mathbb{R}^{T}$,

$\displaystyle V_{t}(aX+b)=aV_{t}(X)+\sum_{s=t+1}^{T}b_{s},\quad
V^{n}_{t}(aX^{n}+b)=aV^{n}_{t}(X^{n})+\sum_{s=t+1}^{T}b_{s}.$ (6)

Theorems 1 and 2 presented below essentially say that if
$\mathcal{L}(X^{n},Y^{n})\stackrel{{\scriptstyle w}}{{\to}}\mathcal{L}(X,Y)$
as $n\to\infty$ and also in terms of conditional distributions (see e.g. (15))
together with a uniform integrability assumption (see e.g. (16)), then
$\lim_{n\to\infty}V^{n}_{0}(X^{n})=V_{0}(X)$, where the values
$V^{n}_{0}(X^{n})$ and $V_{0}(X)$ are defined with respect to the filtrations
generated by $(X^{n},Y^{n})$ and $(X,Y)$, respectively. The essential point is
that $V^{n}_{0}(X^{n})$ is typically impossible to compute analytically and
only with significant difficulties numerically, whereas $V_{0}(X)$ has an
explicit expression that is straightforward to compute in terms of the mean
vector and covariance matrix of the Gaussian weak limit (see e.g. (20)).
Therefore, the theorems we present enable the use of conceptually sound
valuation techniques without the significant efforts otherwise needed to
obtain numerical solutions to backward recursions.

###  3.1 Economic motivation

Current regulatory frameworks for the insurance industry prescribe so-called
cost-of-capital valuation of liability cashflows. Such approaches to valuation
consider capital requirements and the costs stemming from the financing of
buffer capital. In the multi-period setting, with capital requirements
determined one period at the time, the randomness of future capital
requirements and associated costs drive the liability valuation.

Multiperiod cost-of-capital valuation is studied in Salzmann and Wüthrich
[12], Möhr [9], Pelsser and Salahnejhad Ghalehjooghi [10] and Engsner et al.
[5]. A common theme is that multiperiod valuations is constructed through
backward induction of one-period valuations. Pelsser and Salahnejhad
Ghalehjooghi [10] study continuous-time limits of multiperiod valuations
defined in terms of one-step valuations, similar to those considered below and
including cost-of-capital valuation. In [10] the convergence takes place as
the length of the time periods tends to zero and the number of time periods
tends to infinity. Clearly, this is a different kind of convergence than that
studied here. Multiperiod valuations are so-called time-consistent by their
construction through the backward induction of one-period valuations. For
mathematical properties of general multiperiod valuations, we refer to
Cheridito et al [4], Artzner et al. [3] and Jobert and Rogers [7].

We now explain the basic ingredients of multi-period cost-of-capital
valuation. Let $V_{0}$ be the value at time $0$ of the liability cashflow $X$.
This amount should be interpreted as the capital that needs to be transferred
along with the liability to another external agent in order for the external
agent (or capital provider) to accept managing the liability runoff and the
associated capital costs. Let $V_{t}$ denote the value at time $t$ of the
liability cashflows $X_{>t}$. The capital $V_{t}$ is reserved at time $t$ for
managing the liability. However, regulation requires the capital
$R_{t}=\rho(-X_{t+1}-V_{t+1}\mid\mathcal{F}_{t})>V_{t}$ to be set aside at
time $t$. The difference $R_{t}-V_{t}$ is made available by a capital provider
requiring an excess expected return $1+\eta_{t}$ on the provided capital
between time $t$ and $t+1$. The acceptability criterion under which the
capital provider accepts to provide capital gives the equation

$\displaystyle\operatorname{E}[R_{t}-X_{t+1}-V_{t+1}\mid\mathcal{F}_{t}]=(1+\eta_{t})(R_{t}-V_{t}).$
(7)

Solving for $V_{t}$ yields $V_{t}=\varphi_{t}(X_{t+1}+V_{t+1})$, where

$\displaystyle\varphi_{t}(Y)=\frac{1}{1+\eta_{t}}\operatorname{E}[Y\mid\mathcal{F}_{t}]+\frac{\eta_{t}}{1+\eta_{t}}\rho(-Y\mid\mathcal{F}_{t}).$
(8)

A commonly used conditional risk measure for variables with finite variance is
$\rho^{\text{SD}}(Y\mid\mathcal{F}_{t})=\operatorname{E}[-Y\mid\mathcal{F}_{t}]+c^{\text{SD}}\operatorname{var}(Y\mid\mathcal{F}_{t})^{1/2}$.
Note that it can be argued that $\rho^{\text{SD}}(Y\mid\mathcal{F}_{t})$ is an
inappropriate choice because it violates the monotonicity property:
$\widetilde{Y}\geq Y$ does not imply
$\rho^{\text{SD}}(Y\mid\mathcal{F}_{t})\geq\rho^{\text{SD}}(\widetilde{Y}\mid\mathcal{F}_{t})$.
For this choice of conditional risk measure, (8) takes the form

$\displaystyle\varphi_{t}(Y)=\operatorname{E}[Y\mid\mathcal{F}_{t}]+c_{t}\operatorname{var}(Y\mid\mathcal{F}_{t})^{1/2},$
(9)

where $c_{t}=c^{\text{SD}}\eta_{t}/(1-\eta_{t})$, and $\varphi_{t}(Y)\in
L^{2}(\mathcal{F}_{t})$ if $Y\in L^{2}(\mathcal{F}_{T})$.

It can be argued that the capital provider, seen as the share holder of the
company, has limited liability and is not required to continue injecting
capital if the the value $R_{t}$ of available asset turns out insufficient to
match the value $X_{t+1}+V_{t+1}$ of the liability towards the policy holders.
For discussions on limited liability in the context of cost-of-capital
valuation we refer to Albrecher et al. [1] and Möhr [9]. In the setting with
limited liability, (7) is replaced by

$\displaystyle\operatorname{E}[(R_{t}-X_{t+1}-V_{t+1})^{+}\mid\mathcal{F}_{t}]=(1+\eta_{t})(R_{t}-V_{t})$
(10)

and (8) is replaced by

$\displaystyle\varphi_{t}(Y)=\rho(-Y\mid\mathcal{F}_{t})-\frac{1}{1+\eta_{t}}\operatorname{E}[(\rho(-Y\mid\mathcal{F}_{t})-Y)^{+}\mid\mathcal{F}_{t}].$
(11)

Regardless of whether we consider cost-of-capital valuation with or without
limited liability, it follows that
$\varphi_{t}(aY+\widetilde{Y})=a\varphi_{t}(Y)+\widetilde{Y}$ for
$a\in\mathbb{R}_{+}$ and $\widetilde{Y}\in L^{1}(\mathcal{F}_{t})$ if the
conditional risk measure satisfies
$\rho(aY+\widetilde{Y}\mid\mathcal{F}_{t})=a\rho(Y\mid\mathcal{F}_{t})-\widetilde{Y}$.
Moreover, $\varphi_{t}$ inherits monotonicity from the conditional risk
measure: if $\widetilde{Y}\geq Y$ implies
$\rho(Y\mid\mathcal{F}_{t})\geq\rho(\widetilde{Y}\mid\mathcal{F}_{t})$, then
$\varphi_{t}(\widetilde{Y})\geq\varphi_{t}(Y)$.

An alternative to an acceptability criterion based on expected excess return
and a cost-of-capital rate is an acceptability criterion saying that a risk
averse capital provider provides capital if the payoff resulting from
providing capital is preferred, in terms of expected utility, to simply
rolling this capital forward by investing it in a riskless bond (or more
generally, investing it in the numeraire asset). In this setting, the
acceptability criterion (10) is replaced by

$\displaystyle\operatorname{E}[u_{t}((R_{t}-X_{t+1}-V_{t+1})^{+})\mid\mathcal{F}_{t}]=u_{t}(R_{t}-V_{t}),$
(12)

where $u_{t}$ is an increasing and concave (utility) function. Consequently,
(11) is replaced by

$\displaystyle\varphi_{t}(Y)=\rho(-Y\mid\mathcal{F}_{t})-u_{t}^{-1}\Big{(}\operatorname{E}\Big{[}u_{t}((\rho(-Y\mid\mathcal{F}_{t})-Y)^{+})\mid\mathcal{F}_{t}\Big{]}\Big{)}.$
(13)

Also in this case $\varphi_{t}$ inherits monotonicity and the property
$\varphi_{t}(Y+\widetilde{Y})=\varphi_{t}(Y)+\widetilde{Y}$, $\widetilde{Y}\in
L^{1}(\mathcal{F}_{t})$, from a monotone and conditionally cash additive
conditional risk measure. However, the property
$\varphi_{t}(aY+\widetilde{Y})=a\varphi_{t}(Y)+\widetilde{Y}$,
$a\in\mathbb{R}_{+}$, requires that $u_{t}$ is chosen a power utility function
$u_{t}(x)=\alpha_{t}x^{\beta_{t}}$, where $\alpha_{t}>0$ and
$\beta_{t}\in(0,1]$. In this case, $\varphi_{t}$ in (13) takes the form

$\displaystyle\varphi_{t}(Y)=\rho(-Y\mid\mathcal{F}_{t})-\operatorname{E}\Big{[}\big{(}(\rho(-Y\mid\mathcal{F}_{t})-Y)^{+}\big{)}^{\beta_{t}}\mid\mathcal{F}_{t}\Big{]}^{1/\beta_{t}}.$
(14)

The mappings $\varphi_{t}:L^{1}(\mathcal{F}_{T})\to L^{1}(\mathcal{F}_{t})$
given by (8), (11) or (14) all satisfy
$\varphi_{t}(Y+\widetilde{Y})=\varphi_{t}(Y)+\widetilde{Y}$ whenever
$\widetilde{Y}\in L^{1}(\mathcal{F}_{t})$. Therefore, (3) can be expressed as,
with $\circ$ denoting composition,

$\displaystyle V_{T}(X)=0,\quad
V_{t}(X)=\varphi_{t}\circ\dots\circ\varphi_{T-1}\bigg{(}\sum_{s=t+1}^{T}X_{s}\bigg{)},\quad
t<T.$

These properties also hold for the mappings $\varphi_{t}$ in (9) with $L^{1}$
replaced by $L^{2}$.

###  3.2 Main results

The main results consist of Theorems 1, 2 and 3. Theorems 1 and 2 presents
conditions under which we have convergence for multi-period values (that are
typically not computable) to a computable explicit limit. Theorem 3 presents a
monotonicity result for the limit expressions in terms of a partial order
between filtrations.

######  Theorem 1.

Let $X,X^{1},X^{2},\dots$ be random vectors in $\mathbb{R}^{T}$ and let
$Y,Y^{1},Y^{2},\dots$ be random vectors in $(\mathbb{R}^{d})^{T}$,
$d\in\mathbb{N}$. Suppose that
$\mathcal{L}(X^{n},Y^{n})\stackrel{{\scriptstyle w}}{{\to}}\mathcal{L}(X,Y)$
as $n\to\infty$, where $\mathcal{L}(X,Y)$ is Gaussian, and that
$(\mathcal{L}(\|X^{n}\|))_{n\in\mathbb{N}}$ is uniformly integrable. Suppose
also that, for each $t$ and each convergent sequence $(x^{n},y^{n})_{\leq
t}\to(x,y)_{\leq t}$ with $(x^{n},y^{n})_{\leq
t}\in\operatorname{supp}(\mathcal{L}((X^{n},Y^{n})_{\leq t}))$,

$\displaystyle\mathcal{L}\Big{(}(X^{n},Y^{n})\mid(X^{n},Y^{n})_{\leq
t}=(x^{n},y^{n})_{\leq t}\Big{)}\stackrel{{\scriptstyle
w}}{{\to}}\mathcal{L}\Big{(}(X,Y)\mid(X,Y)_{\leq t}=(x,y)_{\leq
t}\Big{)}\quad\text{as }n\to\infty,$ (15)
$\displaystyle\Big{(}\mathcal{L}\big{(}\|X^{n}\|\mid(X^{n},Y^{n})_{\leq
t}=(x^{n},y^{n})_{\leq t}\big{)}\Big{)}_{n\in\mathbb{N}}\quad\text{is
uniformly integrable}.$ (16)

For all $t$, let $V_{t}(X)$ and $V^{n}_{t}(X^{n})$ be given by (3) and (4)
with $\varphi_{t}$ and $\varphi^{n}_{t}$ given by either (8) and

$\displaystyle\varphi^{n}_{t}(Y)=\frac{1}{1+\eta_{t}}\operatorname{E}[Y\mid\mathcal{F}^{n}_{t}]+\frac{\eta_{t}}{1+\eta_{t}}\rho(-Y\mid\mathcal{F}^{n}_{t}),\quad\eta_{t}\in\mathbb{R}_{+},$
(17)

or by (11) and

$\displaystyle\varphi^{n}_{t}(Y)=\rho(-Y\mid\mathcal{F}^{n}_{t})-\frac{1}{1+\eta_{t}}\operatorname{E}\Big{[}\big{(}\rho(-Y\mid\mathcal{F}^{n}_{t})-Y\big{)}^{+}\mid\mathcal{F}^{n}_{t}\Big{]},\quad\eta_{t}\in\mathbb{R}_{+},$
(18)

or by (14) and

$\displaystyle\varphi^{n}_{t}(Y)=\rho(-Y\mid\mathcal{F}^{n}_{t})-\operatorname{E}\Big{[}\big{(}(\rho(-Y\mid\mathcal{F}^{n}_{t})-Y)^{+}\big{)}^{\beta_{t}}\mid\mathcal{F}^{n}_{t}\Big{]}^{1/\beta_{t}},\quad\beta_{t}\in(0,1],$
(19)

where $\mathcal{F}_{t}=\sigma((X,Y)_{\leq t})$ and
$\mathcal{F}^{n}_{t}=\sigma((X^{n},Y^{n})_{\leq t})$, and where
$\rho(-Y\mid\mathcal{F}_{t})$ and $\rho(-Y\mid\mathcal{F}^{n}_{t})$ are of the
form (2). Then $\lim_{n\to\infty}V^{n}_{0}(X^{n})=V_{0}(X)$, where

$\displaystyle
V_{0}(X)=\operatorname{E}\bigg{[}\sum_{t=1}^{T}X_{t}\bigg{]}+\sum_{t=1}^{T}\varphi_{t-1}(\varepsilon_{t})\bigg{(}\operatorname{Var}\bigg{(}\sum_{u=t}^{T}X_{u}\mid(X,Y)_{\leq
t-1}\bigg{)}-\operatorname{Var}\bigg{(}\sum_{u=t}^{T}X_{u}\mid(X,Y)_{\leq
t}\bigg{)}\bigg{)}^{1/2},$ (20)

where $\varepsilon_{t}$ is standard normally distributed and independent of
$\mathcal{F}_{t-1}$.

######  Remark 1.

Notice that the conditional variances in (20) are nonrandom since the (joint)
distribution of $(X,Y)$ is Gaussian. Notice also
$\mathcal{F}_{t}=\sigma((X,Y)_{\leq t})$ generates the natural filtration of
$X$ if $Y$ is chosen as nonrandom (a degenerate Gaussian process). We
emphasize that if the $\eta_{t}$ in (8) and (11) do not depend on $t$, and
similarly if the $\beta_{t}$ in (19) do not depend on $t$, then
$\varphi_{t-1}(\varepsilon_{t})=\varphi_{0}(\varepsilon_{1})$ in (20) does not
depend on $t$.

######  Remark 2.

Joint weak convergence $\mathcal{L}(X^{n},Y^{n})\stackrel{{\scriptstyle
w}}{{\to}}\mathcal{L}(X,Y)$ as $n\to\infty$ does not imply conditional weak
convergence such as the convergence in (15). A counterexample as well as
sufficient conditions for conditional weak convergence are presented in [13].

######  Remark 3.

For $\varepsilon$ is standard normally distributed and independent of
$\mathcal{F}_{t-1}$,

$\displaystyle\operatorname{V@R}_{u}(\varepsilon\mid\mathcal{F}_{t-1})=\Phi^{-1}(1-u),\quad\operatorname{AV@R}_{u}(\varepsilon\mid\mathcal{F}_{t-1})=\frac{1}{u}\varphi\left(\Phi^{-1}(1-u)\right),$

where $\varphi$ and $\Phi$ here denote the standard normal density and
distribution function, respectively. Hence, with $\rho_{0}(\varepsilon)$
denoting either $\operatorname{V@R}_{u}(\varepsilon\mid\mathcal{F}_{t-1})$ or
$\operatorname{AV@R}_{u}(\varepsilon\mid\mathcal{F}_{t-1})$ with values above
that do not depend on $t$, we get

$\varphi_{t-1}(\varepsilon)=\frac{\eta_{t-1}}{1+\eta_{t-1}}\rho_{0}(\varepsilon)$

in the case of (8), and

$\displaystyle\varphi_{t-1}(\varepsilon)=\rho_{0}(\varepsilon)-\frac{1}{1+\eta_{t-1}}\Big{(}\rho_{0}(\varepsilon)\Phi(\rho_{0}(\varepsilon))+\varphi(\rho_{0}(\varepsilon))\Big{)}\leq\frac{\eta_{t-1}}{1+\eta_{t-1}}\rho_{0}(\varepsilon)$

in the case of (11). In case of (14), the expectation
$\operatorname{E}[((\rho_{0}(\varepsilon)-\varepsilon)^{+})^{\beta_{t-1}}]$
has to be computed numerically in order to compute
$\varphi_{t-1}(\varepsilon)$.

######  Remark 4.

As stated in Theorem 1, the result holds when $\varphi_{t}$ and
$\varphi^{n}_{t}$ are given by (8) and (17). However, the theorem is actually
proven for mappings of a more general kind, namely, for $\lambda_{t}\in[0,1]$
and $\mu^{1}_{t},\mu^{2}_{t}\in\mathcal{P}([0,1])^{\prime}$, the mappings
$\varphi_{t}:L^{1}(\mathcal{F}_{T})\to L^{1}(\mathcal{F}_{t})$,
$\varphi^{n}_{t}:L^{1}(\mathcal{F}^{n}_{T})\to L^{1}(\mathcal{F}^{n}_{t})$
given by (21) and (22) below:

$\displaystyle\varphi_{t}(Y)$ $\displaystyle=\lambda_{t}\int
F^{-1}_{Y\mid\mathcal{F}_{t}}(p)\mu_{t}^{1}(dp)+(1-\lambda_{t})\int
F^{-1}_{-Y\mid\mathcal{F}_{t}}(p)\mu_{t}^{2}(dp),$ (21)
$\displaystyle\varphi^{n}_{t}(Y)$ $\displaystyle=\lambda_{t}\int
F^{-1}_{Y\mid\mathcal{F}^{n}_{t}}(p)\mu_{t}^{1}(dp)+(1-\lambda_{t})\int
F^{-1}_{-Y\mid\mathcal{F}^{n}_{t}}(p)\mu_{t}^{2}(dp).$ (22)

Note that (8) and (17) correspond to $\mu_{t}^{1}(dp)=dp$ and
$\lambda_{t}=(1+\eta_{t})^{-1}$.

######  Remark 5.

Both (11), (18) and (14), (19) are special cases of

$\displaystyle\varphi_{t}(Y)$
$\displaystyle=\rho(-Y\mid\mathcal{F}_{t})-\gamma_{t}\operatorname{E}\Big{[}\big{(}(\rho(-Y\mid\mathcal{F}_{t})-Y)^{+}\big{)}^{\beta_{t}}\mid\mathcal{F}_{t}\Big{]}^{1/\beta_{t}},$
(23) $\displaystyle\varphi^{n}_{t}(Y)$
$\displaystyle=\rho(-Y\mid\mathcal{F}^{n}_{t})-\gamma_{t}\operatorname{E}\Big{[}\big{(}(\rho(-Y\mid\mathcal{F}^{n}_{t})-Y)^{+}\big{)}^{\beta_{t}}\mid\mathcal{F}^{n}_{t}\Big{]}^{1/\beta_{t}},$
(24)

as seen by choosing $(\gamma_{t},\beta_{t})=((1+\eta_{t})^{-1},1)$ in case of
(11), (18), and $\gamma_{t}=1$ in case of (14), (19).

Our next result shows convergence of multi-period values in the case of where
the one period valuation mapping is defined as a sum of a conditional
expectation and a constant times a conditional standard deviation. Let
$c_{t}\in\mathbb{R}_{+}$ and $\varphi_{t}:L^{2}(\mathcal{F}_{T})\to
L^{2}(\mathcal{F}_{t})$, $\varphi^{n}_{t}:L^{2}(\mathcal{F}^{n}_{T})\to
L^{2}(\mathcal{F}^{n}_{t})$ be given by

$\displaystyle\varphi_{t}(Y)$
$\displaystyle=\operatorname{E}\big{[}Y\mid\mathcal{F}_{t}\big{]}+c_{t}\operatorname{var}\big{[}Y\mid\mathcal{F}_{t}\big{]}^{1/2},$
(25) $\displaystyle\varphi^{n}_{t}(Y)$
$\displaystyle=\operatorname{E}\big{[}Y\mid\mathcal{F}^{n}_{t}\big{]}+c_{t}\operatorname{var}\big{[}Y\mid\mathcal{F}^{n}_{t}\big{]}^{1/2}.$
(26)

$(V_{t}(X))_{t=0}^{T}$ and $(V^{n}_{t}(X^{n}))_{t=0}^{T}$ are defined backward
recursively by (3) and (4), and (5) and (6) hold.

######  Theorem 2.

Let $X,X^{1},X^{2},\dots$ be random vectors in $\mathbb{R}^{T}$ and let
$Y,Y^{1},Y^{2},\dots$ be random vectors in $(\mathbb{R}^{d})^{T}$,
$d\in\mathbb{N}$. Suppose that
$\mathcal{L}(X^{n},Y^{n})\stackrel{{\scriptstyle w}}{{\to}}\mathcal{L}(X,Y)$
as $n\to\infty$, where $\mathcal{L}(X,Y)$ is Gaussian, and such that
$(\mathcal{L}(\|X^{n}\|^{2}))_{n\in\mathbb{N}}$ is uniformly integrable.
Suppose also that, for each $t$ and each convergent sequence
$(x^{n},y^{n})_{\leq t}\to(x,y)_{\leq t}$ with $(x^{n},y^{n})_{\leq
t}\in\operatorname{supp}(\mathcal{L}((X^{n},Y^{n})_{\leq t}))$,

$\displaystyle\mathcal{L}\Big{(}(X^{n},Y^{n})\mid(X^{n},Y^{n})_{\leq
t}=(x^{n},y^{n})_{\leq t}\Big{)}\stackrel{{\scriptstyle
w}}{{\to}}\mathcal{L}\Big{(}(X,Y)\mid(X,Y)_{\leq t}=(x,y)_{\leq
t}\Big{)}\quad\text{as }n\to\infty,$ (27)
$\displaystyle\Big{(}\mathcal{L}(\|X^{n}\|^{2}\mid(X^{n},Y^{n})_{\leq
t}=(x^{n},y^{n})_{\leq t})\Big{)}_{n\in\mathbb{N}}\quad\text{is uniformly
integrable}.$ (28)

Let $V_{t}(X)$ and $V^{n}_{t}(X^{n})$ be given by (3) and (4) with
$\varphi_{t}$ and $\varphi^{n}_{t}$ given by (25) and (26) with
$\mathcal{F}_{t}=\sigma((X,Y)_{\leq t})$ and
$\mathcal{F}^{n}_{t}=\sigma((X^{n},Y^{n})_{\leq t})$. Then
$\lim_{n\to\infty}V^{n}_{0}(X^{n})=V_{0}(X)$, where

$\displaystyle
V_{0}(X)=\operatorname{E}\bigg{[}\sum_{t=1}^{T}X_{t}\bigg{]}+\sum_{t=1}^{T}\varphi_{t-1}(\varepsilon_{t})\bigg{(}\operatorname{Var}\bigg{(}\sum_{u=t}^{T}X_{u}\mid(X,Y)_{\leq
t-1}\bigg{)}-\operatorname{Var}\bigg{(}\sum_{u=t}^{T}X_{u}\mid(X,Y)_{\leq
t}\bigg{)}\bigg{)}^{1/2},$

where $\varepsilon_{t}$ is standard normally distributed and independent of
$\mathcal{F}_{t-1}$.

The final theorem considers the Gaussian limit models appearing in Theorems 1
and 2 and establishes a partial ordering of values $V_{0}(X)$ assigned to a
given cashflow $X$ depending on a partial order between filtrations to which
$X$ is adapted. One may guess that a larger filtration allows for more
accurate predictions over time of future cashflows and thereby reduces the
value $V_{0}(X)$. We show that this holds if the parameters of the one-step
valuation mappings $\varphi_{t}$ do not depend on $t$ and if the conditional
variance of the aggregate cashflow $\sum_{s=1}^{T}X_{s}$ decays in a convex
manner over time as more information becomes available.

We consider filtrations $\mathbb{F}=(\mathcal{F}_{t})_{t=0}^{T}$ with
$\mathcal{F}_{0}=\\{\Omega,\emptyset\\}$ and
$\mathcal{F}_{t}=\sigma((X,Y)_{\leq t})$, for $t\geq 1$, where $(X,Y)$ is
jointly Gaussian. We assume that the one-step valuation mappings $\varphi_{t}$
do not depend on $t$ such that
$\varphi_{t-1}(\varepsilon_{t})=\varphi_{0}(\varepsilon_{1})$ in (20). We
write

$\displaystyle
V_{0}(X,\mathbb{F})=\operatorname{E}\bigg{[}\sum_{t=1}^{T}X_{t}\bigg{]}+\varphi_{0}(\varepsilon_{1})\sum_{t=1}^{T}\bigg{(}$
$\displaystyle\operatorname{Var}\bigg{(}\sum_{u=t}^{T}X_{u}\mid\mathcal{F}_{t-1}\bigg{)}-\operatorname{Var}\bigg{(}\sum_{u=t}^{T}X_{u}\mid\mathcal{F}_{t}\bigg{)}\bigg{)}^{1/2}.$

The following theorem says that for filtrations $\mathbb{F}$ and $\mathbb{G}$
of the above kind (generated by a process that is jointly Gaussian together
with the cashflow process $X$), if
$t\mapsto\operatorname{Var}(\sum_{s=1}^{T}X_{s}\mid\mathcal{F}_{t})$ is
convex, then $V_{0}(X,\mathbb{F})\geq V_{0}(X,\mathbb{G})$ whenever
$\mathcal{F}_{t}\subseteq\mathcal{G}_{t}$ for every $t$. The convexity
assumption is necessary.

######  Theorem 3.

Let $(X,Y)_{t=1}^{T}$ and $(X,Z)_{t=1}^{T}$ be Gaussian processes. Let
$\mathbb{F}=(\mathcal{F}_{t})_{t=0}^{T}$ and
$\mathbb{G}=(\mathcal{G}_{t})_{t=0}^{T}$ be filtrations with
$\mathcal{F}_{0}=\mathcal{G}_{0}=\\{\Omega,\emptyset\\}$ and
$\mathcal{F}_{t}=\sigma((X,Y)_{\leq t})$ and
$\mathcal{G}_{t}=\sigma((X,Z)_{\leq t})$, for $t\geq 1$. If
$\mathcal{F}_{t}\subseteq\mathcal{G}_{t}$ for every $t$ and if
$t\mapsto\operatorname{Var}(\sum_{s=1}^{T}X_{s}\mid\mathcal{F}_{t})$ is
convex, then $V_{0}(X,\mathbb{F})\geq V_{0}(X,\mathbb{G})$.

######  Remark 6.

We emphasize that the Gaussian assumption in Theorem 3 means that the terms

$\displaystyle
c_{t}=\operatorname{Var}\bigg{(}\sum_{s=1}^{T}X_{s}\mid\mathcal{F}_{t-1}\bigg{)}-\operatorname{Var}\bigg{(}\sum_{s=1}^{T}X_{s}\mid\mathcal{F}_{t}\bigg{)}$

form a nonrandom sequence $(c_{t})_{t=1}^{T}$ and that the convexity property
is equivalent to $c_{1}\geq c_{2}\geq\dots\geq c_{T}$.

##  4 Proofs and auxiliary results

We use the notation $Z^{n}_{\leq t}=(X^{n},Y^{n})_{\leq t}$, $Z_{\leq
t}=(X,Y)_{\leq t}$ and $z^{n}_{\leq t}=(x^{n},y^{n})_{\leq t}$, $z_{\leq
t}=(x,y)_{\leq t}$.

###### Proof of Theorem 1 assuming (21) and (22).

As explained in Remark 4, (8) and (17) are special cases of (21) and (22).

For $t=1,\dots,T$, consider mappings
$\psi^{n}_{t}:\operatorname{supp}(\mathcal{L}(Z^{n}_{\leq t}))\to\mathbb{R}$
given by $\psi^{n}_{T}(z^{n}_{\leq T})=\sum_{s=1}^{T}x^{n}_{s}$ and

$\displaystyle\psi^{n}_{t}(z^{n}_{\leq t})=\lambda_{t}\int
F^{-1}_{\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\mid Z^{n}_{\leq t}=z^{n}_{\leq
t}}(p)\mu_{t}^{1}(dp)+(1-\lambda_{t})\int F^{-1}_{-\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})\mid Z^{n}_{\leq t}=z^{n}_{\leq t}}(p)\mu_{t}^{2}(dp),$ (29)

and let

$\displaystyle\psi^{n}_{0}=\lambda_{0}\int
F^{-1}_{\psi^{n}_{1}(Z^{n}_{1})}(p)\mu_{0}^{1}(dp)+(1-\lambda_{0})\int
F^{-1}_{-\psi^{n}_{1}(Z^{n}_{1})}(p)\mu_{0}^{2}(dp),$ (30)

where $\mu_{t}^{1},\mu_{t}^{2}\in\mathcal{P}([0,1])^{\prime}$. We define
$\psi_{t}:\mathbb{R}^{t}\to\mathbb{R}$ and $\psi_{0}$ analogously without the
superscript $n$. Note that

$\displaystyle\psi^{n}_{t}(Z^{n}_{\leq t})$
$\displaystyle=\sum_{s=1}^{t}X^{n}_{s}+V^{n}_{t}(X^{n})=\varphi^{n}_{t}(\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})),\;t\geq
1,\quad\psi^{n}_{0}=V^{n}_{0}(X)=\varphi^{n}_{0}(\psi^{n}_{1}(Z^{n}_{\leq
1})),$ $\displaystyle\psi_{t}(Z_{\leq t})$
$\displaystyle=\sum_{s=1}^{t}X_{s}+V_{t}(X)=\varphi_{t}(\psi_{t+1}(Z_{\leq
t+1})),\;t\geq 1,\quad\psi_{0}=V_{0}(X)=\varphi_{0}(\psi_{1}(Z_{\leq 1})).$

Therefore, the proof is complete once we show the convergence
$\lim_{n\to\infty}\psi^{n}_{0}=\psi_{0}$.

The argument of the proof is backwards induction. We will show that for
$t=T,\dots,1$,

  * UI($t$):

$(\mathcal{L}(\psi^{n}_{t}(Z^{n}_{\leq t})\mid Z^{n}_{\leq s}=z^{n}_{\leq
s}))_{n}$ is uniformly integrable for each $s=1,\dots,t-1$ and $z^{n}_{\leq
s}\to z_{\leq s}$,

  * CC($t$):

$(\psi^{n}_{t})_{n}$ is continuously convergent, i.e.
$\psi^{n}_{t}(z^{n}_{\leq t})\to\psi_{t}(z_{\leq t})$, whenever $z^{n}_{\leq
t}\to z_{\leq t}$.

Induction base: UI($T$) and CC($T$) hold.

For $s\in\\{1,\dots,T-1\\}$ and $z^{n}_{\leq s}\to z_{\leq s}$, UI($T$)
follows from (16) since norms on Euclidean spaces are equivalent and
$|\psi^{n}_{T}(z^{n}_{\leq
T})|=|\sum_{s=1}^{T}x^{n}_{s}|\leq\sum_{s=1}^{T}|x^{n}_{s}|$. Moreover,

$\psi^{n}_{T}(z^{n}_{\leq
T})=\sum_{s=1}^{T}x^{n}_{s}\to\sum_{s=1}^{T}x_{s}=\psi_{T}(z_{\leq T})$

whenever $z^{n}_{\leq T}\to z_{\leq T}$, yielding CC($T$).

Induction step: UI($t+1$), CC($t+1$) together imply UI($t$), CC($t$).

Fix $s\in\\{1,\dots,t-1\\}$ and sequence $(z^{n}_{\leq s})_{n}$ with
$z^{n}_{\leq s}\to z_{\leq s}$. Let $\kappa^{n}_{s+1,t+1}(z^{n}_{\leq
s},\cdot)$ and $\kappa_{s+1,t+1}(z_{\leq s},\cdot)$ be regular versions of the
conditional distributions

$\operatorname{P}((Z^{n}_{s+1},\dots,Z^{n}_{t+1})^{{\mathrm{T}}}\in\cdot\mid
Z^{n}_{\leq s}=z^{n}_{\leq
s})\quad\text{and}\quad\operatorname{P}((Z_{s+1},\dots,Z_{t+1})^{{\mathrm{T}}}\in\cdot\mid
Z_{\leq s}=z_{\leq s}).$

By assumption (15), $\kappa^{n}_{s+1,t+1}(z^{n}_{\leq
s},\cdot)\stackrel{{\scriptstyle w}}{{\to}}\kappa_{s+1,t+1}(z_{\leq
s},\cdot)$. Since $(\psi^{n}_{t+1}(z^{n}_{\leq s},\cdot))$ is continuously
convergent by the induction assumption CC($t+1$), by the generalized
continuous mapping theorem ([8, Theorem 4.27, Exercise 27]),

$\kappa^{n}_{s+1,t+1}(z^{n}_{\leq s},\cdot)\circ(\psi^{n}_{t+1}(z^{n}_{\leq
s},\cdot))^{-1}\stackrel{{\scriptstyle w}}{{\to}}\kappa_{s+1,t+1}(z_{\leq
s},\cdot)\circ(\psi_{t+1}(z_{\leq s},\cdot))^{-1},$

i.e.

$\displaystyle\mathcal{L}(\psi^{n}_{t+1}(Z_{\leq t+1}^{n})\mid Z^{n}_{\leq
s}=z^{n}_{\leq s})\stackrel{{\scriptstyle
w}}{{\to}}\mathcal{L}(\psi_{t+1}(Z_{\leq t+1})\mid Z_{\leq s}=z_{\leq s}).$
(31)

Lemma 3 shows that the limiting distribution in (31) is Gaussian. By Lemma 2,
there exists a constant $c>0$, such that

$|\psi^{n}_{t}(z)|\leq c\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})|\mid Z^{n}_{\leq t}=z].$

Hence, it is sufficient to show uniform integrability of

$\displaystyle\Big{(}\mathcal{L}\big{(}\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})|\mid Z^{n}_{\leq t}]\mid Z^{n}_{\leq s}=z^{n}_{\leq
s}\big{)}\Big{)}_{n}.$ (32)

Setting $\widetilde{W}^{n}=\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})|\mid Z^{n}_{\leq t}]$ and $W^{n}=|\psi^{n}_{t+1}(Z^{n}_{\leq t+1})|$,
this follows from Lemma 4, using UI($t+1$) and CC($t+1$). Hence, UI($t$)
holds.

We proceed by showing CC($t$), i.e. the convergence of

$\displaystyle\psi^{n}_{t}(z^{n}_{\leq t})$ $\displaystyle=\lambda_{t}\int
F^{-1}_{\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\mid Z^{n}_{\leq t}=z_{\leq
t}^{n}}(p)\mu_{t}^{1}(dp)$ (33) $\displaystyle\quad+(1-\lambda_{t})\int
F^{-1}_{-\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\mid Z^{n}_{\leq t}=z^{n}_{\leq
t}}(p)\mu_{t}^{2}(dp).$ (34)

We will prove the convergence assuming that $\mu^{1}_{t}$ has a bounded
density and that $\operatorname{supp}(\mu^{2}_{t})\subset[a,b]\subset(0,1)$.
Other possibilities for
$\mu^{1}_{t},\mu^{2}_{t}\in\mathcal{P}([0,1])^{\prime}$ are handled by the
same arguments as those shown below.

We start by the integral in (33) and will show convergence by an application
of Pratt’s Lemma [11]. Let $w:(0,1)\to\mathbb{R}$, bounded by some
$c^{\prime}>0$, be the density of $\mu_{t}^{1}$, i.e.
$\mu_{t}^{1}(dp)=w(p)dp$. Let

$f_{n}(p)=F^{-1}_{\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\mid Z^{n}_{\leq
t}=z^{n}_{\leq t}}(p)w(p).$

From (31) follows convergence

$\displaystyle f_{n}(p)\to F^{-1}_{\psi_{t+1}(Z_{\leq t+1})\mid Z_{\leq
t}=z_{\leq t}}(p)w(p)$

for almost every $p\in(0,1)$. By combining (31) and the continuous mapping
theorem applied to the absolute value function follows convergence of upper
and lower bounds $l_{n}(p)\leq f_{n}(p)\leq u_{n}(p)$ for almost every
$p\in(0,1)$:

$\displaystyle l_{n}(p)$
$\displaystyle=c^{\prime}F^{-1}_{-|\psi^{n}_{t+1}(Z^{n}_{\leq t+1})|\mid
Z^{n}_{\leq t}=z^{n}_{\leq t}}(p)\to c^{\prime}F^{-1}_{-|\psi_{t+1}(Z_{\leq
t+1})|\mid Z_{\leq t}=z_{\leq t}}(p)=l(p)$ $\displaystyle u_{n}(p)$
$\displaystyle=c^{\prime}F^{-1}_{|\psi^{n}_{t+1}(Z^{n}_{\leq t+1})|\mid
Z^{n}_{\leq t}=z^{n}_{\leq t}}(p)\to c^{\prime}F^{-1}_{|\psi_{t+1}(Z_{\leq
t+1})|\mid Z_{\leq t}=z_{\leq t}}(p)=u(p).$

The induction assumption UI($t+1$) together with

$\displaystyle\mathcal{L}(|\psi^{n}_{t+1}(Z^{n}_{\leq t+1})|\mid Z^{n}_{\leq
t}=z^{n}_{\leq t})\stackrel{{\scriptstyle
w}}{{\to}}\mathcal{L}(|\psi_{t+1}(Z_{\leq t+1})|\mid Z_{\leq t}=z_{\leq t}),$

allow us to conclude from [8, Theorem 4.11] that

$\displaystyle\int l_{n}(p)dp=$
$\displaystyle-c^{\prime}\operatorname{E}[|\psi^{n}_{t+1}(Z_{\leq
t+1}^{n})|\mid Z_{\leq t}^{n}=z_{\leq t}^{n}]$ $\displaystyle\to$
$\displaystyle-c^{\prime}\operatorname{E}[|\psi_{t+1}(Z_{\leq t+1})|\mid
Z_{\leq t}=z_{\leq t}]=\int l(p)dp$

and

$\displaystyle\int u_{n}(p)dp=\;$ $\displaystyle
c^{\prime}\operatorname{E}[|\psi^{n}_{t+1}(Z_{\leq t+1}^{n})|\mid Z_{\leq
t}^{n}=z_{\leq t}^{n}]$ $\displaystyle\to\;$ $\displaystyle
c^{\prime}\operatorname{E}[|\psi_{t+1}(Z_{\leq t+1})|\mid Z_{\leq t}=z_{\leq
t}]=\int u(p)dp.$

Hence, by Pratt’s Lemma [11, Theorem 1] we have convergence of the integral in
(33), i.e.

$\displaystyle\int F^{-1}_{\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\mid Z^{n}_{\leq
t}=z_{\leq t}^{n}}(p)\mu_{1}(dp)$ $\displaystyle=\int f_{n}(p)dp\to\int
f(p)dp$ $\displaystyle=\int F^{-1}_{\psi_{t+1}(Z_{\leq t+1})\mid Z_{\leq
t}=z_{\leq t}}(p)\mu_{1}(dp).$

We now consider the integral in (34). By Lemma 3, the mapping $p\mapsto
F^{-1}_{-\psi_{t+1}(Z_{\leq t+1})\mid Z_{\leq t}=z_{\leq t}}(p)$ is continuous
for all $p\in(0,1)$. Therefore, from (31) follows pointwise convergence

$F^{-1}_{-\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\mid Z^{n}_{\leq t}=z_{\leq
t}^{n}}(p)\to F^{-1}_{-\psi_{t+1}(Z_{\leq t+1})\mid Z_{\leq t}=z_{\leq t}}(p)$

for all $p\in(0,1)$. In particular, the convergence holds at the points
$a,b\in(0,1)$. Therefore, using the fact that $p\mapsto
F^{-1}_{-\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\mid Z^{n}_{\leq t}=z^{n}_{\leq
t}}(p)$ and $p\mapsto F^{-1}_{-\psi_{t+1}(Z_{\leq t+1})\mid Z_{\leq t}=z_{\leq
t}}(p)$ are increasing, we can find $d>0$, such that

$c^{\prime\prime}=\max\bigg{(}\Big{|}F^{-1}_{-\psi_{t+1}(Z_{\leq t+1})\mid
Z_{\leq t}=z_{\leq t}}(a)\Big{|},\Big{|}F^{-1}_{-\psi_{t+1}(Z_{\leq t+1})\mid
Z_{\leq t}=z_{\leq t}}(b)\Big{|}\bigg{)}+d$

uniformly bounds the integrand, i.e.

$\Big{|}F^{-1}_{-\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\mid Z^{n}_{\leq
t}=z^{n}_{\leq t}}(p)\Big{|}\leq c^{\prime\prime}$

for all $p\in[a,b]$ and for all $n\in\mathbb{N}$. Hence, the bounded
convergence theorem yields

$\displaystyle\int F^{-1}_{-\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\mid Z^{n}_{\leq
t}=z_{\leq t}^{n}}(p)\mu^{2}_{t}(dp)\to\int F^{-1}_{-\psi_{t+1}(Z_{\leq
t+1})\mid Z_{\leq t}=z_{\leq t}}(p)\mu^{2}_{t}(dp).$

Hence, we have shown CC($t$) and the proof of the induction step is complete.

It remains to show convergence of $\psi^{n}_{0}$. By the continuous
convergence of $(\psi^{n}_{1})$ and the generalized continuous mapping theorem
([8, Theorem 4.27]), it follows that
$\mathcal{L}(\psi^{n}_{1}(Z_{1}^{n}))\stackrel{{\scriptstyle
w}}{{\to}}\mathcal{L}(\psi_{1}(Z_{1}))$, where the limit is Gaussian (cf.
Lemma 3). Therefore the convergence of

$\displaystyle\psi^{n}_{0}=\lambda_{0}\int
F^{-1}_{\psi^{n}_{1}(Z^{n}_{1})}(p)\mu_{0}^{1}(dp)+(1-\lambda_{0})\int
F^{-1}_{-\psi^{n}_{1}(Z^{n}_{1})}(p)\mu_{0}^{2}(dp)$

follows from arguments completely analogous to those verifying the induction
step. Lemma 3 together with the variance decomposition

$\displaystyle\operatorname{Var}\bigg{(}\operatorname{E}\bigg{[}\sum_{v=u}^{T}X_{v}\mid
Z_{\leq u}\bigg{]}\mid Z_{\leq u-1}\bigg{)}$
$\displaystyle=\operatorname{Var}\bigg{(}\sum_{v=u}^{T}X_{v}\mid Z_{\leq
u-1}\bigg{)}$
$\displaystyle\quad-\operatorname{E}\bigg{[}\operatorname{Var}\bigg{(}\sum_{v=u}^{T}X_{v}\mid
Z_{\leq u}\bigg{)}\mid Z_{\leq u-1}\bigg{]}$
$\displaystyle=\operatorname{Var}\bigg{(}\sum_{v=u}^{T}X_{v}\mid Z_{\leq
u-1}\bigg{)}-\operatorname{Var}\bigg{(}\sum_{v=u}^{T}X_{v}\mid Z_{\leq
u}\bigg{)}$

completes the proof. ∎

###### Proof of Theorem 1 assuming (23) and (24).

The proofs is similar to the above proof of Theorem 1 assuming (21) and (22).

Fix $\mu_{0},\dots,\mu_{T-1}\in\mathcal{P}([0,1])^{\prime}$. Let
$\nu^{n}_{T}(z^{n}_{\leq T})=\sum_{s=1}^{T}x^{n}_{s}$ and
$\psi^{n}_{T}(z^{n}_{\leq T})=\sum_{s=1}^{T}x^{n}_{s}$. For $t=1,\dots,T-1$,
define mappings
$\nu^{n}_{t},\psi^{n}_{t}:\operatorname{supp}(\mathcal{L}(Z^{n}_{\leq
t}))\to\mathbb{R}$ by

$\displaystyle\nu^{n}_{t}(z^{n}_{\leq t})$ $\displaystyle=\int
F^{-1}_{-\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\mid Z^{n}_{\leq t}=z^{n}_{\leq
t}}(p)\mu_{t}(dp),$ (35) $\displaystyle\psi^{n}_{t}(z^{n}_{\leq t})$
$\displaystyle=\nu^{n}_{t}(z^{n}_{\leq
t})-\gamma_{t}\operatorname{E}\Big{[}\Big{(}\big{(}\nu^{n}_{t}(z^{n}_{\leq
t})-\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\big{)}^{+}\Big{)}^{\beta_{t}}\mid
Z^{n}_{\leq t}=z^{n}_{\leq t}\Big{]}^{1/\beta_{t}}.$ (36)

Let

$\displaystyle\nu^{n}_{0}=\int
F^{-1}_{-\psi^{n}_{1}(Z^{n}_{1})}(p)\mu_{0}(dp),\quad\psi^{n}_{0}=\nu^{n}_{0}-\gamma_{0}\operatorname{E}\Big{[}\Big{(}\big{(}\nu^{n}_{0}-\psi^{n}_{1}(Z^{n}_{1})\big{)}^{+}\Big{)}^{\beta_{0}}\Big{]}^{1/\beta_{0}}.$
(37)

Define $\nu_{t},\psi_{t}:\mathbb{R}^{t}\to\mathbb{R}$ and $\nu_{0},\psi_{0}$
analogously without the superscript $n$. The proof is complete once we show
the convergence $\lim_{n\to\infty}\psi^{n}_{0}=\psi_{0}$.

The argument of the proof is backwards induction. We will show that for
$t=T,\dots,1$,

  * UI($t$):

$(\mathcal{L}(\psi^{n}_{t}(Z^{n}_{\leq t})\mid Z^{n}_{\leq s}=x^{n}_{\leq
s}))_{n}$ is uniformly integrable for each $s=1,\dots,t-1$ and $z^{n}_{\leq
s}\to z_{\leq s}$,

  * CC($t$):

$(\psi^{n}_{t})_{n}$ is continuously convergent, i.e.
$\psi^{n}_{t}(z^{n}_{\leq t})\to\psi_{t}(z_{\leq t})$, whenever $z^{n}_{\leq
t}\to z_{\leq t}$.

Induction base: UI($T$) and CC($T$) hold. The argument verifying the induction
base is identical to that in the proofs of Theorem 1 assuming (21) and (22).

Induction step: UI($t+1$), CC($t+1$) together imply UI($t$), CC($t$). The
argument verifying that

$\displaystyle\mathcal{L}(\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\mid Z^{n}_{\leq
s}=z^{n}_{\leq s})\stackrel{{\scriptstyle
w}}{{\to}}\mathcal{L}(\psi_{t+1}(Z_{\leq t+1})\mid Z_{\leq s}=z_{\leq s}).$
(38)

holds is identical to that in the proofs of Theorem 1 assuming (21) and (22).
By Lemma 5, there exists a constant $c>0$, such that

$|\psi^{n}_{t}(z)|\leq c\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})|\mid Z^{n}_{\leq t}=z].$

Hence, to verify UI($t$) it is sufficient to show uniform integrability of

$\Big{(}\mathcal{L}\Big{(}\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})|\mid Z^{n}_{\leq t}]\mid Z^{n}_{\leq s}=z^{n}_{\leq
s}\Big{)}\Big{)}_{n\in\mathbb{N}}$

and after setting $W^{n}=|\psi^{n}_{t+1}(Z^{n}_{\leq t+1})|$,
$\widetilde{W}^{n}=\operatorname{E}[|\psi^{n}_{t+1}(Z_{\leq t+1}^{n})|\mid
Z^{n}_{\leq t}]$, UI($t$) follows from UI($t+1$) and (38) by applying Lemma 4.

We proceed by showing CC($t$), i.e. the convergence of (36). We first consider
(35). If $\mu_{t}$ admits a bounded density, convergence
$\rho_{t}^{n}(z^{n}_{\leq t})\to\rho_{t}(z_{\leq t})$ can be shown by Pratt’s
Lemma, together with (38) and UI($t+1$) (cf. Proof of Theorem 1 assuming (21)
and (22)). If $\mu_{t}$ satisfies $\operatorname{supp}(\mu_{t})\subset[a,b]$
with $0<a<b<1$, we first note that we have pointwise convergence
$F^{-1}_{-\psi^{n}_{t+1}(Z^{n}_{\leq t})\mid Z^{n}_{\leq t}=z^{n}_{\leq
t}}(p)\to F^{-1}_{-\psi_{t+1}(Z_{\leq t})\mid Z_{\leq t}=z_{\leq t}}(p)$ for
all $p\in(0,1)$ since the limiting distribution in (38) is Gaussian (cf. Lemma
3). There exists $c>0$ such that

$\displaystyle\sup_{n}\sup_{p\in[a,b]}|F^{-1}_{-\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})\mid Z^{n}_{\leq t}=z^{n}_{\leq
t}}(p)|\leq\sup_{n}c\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq t+1})|\mid
Z^{n}_{\leq t}=z^{n}_{\leq t}]$

and the right-hand side is finite. The inequality is due to properties of
quantile functions (cf. Proof of Lemma 2). The finiteness is because of the
convergence of the conditional expectation which is due to (38) and UI($t+1$).
Hence, dominated convergence yields $\nu^{n}_{t}(z^{n}_{\leq
t})\to\nu_{t}(z_{\leq t})$.

We now show convergence of the conditional expectation in (36) which can be
written in terms of

$\displaystyle\operatorname{E}\Big{[}h_{n}\big{(}\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})\big{)}\mid Z^{n}_{\leq t}=z^{n}_{\leq t}\Big{]},$

where $h_{n}(w)=\big{(}\big{(}\nu^{n}_{t}(z^{n}_{\leq
t})-w\big{)}^{+}\big{)}^{\beta_{t}}$ converges continuously towards
$h(w)=\big{(}\big{(}\nu_{t}(z_{\leq t})-w\big{)}^{+}\big{)}^{\beta_{t}}$.
Moreover, since

$|h_{n}(w)|\leq\bigg{(}\sup_{n}(\nu^{n}_{t}(z^{n}_{\leq
t}))+|w|\bigg{)}^{\beta_{t}},\quad\sup_{n}(\nu^{n}_{t}(z^{n}_{\leq
t}))<\infty,$

and $\beta_{t}\in(0,1]$, uniform integrability of
$\mathcal{L}\big{(}h_{n}\big{(}\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\big{)}\mid
Z^{n}_{\leq t}=z^{n}_{\leq t}\big{)}$ follows from UI($t+1$). This means
convergence

$\displaystyle\operatorname{E}\Big{[}h_{n}\big{(}\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})\big{)}\mid Z^{n}_{\leq t}=z^{n}_{\leq
t}\Big{]}\to\operatorname{E}\Big{[}h\big{(}\psi_{t+1}(Z_{\leq t+1})\big{)}\mid
Z_{\leq t}=z_{\leq t}\Big{]}$

which implies CC($t$) and the proof of the induction step is complete.

Finally, the proof of the convergence $\lim_{n\to\infty}\psi^{n}_{0}=\psi_{0}$
follows by arguments analogous to those in the proof of Theorem 1 assuming
(21) and (22). ∎

######  Lemma 1.

Let $W$ be a random variable with quantile function
$F^{-1}(p)=\min\\{x\in\mathbb{R}:F_{W}(x)\geq p\\}$. If
$\mu\in\mathcal{P}([0,1])^{\prime}$, then there exists $c\in(0,\infty)$ such
that

$\displaystyle\Big{|}\int F^{-1}_{W}(p)\mu(dp)\Big{|}\leq
c\operatorname{E}[|W|].$

###### Proof.

We have the general bounds $l,u$ given by

$\displaystyle l=\int F^{-1}_{-|W|}(p)\mu(dp)\leq\int
F^{-1}_{W}(p)\mu(dp)\leq\int F^{-1}_{|W|}(p)\mu(dp)=u.$

Moreover,

$\displaystyle F^{-1}_{-|W|}(p)=-F^{-1}_{|W|}((1-p)+)=-\lim_{v\downarrow
1-p}F^{-1}_{|W|}(v).$

For any $p\in(0,1)$,

$\displaystyle pF^{-1}_{|W|}(1-p)$
$\displaystyle\leq\int_{1-p}^{1}F^{-1}_{|W|}(v)dv\leq\operatorname{E}[|W|],$
$\displaystyle pF^{-1}_{-|W|}(p)$
$\displaystyle\geq\int_{0}^{p}F^{-1}_{-|W|}(v)dv\geq-\operatorname{E}[|W|].$

Consequently, if $\operatorname{supp}(\mu)\subset[a,b]\subset(0,1)$, then

$\displaystyle l\geq-\frac{1}{a}\operatorname{E}[|W|],\quad
u\leq\frac{1}{1-b}\operatorname{E}[|W|],\quad\Big{|}\int
F^{-1}_{W}(p)\mu(dp)\Big{|}\leq\max\Big{(}\frac{1}{a},\frac{1}{1-b}\Big{)}\operatorname{E}[|W|].$

If $\mu$ has a density bounded by $c^{\prime}\in(0,\infty)$, then

$\displaystyle l\geq-c^{\prime}\operatorname{E}[|W|],\quad u\leq
c^{\prime}\operatorname{E}[|W|],\quad\Big{|}\int
F^{-1}_{W}(p)\mu(dp)\Big{|}\leq c^{\prime}\operatorname{E}[|W|].$

∎

######  Lemma 2.

If $\psi^{n}_{t}$ is given by (29) and (30), then exists $c\in(0,\infty)$ such
that

$\displaystyle|\psi^{n}_{0}|\leq
c\operatorname{E}[|\psi_{1}^{n}(Z^{n}_{1})|]\quad\text{and}\quad|\psi^{n}_{t}(z)|\leq
c\operatorname{E}[|\psi_{t+1}^{n}(Z^{n}_{\leq t+1})|\mid Z^{n}_{\leq t}=z]$

for any $z\in\operatorname{supp}(\mathcal{L}(Z^{n}_{\leq t}))$.

###### Proof.

We proof the inequality for $t\geq 1$. The argument for $t=0$ is identical.
Since

$\displaystyle|\psi^{n}_{t}(z)|\leq\Big{|}\int
F^{-1}_{\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\mid Z^{n}_{\leq
t}=z}(p)\mu_{t}^{1}(dp)\Big{|}+\Big{|}\int F^{-1}_{-\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})\mid Z^{n}_{\leq t}=z}(p)\mu_{t}^{2}(dp)\Big{|},$

we can apply Lemma 1, bounding the right-hand side by
$c\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq t})|\mid Z^{n}_{\leq t}=z]$ for
some $c$. ∎

The following result is similar to Proposition 6 in [5].

######  Lemma 3.

Let $(Z_{t})_{t=1}^{T}=(X_{t},Y_{t})_{t=1}^{T}$ be a Gaussian process, let
$\mathcal{F}_{0}=\\{\Omega,\emptyset\\}$, and let
$\mathcal{F}_{t}=\sigma(Z_{\leq t})$ for $t\geq 1$. Let $V_{T}=0$ and, for
$t\in\\{0,1,\dots,T-1\\}$, let $V_{t}=\varphi_{t}(X_{t+1}+V_{t+1})$, where
$\varphi_{t}$ is positive homogeneous and conditionally cash additive. Suppose
that for each $t$,

$\displaystyle\text{if}\quad\operatorname{P}(W\in\cdot\mid\mathcal{F}_{t})=\operatorname{P}(\widetilde{W}\in\cdot\mid\mathcal{F}_{t}),\quad\text{then}\quad\varphi_{t}(W)=\varphi_{t}(\widetilde{W}),$

and if $W$ is independent of $\mathcal{F}_{t}$, then $\varphi_{t}(W)$ is
nonrandom. Then

$\displaystyle
V_{t}=\operatorname{E}\bigg{[}\sum_{u=t+1}^{T}X_{u}\mid\mathcal{F}_{t}\bigg{]}+\sum_{u=t+1}^{T}\varphi_{u-1}(\varepsilon_{u})\operatorname{Var}\bigg{(}\operatorname{E}\bigg{[}\sum_{v=u}^{T}X_{v}\mid\mathcal{F}_{u}\bigg{]}\mid\mathcal{F}_{u-1}\bigg{)}^{1/2},$

where $\varepsilon_{u}$ is standard normal and independent of
$\mathcal{F}_{u-1}$.

###### Proof.

The lemma is proved by backward induction. We prove the induction step.
Suppose that $X_{t+1}+V_{t+1}$ and $Z_{\leq t}$ are jointly Gaussian. Then
there exists a standard normal $\varepsilon_{t+1}$ independent of
$\mathcal{F}_{t}$ such that

$\displaystyle\operatorname{P}(X_{t+1}+V_{t+1}\in\cdot\mid\mathcal{F}_{t})=\operatorname{P}\Big{(}\operatorname{E}[X_{t+1}+V_{t+1}\mid\mathcal{F}_{t}]+\varepsilon_{t+1}\operatorname{Var}(X_{t+1}+V_{t+1}\mid\mathcal{F}_{t})^{1/2}\in\cdot\mid\mathcal{F}_{t}\Big{)}$

and it follows from properties of the multivariate normal distribution that
the conditional variance
$\operatorname{Var}(X_{t+1}+V_{t+1}\mid\mathcal{F}_{t})$ is nonrandom. Hence,
by positive homogeneity and conditional cash additivity,

$\displaystyle\varphi_{t}(X_{t+1}+V_{t+1})=\operatorname{E}[X_{t+1}+V_{t+1}\mid\mathcal{F}_{t}]+\varphi_{t}(\varepsilon_{t+1})\operatorname{Var}(X_{t+1}+V_{t+1}\mid\mathcal{F}_{t})^{1/2}.$

By the induction assumption (in the nontrivial case $t+1<T$),

$\displaystyle
V_{t+1}=\operatorname{E}\bigg{[}\sum_{u=t+2}^{T}X_{u}\mid\mathcal{F}_{t+1}\bigg{]}+\sum_{u=t+2}^{T}\varphi_{u-1}(\varepsilon_{u})\operatorname{Var}\bigg{(}\operatorname{E}\bigg{[}\sum_{v=u}^{T}X_{v}\mid\mathcal{F}_{u}\bigg{]}\mid\mathcal{F}_{u-1}\bigg{)}^{1/2}.$

Summing up the terms and using the tower property of conditional expectations
complete the proof. ∎

######  Lemma 4.

Let $s\in\\{1,\dots,t-1\\}$ and let $(W^{n})_{n\in\mathbb{N}}$ be a sequence
of non-negative random variables such that

$\displaystyle\mathcal{L}(W^{n}\mid Z^{n}_{\leq s}=z^{n}_{\leq
s})\stackrel{{\scriptstyle w}}{{\to}}\mathcal{L}(W\mid Z_{\leq s}=z_{\leq
s})\quad\text{as }n\to\infty$ (39)

and

$\displaystyle\Big{(}\mathcal{L}(W^{n}\mid Z^{n}_{\leq s}=z^{n}_{\leq
s})\Big{)}_{n\in\mathbb{N}}\quad\text{is uniformly integrable.}$ (40)

If $\widetilde{W}^{n}=\operatorname{E}[W^{n}\mid Z^{n}_{\leq t}]$, then
$\big{(}\mathcal{L}(\widetilde{W}^{n}\mid Z^{n}_{\leq s}=z^{n}_{\leq
s})\big{)}_{n\in\mathbb{N}}$ is uniformly integrable.

###### Proof.

First note that

$\displaystyle\operatorname{E}\Bigl{[}\widetilde{W}^{n}I\\{\widetilde{W}^{n}\geq
r\\}\mid Z^{n}_{\leq s}\Bigr{]}$
$\displaystyle=\operatorname{E}\Bigl{[}\operatorname{E}[W^{n}\mid Z^{n}_{\leq
t}]I\\{\widetilde{W}^{n}\geq r\\}\mid Z^{n}_{\leq s}\Bigr{]}$
$\displaystyle=\operatorname{E}\Bigl{[}\operatorname{E}[W^{n}I\\{\widetilde{W}^{n}\geq
r\\}\mid Z^{n}_{\leq t}]\mid Z^{n}_{\leq s}\Bigr{]}$
$\displaystyle=\operatorname{E}\Bigl{[}W^{n}I\\{\widetilde{W}^{n}\geq r\\}\mid
Z^{n}_{\leq s}\Bigr{]},$

i.e. $\operatorname{E}[\widetilde{W}^{n}I\\{\widetilde{W}^{n}\geq r\\}\mid
Z^{n}_{\leq s}=z^{n}_{\leq s}]=\operatorname{E}[W^{n}I\\{\widetilde{W}^{n}\geq
r\\}\mid Z^{n}_{\leq s}=z^{n}_{\leq s}]$. For each $\gamma_{r}>0$,

$\displaystyle\operatorname{E}[W^{n}I\\{\widetilde{W}^{n}\geq r\\}\mid
Z^{n}_{\leq s}=z^{n}_{\leq s}]$
$\displaystyle\quad=\operatorname{E}[W^{n}I\\{\widetilde{W}^{n}\geq
r\\}I\\{W^{n}\leq\gamma_{r}\\}\mid Z^{n}_{\leq s}=z^{n}_{\leq s}]$
$\displaystyle\quad\quad+\operatorname{E}[W^{n}I\\{\widetilde{W}^{n}\geq
r\\}I\\{W^{n}>\gamma_{r}\\}\mid Z^{n}_{\leq s}=z^{n}_{\leq s}]$
$\displaystyle\quad\leq\gamma_{r}\operatorname{P}(\widetilde{W}^{n}\geq r\mid
Z^{n}_{\leq s}=z^{n}_{\leq
s})+\operatorname{E}[W^{n}I\\{W^{n}>\gamma_{r}\\}\mid Z^{n}_{\leq
s}=z^{n}_{\leq s}].$ (41)

By (39) and (40),

$\operatorname{E}[\widetilde{W}^{n}I\\{\widetilde{W}^{n}\geq r\\}\mid
Z^{n}_{\leq s}=z^{n}_{\leq s}]<\infty\quad\text{for all }n\in\mathbb{N},$

and similarly with $W^{n}$ instead of $\widetilde{W}^{n}$. Therefore, uniform
integrability of $\big{(}\mathcal{L}(\widetilde{W}^{n}\mid Z^{n}_{\leq
s}=z^{n}_{\leq s})\big{)}_{n\in\mathbb{N}}$ is equivalent to (cf. [8] p. 67)

$\displaystyle\lim_{r\to\infty}\limsup_{n\to\infty}\operatorname{E}[\widetilde{W}^{n}I\\{\widetilde{W}^{n}\geq
r\\}\mid Z^{n}_{\leq s}=z^{n}_{\leq s}]=0$

which follows if we show that there exists $\gamma_{r}$ such that
$\lim_{r\to\infty}\limsup_{n\to\infty}[\eqref{bound:23}]=0$. By the uniform
integrability assumption (40) and the finiteness of all the terms,

$\displaystyle\lim_{r\to\infty}\limsup_{n\to\infty}\operatorname{E}[W^{n}I\\{W^{n}>\gamma_{r}\\}\mid
Z^{n}_{\leq s}=z^{n}_{\leq s}]=0.$

Therefore, it only remains to show that

$\lim_{r\to\infty}\limsup_{n\to\infty}\gamma_{r}\operatorname{P}(\widetilde{W}^{n}\geq
r\mid Z^{n}_{\leq s}=z^{n}_{\leq s})=0$

for some sequence $\gamma_{r}\to\infty$. By Markov’s inequality,

$\displaystyle\operatorname{P}(\widetilde{W}^{n}\geq r\mid Z^{n}_{\leq
s}=z^{n}_{\leq s})$
$\displaystyle\leq\frac{\operatorname{E}[\widetilde{W}^{n}\mid Z^{n}_{\leq
s}=z^{n}_{\leq s}]}{r}$ $\displaystyle=\frac{\operatorname{E}[W^{n}\mid
Z^{n}_{\leq s}=z^{n}_{\leq s}]}{r}$
$\displaystyle\to\frac{\operatorname{E}[W\mid Z_{\leq s}=z_{\leq s}]}{r},$

where the convergence in the last step follows from Lemma 4.11 in [8], using
(39) and (40). Letting e.g. $\gamma_{r}=\sqrt{r}$, the first term in (4) can
therefore be made arbitrarily small by choosing $r$ sufficiently large. ∎

######  Lemma 5.

If $\nu^{n}_{t},\psi^{n}_{t}$ are given by (35), (36) and (37), then exists
$c\in(0,\infty)$ such that

$\displaystyle|\psi^{n}_{0}|\leq
c\operatorname{E}[|\psi_{1}^{n}(Z^{n}_{1})|]\quad\text{and}\quad|\psi^{n}_{t}(z)|\leq
c\operatorname{E}[|\psi_{t+1}^{n}(Z^{n}_{\leq t+1})|\mid Z^{n}_{\leq t}=z]$

for any $z\in\operatorname{supp}(\mathcal{L}(Z^{n}_{\leq t}))$.

###### Proof of Lemma 5.

Consider $t\geq 1$. Note that, by Lemma 1, there exists $c^{\prime}>0$, such
that

$|\nu^{n}_{t}(z)|\leq c^{\prime}\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})|\mid Z^{n}_{\leq t}=z].$

Hence,

$\displaystyle|\psi^{n}_{t}(z)|$ $\displaystyle\leq
c^{\prime}\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq t+1})|\mid Z^{n}_{\leq
t}=z]$
$\displaystyle\quad+\gamma_{t}\operatorname{E}\Big{[}\Big{(}\big{(}c^{\prime}\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})|\mid Z^{n}_{\leq t}=z]-\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})\big{)}^{+}\Big{)}^{\beta_{t}}\mid Z^{n}_{\leq
t}=z\Big{]}^{1/\beta_{t}}.$

By Jensen’s inequality, the second summand above is bounded by

$\displaystyle\gamma_{t}\operatorname{E}\Big{[}\big{(}c^{\prime}\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})|\mid Z^{n}_{\leq t}=z]-\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\big{)}^{+}\mid
Z^{n}_{\leq t}=z\Big{]}$
$\displaystyle\quad\leq\gamma_{t}\operatorname{E}\Big{[}c^{\prime}\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})|\mid Z^{n}_{\leq t}=z]+|\psi^{n}_{t+1}(Z^{n}_{\leq t+1})|\mid
Z^{n}_{\leq t}=z\Big{]}$
$\displaystyle\quad=(c^{\prime}+1)\gamma_{t}\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})|\mid Z^{n}_{\leq t}=z].$

Hence, the statement holds with $c=c^{\prime}+\gamma_{t}(c^{\prime}+1)$. The
inequality $|\psi^{n}_{0}|\leq c\operatorname{E}[|\psi_{1}^{n}(Z^{n}_{1})|]$
is shown by the same arguments. ∎

###### Proof of Theorem 2.

The proof proceeds in a similar manner as the proof of Theorem 1. For
$t=1,\dots,T$, consider mappings
$\psi^{n}_{t}:\operatorname{supp}(\mathcal{L}(Z^{n}_{\leq t}))\to\mathbb{R}$
given by

$\displaystyle\psi^{n}_{t}(z^{n}_{\leq
t})=\left\\{\begin{array}[]{ll}\sum_{s=1}^{T}x^{n}_{s},&t=T,\\\
\operatorname{E}\big{[}\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\mid Z^{n}_{\leq
t}=z^{n}_{\leq
t}\big{]}+c_{t}\operatorname{Var}\big{(}\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\mid
Z^{n}_{\leq t}=z^{n}_{\leq t}\big{)}^{1/2},&t<T,\end{array}\right.$

and

$\displaystyle\psi^{n}_{0}=\operatorname{E}\big{[}\psi^{n}_{1}(Z^{n}_{1})\big{]}+c_{0}\operatorname{Var}\big{(}\psi^{n}_{1}(Z^{n}_{1})\big{)}^{1/2}.$

We define $\psi_{t}:\mathbb{R}^{t}\to\mathbb{R}$ and $\psi_{0}$ analogously
without the superscript $n$. The proof is complete once we show the
convergence $\lim_{n\to\infty}\psi^{n}_{0}=\psi_{0}$.

The argument of the proof is backwards induction. We will show that for
$t=T,\dots,1$,

  * UI($t$):

$(\mathcal{L}(\psi^{n}_{t}(Z^{n}_{\leq t})^{2}\mid Z^{n}_{\leq s}=z^{n}_{\leq
s}))_{n}$ is uniformly integrable for each $s=1,\dots,t-1$ and $z^{n}_{\leq
s}\to z_{\leq s}$,

  * CC($t$):

$(\psi^{n}_{t})_{n}$ is continuously convergent, i.e.
$\psi^{n}_{t}(z^{n}_{\leq t})\to\psi_{t}(z_{\leq t})$, whenever $z^{n}_{\leq
t}\to z_{\leq t}$.

Induction base: UI($T$) and CC($T$) hold.

For $s\in\\{1,\dots,T-1\\}$ and $z^{n}_{\leq s}\to z_{\leq s}$, UI($T$)
follows immediately from (28) since $\psi^{n}_{T}(z^{n}_{\leq
T})=\sum_{s=1}^{T}x^{n}_{s}$. Moreover,

$\psi^{n}_{T}(z^{n}_{\leq
T})=\sum_{s=1}^{T}x^{n}_{s}\to\sum_{s=1}^{T}x_{s}=\psi_{T}(z_{\leq T})$

whenever $z_{\leq T}^{n}\to z_{\leq T}$, yielding CC($T$).

Induction step:UI($t+1$), CC($t+1$) together imply UI($t$), CC($t$). Fix
$s\in\\{1,\dots,t-1\\}$ and a sequence $(z^{n}_{\leq s})_{n}$ with
$z^{n}_{\leq s}\to z_{\leq s}$. By the same argument as in the proof of
Theorem 1,

$\displaystyle\mathcal{L}(\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\mid Z^{n}_{\leq
s}=z^{n}_{\leq s})\stackrel{{\scriptstyle
w}}{{\to}}\mathcal{L}(\psi_{t+1}(Z_{\leq t+1})\mid Z_{\leq s}=z_{\leq s})$
(42)

which implies

$\displaystyle\mathcal{L}((\psi^{n}_{t+1}(Z^{n}_{\leq t+1}))^{2}\mid
Z^{n}_{\leq s}=z^{n}_{\leq s})\stackrel{{\scriptstyle
w}}{{\to}}\mathcal{L}((\psi_{t+1}(Z_{\leq t+1}))^{2}\mid Z_{\leq s}=z_{\leq
s}).$ (43)

Lemma 3 shows that the limiting distribution in (42) is Gaussian. We can bound

$\displaystyle\psi^{n}_{t}(z)^{2}$ $\displaystyle\leq
2\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq t+1})|\mid Z^{n}_{\leq
t}=z]^{2}+2c_{t}^{2}\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})|^{2}\mid Z^{n}_{\leq t}=z]$
$\displaystyle\leq(2+2c_{t}^{2})\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})|^{2}\mid Z^{n}_{\leq t}=z].$

Therefore,

$\displaystyle\operatorname{E}[|\psi^{n}_{t}(Z^{n}_{\leq t})|^{2}\,\mid
Z^{n}_{\leq s}=z^{n}_{\leq s}]$
$\displaystyle\leq(2+2c_{t}^{2})\operatorname{E}[\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})|^{2}\,\mid Z^{n}_{\leq t}]\mid Z^{n}_{\leq s}=z^{n}_{\leq s}]$
$\displaystyle=(2+2c_{t}^{2})\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})|^{2}\,\mid Z^{n}_{\leq s}=z^{n}_{\leq s}].$

Hence, it is sufficient to show uniform integrability of

$\displaystyle\Big{(}\mathcal{L}\big{(}\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})|^{2}\mid Z^{n}_{\leq t}]\mid Z^{n}_{\leq s}=z^{n}_{\leq
s}\big{)}\Big{)}_{n}.$ (44)

Setting $\widetilde{W}^{n}=\operatorname{E}[|\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})|^{2}\mid Z^{n}_{\leq t}]$ and $W^{n}=|\psi^{n}_{t+1}(Z^{n}_{\leq
t+1})|^{2}$, this follows from Lemma 4 together with (43) and UI($t+1$).
Hence, UI($t$) holds.

We proceed by showing CC($t$). Let $z^{n}_{\leq t}\to z_{\leq t}$. We need to
show convergence of

$\displaystyle\psi^{n}_{t}(z^{n}_{\leq t})$
$\displaystyle=\operatorname{E}[\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\mid
Z^{n}_{\leq t}=z^{n}_{\leq t}]$
$\displaystyle\quad+c_{t}\left(\operatorname{E}[(\psi^{n}_{t+1}(Z^{n}_{\leq
t+1}))^{2}\mid Z^{n}_{\leq t}=z^{n}_{\leq
t}]-\operatorname{E}[\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\mid Z^{n}_{\leq
t}=z^{n}_{\leq t}]^{2}\right)^{1/2}.$

Indeed, by (42) and (43),

$\displaystyle\mathcal{L}(\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\mid Z^{n}_{\leq
t}=z^{n}_{\leq t})\stackrel{{\scriptstyle
w}}{{\to}}\mathcal{L}(\psi_{t+1}(Z_{\leq t+1})\mid Z_{\leq t}=z_{\leq t}),$
$\displaystyle\mathcal{L}((\psi^{n}_{t+1}(Z^{n}_{\leq t+1}))^{2}\mid
Z^{n}_{\leq t}=z^{n}_{\leq t})\stackrel{{\scriptstyle
w}}{{\to}}\mathcal{L}((\psi_{t+1}(Z_{\leq t+1}))^{2}\mid Z_{\leq t}=z_{\leq
t}),$

and by [8, Lemma 4.11] applied to UI($t+1$), it follows that

$\operatorname{E}[\psi^{n}_{t+1}(Z^{n}_{\leq t+1})\mid Z^{n}_{\leq
t}=z^{n}_{\leq t}]\to\operatorname{E}[\psi_{t+1}(Z_{\leq t+1})\mid Z_{\leq
t}=z_{\leq t}]$

and

$\operatorname{E}[(\psi^{n}_{t+1}(Z^{n}_{\leq t+1}))^{2}\mid Z^{n}_{\leq
t}=z^{n}_{\leq t}]\to\operatorname{E}[(\psi_{t+1}(Z_{\leq t+1}))^{2}\mid
Z_{\leq t}=z_{\leq t}],$

yielding CC($t$) which completes the proof of the induction step.

As a result of the induction,

$\mathcal{L}(\psi^{n}_{1}(Z^{n}_{1}))\stackrel{{\scriptstyle
w}}{{\to}}\mathcal{L}(\psi_{1}(Z_{1}))$

and $(\psi^{n}_{1}(Z^{n}_{1}))_{n},((\psi^{n}_{1}(Z^{n}_{1}))^{2})_{n}$ are
uniformly integrable. Therefore,

$\displaystyle\psi^{n}_{0}$
$\displaystyle=\operatorname{E}[\psi^{n}_{1}(Z^{n}_{1})]+c_{0}\left(\operatorname{E}[(\psi^{n}_{1}(Z^{n}_{1}))^{2}]-\operatorname{E}[\psi^{n}_{1}(Z^{n}_{1})]^{2}\right)^{1/2}$
$\displaystyle\to\operatorname{E}[\psi_{1}(Z_{1})]+c_{0}\left(\operatorname{E}[(\psi_{1}(Z_{1}))^{2}]-\operatorname{E}[\psi_{1}(Z_{1})]^{2}\right)^{1/2}=\psi_{0}.$

∎

######  Lemma 6.

Let $c\in\mathbb{R}^{T}_{+}$ satisfy $\sum_{t=1}^{T}c_{t}=1$ and $c_{t+1}\leq
c_{t}$ for every $t$. Then $c$ solves

$\displaystyle\max_{d\in\mathcal{D}_{c,T}}\sum_{t=1}^{T}d_{t}^{1/2},\quad\mathcal{D}_{c,T}=\bigg{\\{}d\in\mathbb{R}^{T}_{+}:\sum_{t=1}^{T}d_{t}=1,\sum_{s=1}^{t}d_{s}\geq\sum_{s=1}^{t}c_{s}\text{
for }t=1,\dots,T\bigg{\\}}.$ (45)

###### Proof.

Since $\mathcal{D}_{c,T}$ is a convex set and the objective function is
concave, we have a convex optimization problem. Consequently, it is sufficient
to verify that $c\in\mathcal{D}_{c,T}$ is a local optimum. By assumption
either $c_{1}=1$ and $c_{t}=0$ for $t\geq 2$ or there exists
$t_{0}\in\\{2,\dots,T\\}$ such that $c_{1}\geq\dots\geq c_{t_{0}}>0$ and
$c_{t_{0}+1}=\dots=c_{T}=0$. The first case has the trivial maximizer
$d_{1}=1$ and $d_{t}=0$ for $t\geq 2$. Hence, it is sufficient to consider
only the second case for a fixed $c$ and $t_{0}\in\\{2,\dots,T\\}$. In this
case, (45) is equivalent to, with $c=(c_{1},\dots,c_{t_{0}})$,

$\displaystyle\max_{d\in\mathcal{D}_{c}}\sum_{t=1}^{t_{0}}d_{t}^{1/2},\quad\mathcal{D}_{c}=\bigg{\\{}d\in\mathbb{R}^{t_{0}}_{+}:\sum_{t=1}^{t_{0}}d_{t}=1,\sum_{s=1}^{t}d_{s}\geq\sum_{s=1}^{t}c_{s}\text{
for }t=1,\dots,t_{0}\bigg{\\}}.$ (46)

Since $\mathcal{D}_{c}$ is a convex set and the objective function is concave,
we have a convex optimization problem. Consequently, it is sufficient to
verify that $c\in\mathcal{D}_{c}$ is a local optimum. Let

$\displaystyle\mathcal{X}=\bigg{\\{}x\in\mathbb{R}^{t_{0}}:\sum_{t=1}^{t_{0}}x_{t}=0,\sum_{s=1}^{t}x_{s}\geq
0\text{ for }t=1,\dots,t_{0}\bigg{\\}}$

and note that $\mathcal{D}_{c}\subset c+\mathcal{X}$. Consider the set of
vectors $\\{b_{1},\dots,b_{t_{0}-1}\\}\subset\mathbb{R}^{t_{0}}$ given by

$\displaystyle b_{i,i}=1,\;b_{i,i+1}=-1,\;b_{i,j}=0\text{ for
}j\notin\\{i,i+1\\}.$

We claim that $\mathcal{X}=\textrm{span}_{+}\\{b_{1},\dots,b_{t_{0}-1}\\}$,
where

$\displaystyle\textrm{span}_{+}\\{b_{1},\dots,b_{t_{0}-1}\\}=\bigg{\\{}\sum_{k=1}^{t_{0}-1}\lambda_{k}b_{k}:\lambda_{1},\dots,\lambda_{t_{0}-1}\geq
0\bigg{\\}}.$

To show that
$\textrm{span}_{+}\\{b_{1},\dots,b_{t_{0}-1}\\}\subseteq\mathcal{X}$ it is
sufficient to note that

$\displaystyle\sum_{s=1}^{t_{0}}\sum_{k=1}^{t_{0}-1}\lambda_{k}b_{k,s}=\sum_{k=1}^{t_{0}-1}\lambda_{k}\sum_{s=1}^{t_{0}}b_{k,s}=0,$
$\displaystyle\sum_{s=1}^{t}\sum_{k=1}^{t_{0}-1}\lambda_{k}b_{k,s}=\sum_{k=1}^{t_{0}-1}\lambda_{k}\sum_{s=1}^{t}b_{k,s}=\lambda_{t}\geq
0.$

We now show that
$\mathcal{X}\subseteq\textrm{span}_{+}\\{b_{1},\dots,b_{t_{0}-1}\\}$. Take
$x\in\mathcal{X}$ and set $\lambda_{1}=x_{1}$ (noting that $x_{1}\geq 0$) and
$\lambda_{k}=\lambda_{k-1}+x_{k}$ for $k\geq 2$. Hence, $\lambda_{k}\geq 0$
for every $k$ and $\sum_{k=1}^{t_{0}-1}\lambda_{k}b_{k}=x$.

Let $g(d)=\sum_{t=1}^{t_{0}}d_{t}^{1/2}$ and note that $g$ is well defined and
concave on $\mathcal{D}_{c}$, and has a well defined gradient at $c$:

$\displaystyle\nabla
g(c)^{{\mathrm{T}}}=\frac{1}{2}(c_{1}^{-1/2},\dots,c_{t_{0}}^{-1/2}).$

Take $d\in\mathcal{D}_{c}$ and notice that $d=c+x$ for $x\in\mathcal{X}$.
Since $g$ is concave,

$\displaystyle g(d)-g(c)\leq\nabla
g(c)^{{\mathrm{T}}}x=\sum_{k=1}^{t_{0}-1}\lambda_{k}\nabla
g(c)^{{\mathrm{T}}}b_{k}=\frac{1}{2}\sum_{k=1}^{t_{0}-1}\lambda_{k}(c_{k}^{-1/2}-c_{k+1}^{-1/2})\leq
0,$

where the last inequality holds since, for every $k$, $\lambda_{k}\geq 0$ and
$c_{k}^{-1/2}-c_{k+1}^{-1/2}\leq 0$. Hence, we have shown that $g(d)\leq g(c)$
and the proof is complete. ∎

###### Proof of Theorem 3.

For $t=1,\dots,T$, set

$\displaystyle c_{t}$
$\displaystyle=\operatorname{Var}\bigg{(}\sum_{s=1}^{T}X_{s}\bigg{)}^{-1}\bigg{(}\operatorname{Var}\bigg{(}\sum_{s=1}^{T}X_{s}\mid\mathcal{F}_{t-1}\bigg{)}-\operatorname{Var}\bigg{(}\sum_{s=1}^{T}X_{s}\mid\mathcal{F}_{t}\bigg{)}\bigg{)},$
$\displaystyle d_{t}$
$\displaystyle=\operatorname{Var}\bigg{(}\sum_{s=1}^{T}X_{s}\bigg{)}^{-1}\bigg{(}\operatorname{Var}\bigg{(}\sum_{s=1}^{T}X_{s}\mid\mathcal{G}_{t-1}\bigg{)}-\operatorname{Var}\bigg{(}\sum_{s=1}^{T}X_{s}\mid\mathcal{G}_{t}\bigg{)}\bigg{)}.$

By construction, $\sum_{t=1}^{T}c_{t}=\sum_{t=1}^{T}d_{t}=1$. Since
$\mathcal{F}_{t}\subseteq\mathcal{G}_{t}$ for every $t$,
$\sum_{s=t+1}^{T}c_{s}\geq\sum_{s=t+1}^{T}d_{s}$ which is equivalent to
$\sum_{s=1}^{t}d_{s}\geq\sum_{s=1}^{t}c_{s}$ for every $t$ since
$\sum_{t=1}^{T}c_{t}=\sum_{t=1}^{T}d_{t}$. Applying Lemma 6 completes the
proof. ∎

## Acknowledgements

F. Lindskog would like to acknowledge financial support from the Swedish
Research Council, Project 2020-05065, and from Länsförsäkringars
Forskningsfond, Project P9.20.

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