#  Expectiles In Risk Averse Stochastic Programming and Dynamic Optimization

Rajmadan Lakshmanan University of Technology, Chemnitz, Faculty of
mathematics. 90126 Chemnitz, Germany Alois Pichler11footnotemark: 1
orcid.org/0000-0001-8876-2429. Contact<EMAIL_ADDRESS>
DFG, German Research Foundation – Project-ID 416228727 – SFB 1410
0000-0001-8876-2429

###### Abstract

This paper features expectiles in dynamic and stochastic optimization.
Expectiles are a family of risk functionals characterized as minimizers of
optimization problems. For this reason, they enjoy various unique stability
properties, which can be exploited in risk averse management, in stochastic
optimization and in optimal control.

The paper provides tight relates of expectiles to other risk functionals and
addresses their properties in regression. Further, we extend expectiles to a
dynamic framework. As such, they allow incorporating a risk averse aspect in
continuous-time dynamic optimization and a risk averse variant of the
Hamilton–Jacobi–Bellman equations.

Keywords: Expectiles • multistage stochastic optimization • dynamic
optimization • stochastic processes

Classification: 90C08, 90C15, 60G07

##  1 Introduction

Classical dynamic programming problems involve the expectation in the
objective. The expectation is a risk neutral assessment of random outcomes. In
many situations, specifically in economic environments, a risk averse
assessment or risk management is much more favorable and desirable. For this
reason there have been attempts to develop risk averse dynamic programming
principles and risk averse Hamilton–Jacobi–Bellman equations.

Non-linear expectations ($g$-expectations, cf. Pardoux and Peng (1990); Coquet
et al. (2002); Peng (1992, 2004, 2010)) have been considered, e.g., to
incorporate the aspect of risk to dynamic equations. A seemingly simpler
approach involves risk measures (or risk functionals) instead of non-linear
expectations, as risk measures are able to assess the risk associated with a
random outcome (cf. Ruszczyński and Yao (2015, 2020)). By construction, risk
measures are defined on random variables. For dynamic programming, they need
to be extended to stochastic processes. The increments of stochastic processes
are random variables so that composing risk measures over time and
accumulating the corresponding risk is a promising approach to extend risk
functionals from random variables to stochastic processes.

Specifically, this paper addresses expectiles in stochastic and dynamic
optimization. Expectiles constitute a family of risk measure with unique
properties. We demonstrate how they can be employed to incorporate risk
aversion in dynamic programming and to develop risk averse
Hamilton–Jacobi–Bellman equations.

Cont et al. (2008) point out the importance of estimating risk measures in a
robust way. In this context, Gneiting (2011) proves that the Average Value-at-
Risk, the most important risk measure in theory and practice, is not
elicitable, that is, it is not possible to describe the risk measure as
minimizer. More generally, Ziegel (2014) proves that the only elicitable
spectral risk measure is the (trivial) expectation. Bellini et al. (2014)
finally provide a proof that only expectiles constitute elicitable risk
measures.

Expectiles have been introduced earlier in Newey and Powell (1987) as

$e_{\alpha}(X)\coloneqq\operatorname*{arg\,min}_{x\in\mathbb{R}}\operatorname{{\mathds{E}}}\ell_{\alpha}(X-x),$
(1.1)

where $X$ is a $\mathbb{R}$-valued random variable, $\alpha\in(0,1)$ and the
scoring function (loss function) is111$x_{+}\coloneqq\max(0,x)$

$\ell_{\alpha}(x)\coloneqq\alpha\cdot
x_{+}^{2}+(1-\alpha)(-x)_{+}^{2}=\begin{cases}\quad\alpha\cdot x^{2}&\text{if
}x\geq 0,\\\ (1-\alpha)\,x^{2}&\text{if }x\leq 0.\end{cases}$ (1.2)

The characterization as a minimizer in the definition (1.1) applies for $X\in
L^{2}$. The first order condition (cf. (1.3) below) is an equivalent
characterization of the expectile, which applies – more generally – for $X\in
L^{1}\supset L^{2}$.

######  Definition 1.1 (Expectiles, cf. (Newey and Powell, 1987)).

For $X\in L^{1}$ and a risk level $\alpha\in[0,1]$, the expectiles of a random
variable $X$ is the unique solution of the equation

$\alpha\,\operatorname{{\mathds{E}}}(X-x)_{+}=(1-\alpha)\operatorname{{\mathds{E}}}(x-X)_{+},$
(1.3)

where $x\in\mathbb{R}$.

######  Remark 1.2.

In an alternative way, replacing the objective in (1.1) by
$\operatorname{{\mathds{E}}}\bigl{(}\ell_{\alpha}(X-x)-\ell_{\alpha}(X-x_{0})\bigr{)}$
for some fixed $x_{0}\in\mathbb{R}$ extends the definition to $X\in L^{1}$ as
well, so that expectiles are well-defined for $X\in L^{1}$, even as
minimizers.

For $\alpha=\nicefrac{{1}}{{2}},$ the expectile is the expectation,
$e_{\nicefrac{{1}}{{2}}}(X)=\operatorname{{\mathds{E}}}X$. It follows from
symmetry of the loss function $\ell_{\alpha}$ (i.e.,
$\ell_{\alpha}(x)=\ell_{1-\alpha}(-x)$) that

$e_{\alpha}(X)=-e_{1-\alpha}(-X),$ (1.4)

so that the expectile involves both tails, the lower and the upper tail of the
distribution of the random variable $X$. For $X\in L^{\infty}$, the expectile
approaches the essential supremum for increasing risk level,
$e_{\alpha}(X)\to\operatorname*{ess\,sup}X$ as $\alpha\to 1$.222The essential
supremum of $X$ is the smallest _number_ $c\in\mathbb{R}$ so that $X\leq c$
a.s. More generally, we have the monotone behavior

$\operatorname{{\mathds{E}}}X\leq e_{\alpha}(X)\leq
e_{\alpha^{\prime}}(X)\leq\operatorname*{ess\,sup}X$ (1.5)

for $\nicefrac{{1}}{{2}}\leq\alpha\leq\alpha^{\prime}\leq 1$.

#### Outline of the paper.

In the following Section 2 we elaborate that expectiles constitute a risk
measure, and we provide tight relations to other risk measures. Next, we
introduce conditional risk functionals in Section 3. These are important for
risk management in discrete and in continuous time. In continuous time
(Section 4), we consider the risk-averse generator, which turns out to be a
non-linear differential operator. We finally employ expectiles for dynamic
optimization problems in Section 5 and conclude in Section 6.

##  2 Elicitable risk measures

The expectile $e_{\alpha}(\cdot)$ is a risk measure as introduced in Artzner
et al. (1999). That is, the mapping $X\mapsto e_{\alpha}(X)$, provided that
$\alpha\geq\nicefrac{{1}}{{2}}$, satisfies the following four axioms
formulated for (convex) risk measures
$\mathcal{R}\colon\mathcal{Y}\to\mathbb{R}$, where $\mathcal{Y}$ is an
appropriate linear space of $\mathbb{R}$-valued random variables (for example
$\mathcal{Y}=L^{1}(P)$) on the probability space $(\Omega,\mathcal{F},P)$:

  1. (i)

$\mathcal{R}(X)\leq\mathcal{R}(Y)$ for all $X\leq Y$ almost everywhere,

  2. (ii)

$\mathcal{R}(X+Y)\leq\mathcal{R}(X)+\mathcal{R}(Y)$ for all $X$,
$Y\in\mathcal{Y}$,

  3. (iii)

$\mathcal{R}(\lambda\,X)=\lambda\,\mathcal{R}(X)$ for all $\lambda>0$, and

  4. (iv)

$\mathcal{R}(c+X)=c+\mathcal{R}(X)$ for all $c\in\mathbb{R}$.

The expectile is a risk functional satisfying the Axioms (i)–(iv) above
(Appendix 7 presents a brief proof for the subadditivity (ii), while the other
assertions are evident). Further, the expectile $e_{\alpha}(\cdot)$ is the
only risk measure which can be expressed as a minimizer – as in (1.1) – in
addition. We will elaborate below that the expectile is not a spectral risk
measure. The natural space (cf. Pichler (2013)) of expectiles is
$\mathcal{Y}=L^{1}$, cf. also the discussion in Section 1 above. In what
follows – unless stated differently – we will always assume that
$\mathcal{Y}=L^{1}$.

Explicit expressions for the expectiles are available only in exceptional
cases. For the uniform distribution in the interval $[0,1]$,
$U\sim\mathcal{U}[0,1]$, e.g., the expectile is
$e_{\alpha}(U)=\frac{\alpha-\sqrt{\alpha(1-\alpha)}}{2\alpha-1}$.

To extend expectiles to a risk measure in continuous time employing the Wiener
process (Brownian motion), we shall frequently need the expectile of the
normal distribution, for which at least the following series expansion is
available.

######  Example 2.1.

An explicit expression for the expectile of normally distributed random
variables, $X\sim\mathcal{N}(\mu,\sigma^{2})$, is not available. It holds that

$e_{\alpha}(X)=\mu+\sigma\sqrt{\frac{8}{\pi}}\Bigl{(}\alpha-\frac{1}{2}\Bigr{)}+\sigma\frac{8\sqrt{2}}{\sqrt{\pi}^{3}}\Bigl{(}\alpha-\frac{1}{2}\Bigr{)}^{3}+\mathcal{O}\Bigl{(}\alpha-\frac{1}{2}\Bigr{)}^{5}.$
(2.1)

###### Proof.

The general assertion derives from the standard normal distribution. Denoting
the density of the standard normal distribution by
$\varphi(t)=\frac{1}{\sqrt{2\pi}}e^{-t^{2}/2}$ and by
$\Phi(x)=\int_{-\infty}^{x}\varphi(t)\,dt$ its antiderivative, it holds that

$\displaystyle\operatorname{{\mathds{E}}}(X-t)_{+}$
$\displaystyle=\int_{t}^{\infty}(x-t)\varphi(x)\,dx=\varphi(t)-t\bigl{(}1-\Phi(t)\bigr{)}$
and $\displaystyle\operatorname{{\mathds{E}}}(t-X)_{+}$
$\displaystyle=\int_{-\infty}^{t}(t-x)\varphi(x)\,dx=t\,\Phi(t)+\varphi(t),$

which follows readily by employing the identity
$\varphi^{\prime}(x)=-x\,\varphi(x)$. Based on (1.3) define now

$\displaystyle f(\alpha,e)$
$\displaystyle\coloneqq\alpha\cdot\bigl{(}\varphi(e)-e\bigl{(}1-\Phi(e)\bigr{)}\bigr{)}-(1-\alpha)\cdot\bigl{(}e\,\Phi(e)+\varphi(e)\bigr{)}$
(2.2)
$\displaystyle=(2\alpha-1)\bigl{(}\varphi(e)+e\,\Phi(e)\bigr{)}-\alpha\,e$

so that the expectile $e_{\alpha}$ of the normally distributed random variable
$X$ satisfies $f(\alpha,e_{\alpha})=0$ for every $\alpha\in(0,1)$. We now
apply the implicit function theorem.

As $e_{\nicefrac{{1}}{{2}}}(X)=\operatorname{{\mathds{E}}}X=0$ for the normal
distribution it holds that $f(\nicefrac{{1}}{{2}},e_{\nicefrac{{1}}{{2}}})=0$.
Further, the partial derivatives of $f$ at $(\alpha,e)$ are
$f_{\alpha}(\nicefrac{{1}}{{2}},0)=\sqrt{\frac{2}{\pi}}$ and
$f_{e}(\alpha,e)=-\frac{1}{2}$ so that the first term in assertion (2.1)
follows with the implicit function theorem. The coefficient for the next term
$\bigl{(}\alpha-\nicefrac{{1}}{{2}}\bigr{)}^{2}$ is zero, because the function
(2.2) is odd with respect to the center $\nicefrac{{1}}{{2}}$, as the normal
distribution is symmetric, cf. (1.4). The remaining coefficient is found by
differentiating the function (2.2) further. We omit the rather technical
computations here, as our further results build on the first two terms only. ∎

######  Example 2.2.

For a log-normal random variable $X$ with $\log
X\sim\mathcal{N}(\mu,\sigma^{2})$, the expectiles are

$e_{\alpha}(X)=e^{\mu+\frac{\sigma^{2}}{2}}+\left(e^{\sigma^{2}}-1\right)e^{2\mu+\sigma^{2}}\,\Bigl{(}\alpha-\frac{1}{2}\Bigr{)}\,4\sqrt{e}\bigl{(}2\Phi(\nicefrac{{1}}{{2}})-1\bigr{)}+\mathcal{O}\Bigl{(}\alpha-\frac{1}{2}\Bigr{)}^{2}.$

###### Proof.

As above, the proof again relies on explicitly available expressions

$\displaystyle\operatorname{{\mathds{E}}}(X-t)_{+}$ $\displaystyle=\int_{\log
t}^{\infty}\bigl{(}e^{x}-t\bigr{)}\varphi(x)\,dx=\sqrt{e}\,\Phi(1-\log
t)-t\,\Phi(-\log t)$ and $\displaystyle\operatorname{{\mathds{E}}}(t-X)_{+}$
$\displaystyle=\int_{-\infty}^{\log
t}\bigl{(}t-e^{x}\bigr{)}\varphi(x)\,dx=\sqrt{e}\,\Phi(1-\log
t)-\sqrt{e}+t\,\Phi(\log t).$

The statement follows again by the implicit function theorem. ∎

###  2.1 Tight comparison with important risk measures

In what follows, we shall compare expectiles with important risk measures and
give the tightest-possible estimates and the smallest spectral risk measure
enveloping the expectiles.

The Average Value-at-Risk is the smallest convex envelope of the Value-at-Risk
(cf. Föllmer and Schied (2004)). The Average Value-at-Risk can be stated in
the equivalent forms (cf. Pflug (2000))

$\displaystyle\operatorname{\mathsf{AV@R}}_{\alpha}(X)$
$\displaystyle\coloneqq\frac{1}{1-\alpha}\int_{\alpha}^{1}F_{X}^{-1}(\alpha)\mathrm{d}\alpha$
$\displaystyle=\min\left\\{q+\frac{1}{1-\alpha}\operatorname{{\mathds{E}}}(X-q)_{+}\colon
q\in\mathbb{R}\right\\},$ (2.3) where
$\displaystyle\operatorname{\mathsf{V@R}}_{\alpha}(X)$ $\displaystyle\coloneqq
F_{X}^{-1}(\alpha)\coloneqq\inf\bigl{\\{}x\colon P(X\leq
x)\geq\alpha\bigr{\\}}$ (2.4)

is the Value-at-risk.

The Average Value-at-risk is the fundamental building block in the Kusoka
representation (cf. Kusuoka (2001)) and the most important risk functional in
actuarial practice. Notice as well that the Average Value-at-Risk is the
_minimum objective_ of an optimization problem (problem (2.3)), while the
expectile in (1.1) is the _minimizer_ of an optimization problem.

######  Remark 2.3 (Quantiles).

Similarly to the expectile, the Value-at-Risk defined in (2.4) is a minimizer
of an optimization problem, specifically the problem

$\displaystyle\min_{q\in\mathbb{R}}\operatorname{{\mathds{E}}}\tilde{\ell}_{\alpha}$
$\displaystyle(X-q)$ with scoring function
$\displaystyle\tilde{\ell}_{\alpha}(x)$
$\displaystyle\coloneqq\begin{cases}-(1-\alpha)\,x&\text{if }x\leq 0,\\\
\quad\alpha\cdot x&\text{if }x\geq 0\end{cases}\
=\Big{(}\alpha-\frac{1}{2}\Big{)}x+\frac{1}{2}\left|x\right|,$ (2.5)

well-known from quantile regression. Indeed, the first order condition is
$0=\frac{\partial}{\partial
q}\operatorname{{\mathds{E}}}\ell_{\alpha}(X-q)=\alpha\operatorname{{\mathds{E}}}{\mathds{1}}_{\left\\{X>q\right\\}}-(1-\alpha)\operatorname{{\mathds{E}}}{\mathds{1}}_{\left\\{X\leq
q\right\\}}=\alpha-P(X\leq q)$ and hence the assertion. However, by violating
(ii) above, the Value-at-Risk is _not_ a convex risk functional.

######  Definition 2.4 (Spectral risk measure, cf. Acerbi and Simonetti
(2002); Acerbi (2002)).

Let $\sigma\colon[0,1)\to\mathbb{R}_{\geq 0}$ be a non-negative, non-
decreasing function with $\int_{0}^{1}\sigma(u)\,\mathrm{d}u=1$. Then

$\mathcal{R}_{\sigma}(X)=\int_{0}^{1}F_{X}^{-1}(\alpha)\sigma(\alpha)\,\mathrm{d}\alpha,\qquad
X\in\mathcal{Y},$

is a risk measure. $\mathcal{R}_{\sigma}$ is called a the _spectral risk
measure_ and the function $\sigma$ is called the _spectrum_ of
$\mathcal{R}_{\sigma}$.

The expectiles are not a spectral risk measure themselves. But for every
expectile, there is a smallest spectral risk measure.

######  Proposition 2.5 (Enveloping risk measure).

If $\mathcal{R}_{\sigma}(X)$ is any spectral risk measure with

$e_{\alpha}(X)\leq\mathcal{R}_{\sigma}(X)$ (2.6)

for every random variable $X\in\mathcal{Y}$, then $e_{\alpha}(X)\leq
s_{\alpha}(X)\leq\mathcal{R}_{\sigma}(X)$ for all $X$, where

$s_{\alpha}(X)\coloneqq\int_{0}^{1}F_{X}^{-1}(u)\frac{\alpha(1-\alpha)}{\big{(}\alpha-u(2\alpha-1)\big{)}^{2}}\,\mathrm{d}u;$
(2.7)

that is, $s_{\alpha}$ is the smallest spectral risk measure larger than
$e_{\alpha}$.

Figure 1: The function $\Sigma_{\alpha}$ and $\tilde{\Sigma}_{\gamma}$,
exemplified for $\alpha=70\,\%$ and $\gamma=60\,\%$

###### Proof.

Above all, $s_{\alpha}(\cdot)$ is a spectral risk functional, as
$u\mapsto\frac{\alpha(1-\alpha)}{\bigl{(}\alpha-u(2\alpha-1)\big{)}^{2}}$ is a
non-negative, increasing function and
$\int_{0}^{1}\frac{\alpha(1-\alpha)}{\big{(}\alpha-u(2\alpha-1)\big{)}^{2}}\,\mathrm{d}u=1$.

Bellini et al. (2014, Proposition 9) provide the Kusuoka representation

$e_{\alpha}(X)=\max_{\gamma\in[\nicefrac{{1}}{{\beta}},1]}\gamma\operatorname{{\mathds{E}}}X+(1-\gamma)\operatorname{\mathsf{AV@R}}_{\frac{\beta-\frac{1}{\gamma}}{\beta-1}}(X)$
(2.8)

for expectiles, where $\beta=\frac{\alpha}{1-\alpha}$. Define the functions
$\Sigma_{\gamma}(u)\coloneqq\gamma(1-u)+(1-\gamma)\min\bigg{(}1,\frac{1-u}{1-\frac{\beta-\frac{1}{\gamma}}{\beta-1}}\bigg{)}$
and $\Sigma(u)\coloneqq\frac{\alpha(1-u)}{\alpha-u(2\alpha-1)}$. Both
functions coincide at $u=0$, $u=1$ and
$u=\frac{\alpha(1+\gamma)-1}{(2\alpha-1)\gamma}$; indeed
$\Sigma_{\gamma}(0)=\Sigma(0)=1$, $\Sigma_{\gamma}(1)=\Sigma(1)=0$ and

$\Sigma\left(\frac{\alpha(1+\gamma)-1}{(2\alpha-1)\gamma}\right)=\Sigma_{\gamma}\left(\frac{\alpha(1+\gamma)-1}{(2\alpha-1)\gamma}\right)=\frac{\alpha(1-\gamma)}{2\alpha-1}.$
(2.9)

As $\Sigma_{\gamma}$ is piece wise linear and $\Sigma$ concave, it follows
that $\Sigma_{\gamma}(u)\leq\Sigma(u)$ for all $u\in[0,1]$. With integration
by parts it follows further that

$\displaystyle\gamma\operatorname{{\mathds{E}}}X+(1-\gamma)\operatorname{\mathsf{AV@R}}_{\frac{\beta-\frac{1}{\gamma}}{\beta-1}}(X)$
$\displaystyle=-\int_{0}^{1}F_{X}^{-1}(u)\,\mathrm{d}\Sigma_{\gamma}(u)$
$\displaystyle=-\left.F_{X}^{-1}(u)\Sigma_{\gamma}(u)\right|_{u=0}^{1}+\int_{0}^{1}\Sigma_{\gamma}(u)\,\mathrm{d}F_{X}^{-1}(u)$
$\displaystyle\leq-\left.F_{X}^{-1}(u)\Sigma(u)\right|_{u=0}^{1}+\int_{0}^{1}\Sigma(u)\,\mathrm{d}F_{X}^{-1}(u)$
(2.10) $\displaystyle=-\int_{0}^{1}F_{X}^{-1}(u)\,\mathrm{d}\Sigma(u)$
$\displaystyle=s_{\alpha}(X)$

and thus $e_{\alpha}\leq s_{\alpha}$. The assertion follows, as for every
$u\in(0,1)$ there is
$\gamma\in\bigl{(}\frac{1-\alpha}{\alpha\thinspace},1\bigr{)}$
($\gamma=\frac{1-\alpha}{u(1-2\alpha)+\alpha}$) so that
$\Sigma_{\gamma}(u)=\Sigma(u)$ by (2.9) above (cf. Figure 1 for illustration).
∎

We have the following comparison with the Average Value-at-Risk. The
comparison is sharp in the sense that the risk rates cannot be improved.

######  Corollary 2.6.

For every random variable $X\in L^{1}$ it holds that

$\displaystyle e_{\frac{1}{2-\alpha}}(X)$
$\displaystyle\leq\operatorname{\mathsf{AV@R}}_{\alpha}(X),\qquad\alpha\in[0,1],$
(2.11) and
$\displaystyle\frac{\alpha}{3\alpha-1}\operatorname{{\mathds{E}}}X+\frac{2\alpha-1}{3\alpha-1}\operatorname{\mathsf{AV@R}}_{2-\frac{1}{\alpha}}(X)$
$\displaystyle\leq
e_{\alpha}(X)\leq\operatorname{\mathsf{AV@R}}_{2-\frac{1}{\alpha}}(X)$ (2.12)

for every $\alpha\in\left[\nicefrac{{1}}{{2}},1\right]$.

For non-negative random variables ($X\geq 0$ a.s.) we further have

$\displaystyle\operatorname{\mathsf{AV@R}}_{\alpha}(X)$
$\displaystyle\leq\frac{1}{1-\alpha}e_{\frac{1}{2-\alpha}}(X)$ (2.13) and
$\displaystyle e_{\alpha}(X)$
$\displaystyle\leq\frac{\alpha}{1-\alpha}\operatorname{{\mathds{E}}}X.$ (2.14)

The risk rates in the preceding equations (2.11)–(2.14) are optimal, they
cannot be improved.

######  Remark 2.7.

The preceding corollary might give the impression that $e_{\alpha}$ is ‘weak’
in the sense that it attains smaller values than the average value at risk and
is comparable to the risk neutral expectation. However, it holds that
$e_{\alpha}(X)\to 1$ for $\alpha\to 1$, as follows readily from (1.3).
Further, we have that the Average Value-at-Risk is a lower bound for the
expectiles in view of (2.13), so that expectiles are at least as ‘strong’ as
the Average Value-at-Risk.

###### Proof of Corollary 2.6.

Employing the notation of the proof of Proposition 2.5 and
$\Sigma_{\alpha}(u)\coloneqq\min\left(1,\frac{1-u}{\frac{1}{\alpha}-1}\right)$,
we have that $\Sigma(u)\leq\Sigma_{\alpha}(u)$. As in the proof above we
conclude that
$\mathcal{R_{\alpha}}(X)\leq\operatorname{\mathsf{AV@R}}_{\alpha}(X)$ and with
(2.6) that (2.11). The inequality (2.11) is tight, as
$\Sigma_{\gamma}^{\prime}(1)\xrightarrow[\gamma\to 1]{}\Sigma^{\prime}(1)$.

As for the remaining inequality choose $\gamma=\frac{\alpha}{3\alpha-1}$ in
(2.8), and replace $\alpha$ by $\frac{1}{2-\alpha}$ in (2.12) to obtain
(2.11).

The inequality
$\min\left(1,\frac{1-u}{1-\alpha}\right)\leq\frac{1}{1-\alpha}\Sigma_{\gamma}(u)$
is evident for every $u\in[0,1]$, and the remaining assertion (2.13) follows
by the same reasoning as above. However, for inequality (2.10) to hold true it
is essential that $X\geq 0$ a.s.

Hölder’s inequality, applied to (2.7), gives

$\operatorname{{\mathds{E}}}X\leq
s_{\alpha}(X)\leq\int_{0}^{1}F_{X}^{-1}(u)\,\mathrm{d}u\cdot\max_{u\in[0,1]}\frac{\alpha(1-\alpha)}{\bigl{(}\alpha-u(2\alpha-1)\bigr{)}^{2}}=\operatorname{{\mathds{E}}}X\cdot\frac{\alpha}{1-\alpha}$

and thus (2.14). ∎

##  3 Conditional and Dynamic Risk Measure

Risk functionals – as discussed above – are employed to assess the risk of a
random outcome. For this reason, they have the economic interpretation of an
insurance premium, while the random outcome is the random insurance benefit
(the random variable). While the premium is known beforehand, the insurance
benefit (the random outcome) is not, it is revealed later.

Conditional risk measures are employed in risk management over time, they
address stochastic processes instead of random variables. Nested risk
measures, which are compositions of risk functionals over time, enjoy the
economic interpretation of risk premiums for insurance on a rolling horizon
basis. For a discussion of nested risk functionals we may refer to Cheridito
and Kupper (2011); Riedel (2004); Shapiro (2012); Ruszczyński and Shapiro
(2006) and Pichler and Schlotter (2019).

###  3.1 The conditional expectile

Definition 1.1 allows extending the expectile to conditional expectiles, which
are conditioned on some $\sigma$-algebra. This constitutes a major building
block to extend the definition of expectiles from random variables to
stochastic processes.

######  Definition 3.1 (Conditional expectiles).

Let $X\in L^{1}$ be a random variable and $\mathcal{G}$ be a sub
$\sigma$-algebra of $\mathcal{F}$, $\mathcal{G}\subset\mathcal{F}$ and
$\alpha$ a $\mathcal{G}$-measureable variable with values in $[0,1]$. The
$\mathcal{G}$-measureable random variable $Z$ satisfying

$\alpha\cdot\operatorname{{\mathds{E}}}\bigl{(}\bigl{(}X-Z\bigr{)}_{+}\mid\mathcal{G}\bigr{)}=(1-\alpha)\cdot\operatorname{{\mathds{E}}}\bigl{(}\bigl{(}Z-X\bigr{)}_{+}\mid\mathcal{G}\bigr{)}\qquad\text{a.s.}$
(3.1)

is called the _conditional expectile_ (i.e., the conditional version of (1.3))
and denoted $Z=e_{\alpha}(X\mid\mathcal{G})$. As usual for the conditional
expectation, we shall also write $e^{\mathcal{G}}(X)\coloneqq
e(X\mid\mathcal{G})$ and $e^{Y=y}(X)\coloneqq e(X\mid Y=y)$ for the
conditional expectile and its versions.

The solution of the problem (3.1) exists and is unique for the same reasons as
for the usual expectile, and $e_{\alpha}(X\mid\mathcal{G})\in L^{1}$, as
$\bigl{(}e_{\alpha}(X\mid\mathcal{G})-X\bigr{)}_{+}$ and
$\operatorname{{\mathds{E}}}\bigl{(}\bigl{(}X-e_{\alpha}(X\mid\mathcal{G})\bigr{)}_{+}\mid\mathcal{G}\bigr{)}$
exist in (3.1).

######  Remark 3.2.

Based on the properties of the conditional expectation (cf. Section 2), we
have the following properties of the conditional expectile.

  1. (i)

$e_{\alpha}^{\mathcal{G}}(X)\leq e_{\alpha}^{\mathcal{G}}(Y)$ a.e. for all
$X\leq Y$ almost everywhere,

  2. (ii)

$e_{\alpha}^{\mathcal{G}}(X+Y)\leq
e_{\alpha}^{\mathcal{G}}(X)+e_{\alpha}^{\mathcal{G}}(Y)$ a.e.,

  3. (iii)

$e_{\alpha}^{\mathcal{G}}(\lambda\,X)=\lambda\,e_{\alpha}^{\mathcal{G}}(X)$
for all $\lambda>0$ and $\lambda$ which is $\mathcal{G}$-measurable,

  4. (iv)

$e_{\alpha}^{\mathcal{G}}(c+X)=c+e_{\alpha}^{\mathcal{G}}(X)$ for all
$\mathbb{R}$-valued $c$ measurable with respect to $\mathcal{G}$.

In what follows, we shall consider the conditional expectile for a single
$\sigma$-algebra first and discuss regression. Next, we consider filtrations
$\mathcal{F}=(\mathcal{F}_{t})_{t\in\mathcal{T}}$, typically generated by a
stochastic process $X=(X_{t})_{t\in\mathcal{T}}$.

###  3.2 Conditional expectiles in stochastic optimization and regression

Stochastic optimization and most typical problems in machine learning (as the
training of neural networks) as well as specific problems in inverse problems
(cf. Lu and Pereverzev (2013)) consider the problem

minimize $\displaystyle f_{0}(x)\coloneqq\operatorname{{\mathds{E}}}f(x,\xi)$
(3.2) subject to $\displaystyle x\in\mathcal{X},$

where the objective is a risk neutral expectation,
$f\colon\mathcal{X}\times\mathbb{R}^{m}\to\mathbb{R}$ is a function,
$\mathcal{X}\subset\mathbb{R}^{d}$ is closed and $\xi$ is a random variable
with values in $\mathbb{R}^{m}$. Sample average approximation builds on
independent realizations $\xi_{i}$ of identically distributed random variable
$\xi$, $i=1,\dots$, to solve (3.2) in real world applications. To this end,
the empirical version

$\hat{f}_{n}(x)\coloneqq\frac{1}{n}\sum_{i=1}^{n}f(x,\xi_{i})$

is considered instead of the expectation $\operatorname{{\mathds{E}}}f(x,\xi)$
in (3.2) for varying $x\in\mathcal{X}$.

We consider the measure points (observations) $X\in\mathcal{X}$ to be random
(with measure $P$) as well and intend to ‘learn’ the function $f_{0}$ based on
observations

$\bigl{(}X_{i},f(X_{i},\xi_{i})\bigr{)},\quad i=1,\dots,n,$ (3.3)

where $(X_{i},\xi_{i})$ are revealed jointly (cf. Dentcheva and Lin (2021) for
further motivation in stochastic optimization and an alternative approach);
even more generally, we consider the iid observations

$(X_{i},f_{i}),\quad i=1,\dots n,$ (3.4)

which is (3.3) with $f_{i}\coloneqq f(X_{i},\xi_{i})$.

To model (3.4), let $\rho$ be the probability measure of the joint
distribution $(X,f)$ and denote the marginal measure by
$P(A)\coloneqq\rho(A\times\mathbb{R})$. Then there exists a regular
conditional probability kernel (cf. Kallenberg (2002)) so that

$\rho(A\times B)=\int_{A}\rho(f\in B|\,x)\,P(dx).$ (3.5)

The bivariate measure $\rho$ in (3.5) is not an artifact. Indeed, denote the
conditional measures of $f$ given $X$ by the Markov kernel
$\rho\colon\mathcal{X}\times\mathcal{B}(\mathbb{R})\to[0,1]$, that is,
$\rho(f\in A\mid X=x)=\rho(x,A)$, then $(X,f)$ jointly follow the composed
measure (3.5),

$(X,f)\sim\rho,$

and hence both approaches are equivalent.

For a random vector $(X,f)\in\mathbb{R}^{d}\times\mathbb{R}$ with law $\rho$
set

$f_{0}(x)\coloneqq\operatorname{{\mathds{E}}}(f\mid X=x);$ (3.6)

this definition notably corresponds to

$f_{0}(x)=\operatorname{{\mathds{E}}}(f(X,\xi)\mid X=x)$

in the setting (3.3) above. For this reason, the stochastic optimization
problem (3.2) is equivalent to333The essential
infimum$\operatorname*{ess\,inf}(f\mid X)$ is the largest random variable $g$,
measurable with respect to $\sigma(X)$ (the $\sigma$-algebra generated by
$X$), so that $g\leq f$, cf. Föllmer and Schied (2004, Definition A.34).
Measurability is the crucial difference in comparison to the (unconditional)
essential supremum in Footnote 2.

$\operatorname*{ess\,inf}_{x\in\mathcal{X}}\operatorname{{\mathds{E}}}(f\mid
X=x),$ (3.7)

where $(X,f)$ is a random variable with law $\rho$, provided that
$\operatorname{supp}P=\mathcal{X}$, where

$\operatorname{{supp}}P\coloneqq\bigcap\big{\\{}A\colon A\text{ is closed and
}P(A)=1\big{\\}}$

is the support.444Cf. Rüschendorf (2014) for the support of the marginal
measure $P$.

Note, however, that not every random vector $(X,f)$ can be recast as in (3.3)
for a function $f$ and a random $\xi$. For this reason, the problem
formulation (3.7) is more general than the genuine problem (3.2).

###  3.3 Risk assessment with conditional expectiles

To incorporate risk in the assessment, consider the conditional expectation
(3.6) and define

$f_{\alpha}(x)\coloneqq e_{\alpha}(f\mid X=x),$

where $e_{\alpha}^{\sigma(X)}$ is the conditional expectile introduced in
Section 3.1 above. Based on (1.5), we have that

$f_{\alpha}(x)\geq f_{0}(x),\qquad\text{for }\alpha\geq\nicefrac{{1}}{{2}},\
x\in\mathcal{X}.$

The function $f_{\alpha}$ intentionally _overestimates_ (overrates) the risk-
free assessment $f_{0}$ and the surplus $f_{\alpha}-f_{0}$ is the amount
attributed to risk aversion.

To solve the risk averse version of the stochastic optimization problem (3.7),

minimize $\displaystyle e_{\alpha}(f\mid X=x)$ (3.8) subject to $\displaystyle
x\in\mathcal{X},$

just find an estimator for $\hat{e}_{\alpha}$ for $e_{\alpha}$ first and then
solve

minimize $\displaystyle\hat{e}_{\alpha}(x)$ (3.9) subject to $\displaystyle
x\in\mathcal{X}.$

The substitute $\hat{e}_{\alpha}(\cdot)$ is chosen in an adequate space of
functions. Dentcheva and Lin (2021) consider the Nadaraya–Watson kernel
estimator to solve the problem. Here, we exploit the problem by using
reproducing kernel Hilbert spaces (RKHS) with kernel function $k$, where we
may refer to Berlinet and Thomas-Agnan (2004) for details.

######  Definition 3.3.

For a kernel function $k\colon\mathcal{X}\times\mathcal{X}\to\mathbb{R}$, the
RKHS space $\mathcal{H}_{k}$ is the completion of the functions
$f(x)=\sum_{i=1}^{\ell}w_{i}\,k(x,x_{i})$ with respect to the inner product

$\left\langle k(\cdot,x_{i})\mid
k(\cdot,x_{j})\right\rangle=k(x_{i},x_{j}),\qquad i,j=1,\dots,\ell,$

where $x_{i}$ and $x_{j}\in\mathcal{X}$.

The regularized problem is

$\text{minimize
}\frac{1}{n}\sum_{i=1}^{n}\ell_{\alpha}\bigl{(}\hat{e}_{\alpha}(X_{i})-f_{i}\bigr{)}+\lambda\|\hat{e}_{\alpha}\|_{k}^{2},$
(3.10)

where $\hat{e}_{\alpha}(\cdot)\in\mathcal{H}_{k}$. It follows from the
generalized representer theorem (cf. Schölkopf et al. (2001)), that the
function $e_{\alpha}$ is given by
$e_{\alpha}(\cdot)=\frac{1}{n}\sum_{i=1}^{n}w_{i}\,k(\cdot,X_{i})$, that is,
the supporting points are exactly the points $X_{i}$, $i=1,\dots,n$, where
measurements $f_{i}$, $i=1,\dots,n$, are available. It might be convenient in
some situations to find the best approximation located at the points
$\tilde{x}_{j}$, $j=1,\dots,\tilde{n}$, that is, the function

$\hat{e}_{\alpha}(\cdot)=\frac{1}{\tilde{n}}\sum_{j=1}^{\tilde{n}}w_{j}k(\cdot,\tilde{x}_{j}),$

for fewer or special design points $\tilde{x}_{j}$, $j=1,\dots,\tilde{n}$. We
describe the equations for this generalized problem.

Input: Measurements $(f_{i},X_{i})$, $i=1,\dots n$, and support points
$\tilde{x}_{j}$, $j=1,\dots,\tilde{n}$.

Output: The weights $w_{j}$, $j=1,\dots,\tilde{n}$, of the function
$\hat{e}_{\alpha}(\cdot)=\frac{1}{\tilde{n}}\sum_{j=1}^{\tilde{n}}w_{j}k(\cdot,\tilde{x}_{j})$
(3.11) minimizing (3.10).

Set  $K_{ij}\coloneqq k(X_{i},\tilde{x}_{j})$ (3.12) for $i=1,\dots n$ and
$j=1,\dots\tilde{n}$, and  $\tilde{K}_{ij}\coloneqq
k(\tilde{x}_{i},\tilde{x}_{j})$ (3.13) for $i,j=1,\dots,\tilde{n}$.

while  _change of the weights $w$ encountered_ do

for  _$i=1$ to $n$_ do

update $A_{ii}\leftarrow\begin{cases}\alpha&\text{if
}f_{i}\leq\frac{1}{\tilde{n}}\sum_{j=1}^{\tilde{n}}w_{j}\,k(X_{i},\tilde{x}_{j}),\\\
1-\alpha&\text{else}\end{cases}$

end for

update  $w\leftarrow
w-\Bigl{(}\frac{\lambda}{\tilde{n}^{2}}\tilde{K}+\frac{1}{n^{2}\tilde{n}}K^{\top}AK\Bigr{)}^{-1}\cdot\Bigl{(}\frac{\lambda}{\tilde{n}^{2}}\tilde{K}w+\frac{1}{n^{2}\tilde{n}}K^{\top}AKw-\frac{1}{n\tilde{n}}K^{\top}Af\Bigr{)}$
(3.14)

end while

Result: The best approximating function (3.11).

Algorithm 1 Newton-like iteration to solve (3.10)

The first order conditions of problem (3.10) for the weights $w_{j}$,
$j=1,\dots,\tilde{n}$, are

$\displaystyle 0=\frac{1}{n}\sum_{i=1}^{n}2$
$\displaystyle\cdot\left(\frac{1}{\tilde{n}}\sum_{j^{\prime}=1}^{\tilde{n}}w_{j^{\prime}}k(X_{i},\tilde{x}_{j^{\prime}})-f_{i}\right)\cdot\left\\{\begin{array}[]{cc}\alpha&\text{if
}f_{i}\leq\hat{e}_{\alpha}(X_{i})\\\ 1-\alpha&\text{if
}f_{i}\geq\hat{e}_{\alpha}(X_{i})\end{array}\right\\}\cdot\frac{1}{\tilde{n}}k(\tilde{x}_{j},X_{i})\
+$ (3.17)
$\displaystyle+2\frac{\lambda}{\tilde{n}^{2}}\sum_{j=1}^{\tilde{n}}w_{j}k(\tilde{x}_{i},\tilde{x}_{j}).$
(3.18)

Define
$\tilde{K}\coloneqq\bigl{(}k(\tilde{x}_{\ell},\tilde{x}_{j})\bigr{)}_{\ell,j=1}^{n}$,
$K\coloneqq\bigl{(}k(X_{i},\tilde{x}_{j})\bigr{)}_{i=1,j=1}^{n,\tilde{n}}$ and

$\displaystyle A(w)$
$\displaystyle\coloneqq\operatorname{diag}\bigl{(}a_{i}(w),\
i=1,\dots,n\bigr{)}$ with entries $\displaystyle a_{i}(w)$
$\displaystyle=\begin{cases}\alpha&\text{if
}f_{i}\leq\frac{1}{\tilde{n}}\sum_{j=1}^{n}w_{j}k(X_{i},\tilde{x}_{j}),\\\
1-\alpha&\text{if
}f_{i}\geq\frac{1}{\tilde{n}}\sum_{j=1}^{n}w_{j}k(X_{i},\tilde{x}_{j})\end{cases}$

on the diagonal. Then the equations (3.18) rewrite as

$\left(\frac{\lambda}{\tilde{n}^{2}}\tilde{K}+\frac{1}{n^{2}\tilde{n}}K^{\top}A(w)K\right)w=\frac{1}{n\tilde{n}}K^{\top}A(w)f.$
(3.19)

This equation is not linear in $w$, as $A(w)$ depends in a nonlinear way on
$w$. However, the problem can be solved by inverting the matrix to obtain a
fixed point equation. With that, the equation can be iterated, and the
algorithm converges after finitely many iterations, cf. (3.14) in Algorithm 1.
Figure 2 displays a typical result of expectile regression. Farooq and
Steinwart (2018) is a starting point in investigating convergence properties
of the expectile regression problem.

Figure 2: The expectile $\hat{e}_{90\,\%}(\cdot)$ based on $n=1000$
observations overestimates the conditional expectation

######  Remark 3.4.

Note that the inverted matrix in (3.14) is the derivative of the right-hand
side with respect to $w$, as $A$ is constant for small changes in $w$. For
this reason, the iteration in Algorithm 1 is a Newton iteration in essence,
although the function (1.2) is not differentiable. As $A(w)$ is constant for
small variations of $w$, thus (3.14) vanishes locally.

##  4 Risk aversion in stochastic processes

The considerations on the expectile in the preceding sections are based on
random variables. The conditional variant in the expectile regression is
achieved with a single $\sigma$-algebra. In what follows, we generalize the
expectile for stochastic processes – in a discrete time setting first, and
then in continuous time.

###  4.1 Nested expectile in discrete time

Consider a stochastic process $X=(X_{t_{i}})_{i=0}^{n}$ in discrete time,
where $0\eqqcolon t_{0}<t_{1}<\dots<t_{n}=T$. For a dissection in time
consider the increments

$X_{T}=X_{t_{0}}+(X_{t_{1}}-X_{t_{0}})+\dots+(X_{t_{n}}-X_{t_{n-1}}).$

The stochastic process $X$ is adapted to the filtration $\mathcal{F}$, that
is, $X_{t}$ is measurable for every $\mathcal{F}_{t}$, $t\geq 0$, so most
often we just may choose $\mathcal{F}_{t_{i}}\coloneqq\sigma(X_{t_{j}}\colon
j\leq i)$. As well, we shall denote the sequence of $\sigma$-algebras by
$\mathcal{F}_{t_{0}\colon t_{n}}$.

In what follows, we shall associate a certain risk for the time period $\Delta
t\coloneqq t_{i+1}-t_{i}$ to come. For convenience in the presentation in what
follows, we introduce the _rescaled_ version of the expectile as

$\tilde{e}_{\beta}(\cdot)\coloneqq e_{\frac{1+\sqrt{\beta}}{2}}(\cdot)$

(i.e., $\alpha-\frac{1}{2}=\frac{\sqrt{\beta}}{2}$). The main reason for the
rescaling is that $e_{\nicefrac{{1}}{{2}}}(X)=\operatorname{{\mathds{E}}}X$,
while $\operatorname{\mathsf{AV@R}}_{0}(X)=\operatorname{{\mathds{E}}}X$, e.g.
To ensure consistent parametrizations with other risk measures, we rescale the
risk level so that $\tilde{e}_{0}(X)=\operatorname{{\mathds{E}}}X$ is
associated with the risk-free assessment, while
$\tilde{e}_{1}(X)=\operatorname*{ess\,sup}X$ is the total risk averse
assessment. The varying dynamic ($\sqrt{\beta}$ instead of $\beta$) turns out
to be the natural choice in the continues time situation addressed below.

######  Definition 4.1 (Nested expectile).

Let $\bigl{(}\Omega,\mathcal{F}=(\mathcal{F}_{t_{i}})_{i=1}^{n},P\bigr{)}$ be
a filtered probability space and $\beta\colon\\{t_{0},\dots,t_{n}\\}\to[0,1]$
be stochastic process adapted to the filtration
$\mathcal{F}=(\mathcal{F}_{t_{i}})_{i=1}^{n}$. The nested expectile of the
process with respect to the filtration $\mathcal{F}_{t_{0}:t_{n}}$, denoted
$\tilde{e}_{\beta(\cdot)}^{\mathcal{F}_{t_{0}:t_{n}}}$, is

$\tilde{e}_{\beta(\cdot)}^{\mathcal{F}_{t_{0}:t_{n}}}(X)\coloneqq
X_{0}+\tilde{e}_{\beta(t_{0},X_{t_{o}})\cdot(t_{1}-t_{0})}^{\mathcal{F}_{0}}\left(X_{t_{1}}-X_{t_{0}}+\dots+\tilde{e}_{\beta(t_{n-1},X_{t_{n-1}})\cdot(t_{n}-t_{n-1})}^{\mathcal{F}_{t_{n-1}}}\left(X_{T}-X_{t_{n-1}}\right)\right),$
(4.1)

or slightly more explicitly

$\tilde{e}_{\beta(\cdot)}^{\mathcal{F}_{t_{0}:t_{n}}}(X)=X_{0}+\tilde{e}_{\beta(t_{0},X_{t_{0}})\cdot(t_{1}-t_{0})}^{\mathcal{F}_{0}}\begin{pmatrix}\begin{array}[]{l}X_{t_{1}}-X_{t_{0}}+\\\
\dots+\tilde{e}_{\beta(t_{n-2},X_{t_{n-2}})\cdot(t_{n-1}-t_{n-2})}^{\mathcal{F}_{t_{n-2}}}\begin{pmatrix}\begin{array}[]{l}X_{t_{n-1}}-X_{t_{n-2}}\\\
+\tilde{e}_{\beta(t_{n-1},X_{t_{n-1}})\cdot(t_{n}-t_{n-1})}^{\mathcal{F}_{t_{n-1}}}\left(X_{T}-X_{t_{n-1}}\right)\end{array}\end{pmatrix}\end{array}\end{pmatrix}.$

Nested Risk measures have been considered by Philpott et al. (2013); Philpott
and de Matos (2012), e.g. In discrete time, fundamental properties of the
Average Value-at-Risk have been elaborated by Xin and Shapiro (2012), although
for deterministic risk rates only and for random variables instead of
stochastic processes. The definition above is dynamic, as the risk rate
$\beta$ is an adapted process itself. Note that the risk rate at time $t$ may
be chosen to reflect the history of observations up to $t$, it may depend on
$\\{t_{i}\leq t\colon i=1,\dots,n\\}$.

We consider the following example, which prepares for the Wiener process.

######  Example 4.2 (Random walk, cf. Pichler and Schlotter (2022)).

Consider a random walk process starting at $X_{0}$ with independent Markovian
increments

$X_{t_{i+1}}-X_{t_{i}}\sim\mathcal{N}(0,t_{i+1}-t_{i})$ (4.2)

and constant risk rate $\beta(t,x)=\beta$. With (4.1) and the asymptotic
formula (4.1) for the normal distribution, we have that

$\displaystyle
X_{t_{1}}+\tilde{e}_{\beta}^{\mathcal{F}_{t_{1}}}(X_{t_{2}}-X_{t_{1}})$
$\displaystyle=X_{t_{1}}+\sqrt{t_{i+1}-t_{i}}\sqrt{\frac{2}{\pi}}\sqrt{\beta(t_{i+1}-t_{i})}+o\bigl{(}t_{t+1}-t_{i}\bigr{)}$
$\displaystyle=X_{t_{1}}+\sqrt{\frac{2\beta}{\pi}}(t_{i+1}-t_{i})+o\bigl{(}t_{t+1}-t_{i}\bigr{)}.$
(4.3)

Nesting these expressions as in Definition 4.1 gives the explicit expression

$\tilde{e}_{\beta(\cdot)}^{\mathcal{F}_{t_{0}:T}}(X)=X_{0}+\sqrt{\frac{2\beta}{\pi}}T+o(T),$
(4.4)

where $T$ is the terminal time, while

$\tilde{e}_{0}^{\mathcal{F}_{t_{0}:T}}(X)=X_{0}$

for the risk rate $\beta=0$. The amount attributed to the risk averse
assessment in (4.4) thus accumulates linearly with time.

######  Remark 4.3 (Tower property).

We emphasize as well that Definition 4.1 explicitly involves time, the risk
$\beta(t_{i})\cdot(t_{i+1}-t_{i})$ is associated to the time interval starting
at $t_{i}$ and ending at $t_{i+1}$. With a further point in between,
$t_{i+\nicefrac{{1}}{{2}}}$, the components of the risk functionals above are

$\tilde{e}_{\beta(t_{i})\cdot(t_{i+\nicefrac{{1}}{{2}}}-t_{i})}^{\mathcal{F}_{0}}\left(X_{t_{i+\nicefrac{{1}}{{2}}}}-X_{t_{i}}+\tilde{e}_{\beta(t_{i+\nicefrac{{1}}{{2}}})\cdot(t_{i+1}-t_{i+\nicefrac{{1}}{{2}}})}^{\mathcal{F}_{t_{i+\nicefrac{{1}}{{2}}}}}\left(X_{t_{i+1}}-X_{t_{i+\nicefrac{{1}}{{2}}}}\right)\right)$

instead of

$\tilde{e}_{\beta(t_{i})\cdot(t_{i+1}-t_{i})}^{\mathcal{F}_{0}}\left(X_{t_{i+1}}-X_{t_{i}}\right).$

With that, the risk rates accumulate over time: accumulated risk rates are
$\beta(t_{i})(t_{i+\nicefrac{{1}}{{2}}}-t_{i})+\beta(t_{i+\nicefrac{{1}}{{2}}})(t_{i+1}-t_{i-\nicefrac{{1}}{{2}}})$
in the first case. This amount indeed coincides with
$\beta(t_{i})(t_{i+1}-t_{i})$ (this is the risk rate in the second case),
provided that $\beta(t_{i})=\beta(t_{i+\nicefrac{{1}}{{2}}})$, i.e., the risk
assessment does not vary over time.

For the expectation, the corresponding property is the tower property, that
is,
$\operatorname{{\mathds{E}}}\bigl{(}\operatorname{{\mathds{E}}}X\mid\mathcal{G})\bigr{)}=\operatorname{{\mathds{E}}}X$.

###  4.2 The nested expectile in continuous time

In order to assign risk to a stochastic process in continuous time, we
consider the nested formulation introduced above for decreasing time-steps.

######  Definition 4.4 (Nested expectile).

Let $X=(X_{t})_{t\leq T}$ be a stochastic process adapted to
$\mathcal{F}=(\mathcal{F}_{t})_{t\leq T}$ and $\beta=(\beta_{t})_{t\leq T}$ be
càdlàg (i.e., right continuous, with left limits) and adapted. With the nested
expectile defined in Definition 4.1, the nested expectile is

$\tilde{e}_{\beta}^{\mathcal{F}}(X)=\lim_{\max\Delta t\to
0}\tilde{e}_{\beta_{t_{0}:t_{n}}}^{\mathcal{F}_{t_{0}:t_{n}}}(X),$ (4.5)

provided that the limit with respect to decreasing mesh sizes $\max\Delta
t\coloneqq\max_{i=1}^{n}t_{i+1}-t_{i}$ exists.

######  Example 4.5 (State independent risk rates).

Example 4.2 generalizes for a state independent, but time dependent Riemann
integrable risk rate $\beta(x,t)=\beta(t)$. As above, we obtain that

$X_{t_{1}}+\tilde{e}_{\beta}^{\mathcal{F}_{t_{1}}}(X_{t_{2}}-X_{t_{1}})=X_{t_{1}}+\sqrt{\frac{2\beta(t_{i})}{\pi}}(t_{i+1}-t_{i})+o\bigl{(}t_{t+1}-t_{i}\bigr{)}$

and thus

$\tilde{e}_{\beta(\cdot)}^{\mathcal{F}_{t_{0}:T}}(X)=X_{0}+\sqrt{\frac{2}{\pi}}\int_{0}^{T}\sqrt{\beta(t)}\,\mathrm{d}t$

for $\Delta t\to 0$, as $\beta$ is Riemann integrable. Again, this is an
explicit expression for the total risk aversion of the entire random walk
process with increments (4.2).

######  Definition 4.6 (Risk generator).

Let $(X_{t})_{t\geq 0}$ be a stochastic process adapted to the filtration
$\sigma(X)$ and $\beta(t,x)$ be a risk rate. The risk generator is

$\mathcal{G}_{\beta}f(x,t)\coloneqq\lim_{h\to
0}\frac{\tilde{e}_{\beta(t,X_{t})}^{\sigma(X)}\bigl{(}f(X_{t+h})|\,X_{t}=x\bigr{)}-f(x)}{h},$

provided that the limit exists.

Note, that $\mathcal{G}$ is an operator, which maps the (smooth) function $f$
to $\mathcal{G}_{\beta}f$, which is a function again. In contrast to the risk-
neutral generator, the risk generator $\mathcal{G}_{\beta}$ is possibly not
linear, as we will see in what follows.

######  Proposition 4.7.

Let $X_{t}$ follow the stochastic differential equation

$\mathrm{d}X_{t}=\mu(t,X_{t})\,\mathrm{d}t+\sigma(t,X_{t})\,\mathrm{d}W_{t}$
(4.6)

with respect to the Wiener process (Brownian motion) $(W_{t})_{t\geq 0}$ and
the functions $\mu$ and $\sigma$ be Lipschitz, i.e.,
$|\mu(t,x)-\mu(t,y)|+|\sigma(t,x)-\sigma(t,y)|\leq K|x-y|$ so that strong
solutions of (4.6) exist. For a smooth function $f$, the risk generator is

$\displaystyle\mathcal{G}_{\beta}f(t,x)$ $\displaystyle=\frac{\partial
f(t,x)}{\partial t}+\mu(t,x)\cdot\frac{\partial f(t,x)}{\partial
x}+\frac{1}{2}\sigma(t,x)^{2}\cdot\frac{\partial^{2}f(t,x)}{\partial x^{2}}$
$\displaystyle\qquad+\sqrt{\frac{2}{\pi}\beta(t,x)}\cdot\left|\sigma(x,t)\cdot\frac{\partial
f(t,x)}{\partial x}\right|.$ (4.7)

###### Proof.

The proof follows Øksendal (2003, Section 7.3) (another valuable reference is
Karatzas and Shreve (1991)).

Consider the stochastic process $Y_{t}\coloneqq f(t,X_{t})$. From Ito’s rule
we deduce that

$\displaystyle Y_{t+\Delta t}$ $\displaystyle=Y_{t}+\int_{t}^{t+\Delta
t}\left(\frac{\partial f}{\partial t}(s,X_{s})+\mu(s,X_{s})\frac{\partial
f}{\partial
x}(s,X_{s})+\frac{1}{2}\sigma(s,X_{s})^{2}\frac{\partial^{2}f}{\partial
x^{2}}(s,X_{s})\right)\mathrm{d}s$ $\displaystyle\qquad+\int_{t}^{t+\Delta
t}\sigma(s,X_{s})\frac{\partial f}{\partial x}(s,X_{s})\mathrm{d}W_{s},$

where the second part is a martingale with increments following the Wiener
process. Following the proof of the Ito formula in Øksendal (2003, p. 46ff),
the functions $\mu$ and $\sigma$ are approximated by the constants
$\mu(s,X_{s})\approx\mu(t,X_{t})$ and $\sigma(s,X_{s})\approx\sigma(t,X_{t})$
for $s\in[t,t+\Delta t)$ so that

$\displaystyle Y_{t+\Delta t}-Y_{t}=$ $\displaystyle\left(\frac{\partial
f}{\partial t}(t,X_{t})+\mu(t,X_{t})\frac{\partial f}{\partial
x}(t,X_{t})+\frac{1}{2}\sigma(t,X_{t})^{2}\frac{\partial^{2}f}{\partial
x^{2}}(t,X_{t})\right)\Delta t$
$\displaystyle\qquad+\sigma(t,X_{t})\frac{\partial f}{\partial
x}(t,X_{t})\cdot\bigl{(}W_{t+\Delta t}-W_{t}\bigr{)}.$

$Y_{t+\Delta t}-Y_{t}$ is a normally distributed random variable with mean

$Y_{t}+\left(\frac{\partial f}{\partial t}(t,X_{t})+\mu(t,X_{t})\frac{\partial
f}{\partial
x}(t,X_{t})+\frac{1}{2}\sigma(t,X_{t})^{2}\frac{\partial^{2}f}{\partial
x^{2}}(t,X_{t})\right)\Delta t$

and variance

$\left(\sigma(t,X_{t})\frac{\partial f}{\partial x}(t,X_{t})\right)^{2}\Delta
t.$

We deduce from (2.1) that

$\displaystyle\tilde{e}_{\beta\cdot\Delta t}^{X_{t}}(Y_{t+\Delta t})-Y_{t}$
$\displaystyle=\left(\frac{\partial f}{\partial
t}(t,X_{t})+\mu(t,X_{t})\frac{\partial f}{\partial
x}(t,X_{t})+\frac{1}{2}\sigma(t,X_{t})^{2}\frac{\partial^{2}f}{\partial
x^{2}}(t,X_{t})\right)\Delta t$
$\displaystyle\qquad+\left|\sigma(t,X_{t})\frac{\partial f}{\partial
x}(t,X_{t})\right|\sqrt{\Delta
t}\cdot\sqrt{\frac{8}{\pi}}\left(\frac{1+\sqrt{\beta(t,x)\,\Delta
t}}{2}-\frac{1}{2}\right).$

Now, by the definition of the risk generator (4.5), we get the assertion. ∎

######  Remark 4.8.

The drift (4.3) in Example 4.2 now turns out to be a specific case of the
general relation revealed by (4.7), both reveal the same pattern: any risk
averse assessment adds the additional drift term

$\sqrt{\frac{2}{\pi}\beta(t,x)}\cdot\left|\sigma(x,t)\cdot\frac{\partial
f(t,x)}{\partial x}\right|.$

For the absolute value $|\cdot|$ in the expression, the additional drift term
cannot be negative and always points in one direction, the direction of risk.
This is in line with risk aversion, as deviations in the different directions
are associated with profits and (for the other direction) losses. Further, the
coefficient $\beta$ models the amount of local risk aversion.

The behavior (4.7) has been found with other risk measures as well, for
example for the Entropic Value-at-Risk, cf. Pichler and Schlotter (2022). For
this reason, various results from the literature extend to the nested
expectile.

##  5 The risk averse control problem

While the classical theory on dynamic optimization builds on the risk-neutral
expectation (cf. Fleming and Soner (2006)), we take risk into consideration to
the optimal control problem and derive a risk averse variant of the
Hamilton–Jacobi–Bellman equation. In what follows we derive the governing
equations formally by adapting the presentation from Pichler and Schlotter
(2022) for expectiles.

Consider the stochastic differential equation

$\mathrm{d}X_{t}^{u}=\mu\bigl{(}t,X_{t}^{u},u(t,X_{t}^{u})\bigr{)}\mathrm{d}t+\sigma\bigl{(}t,X_{t}^{u},u(t,X_{t}^{u})\bigr{)}\mathrm{d}W_{t}$
(5.1)

driven by an adapted control policy $u(t,X_{t})$, where $u$ is a measurable
function. It is the objective to minimize the risk-averse expectation of the
accumulated costs,

$\int_{t}^{T}c\bigl{(}s,X_{s},u(s,X_{s})\bigr{)}\mathrm{d}s+\Psi\bigl{(}X_{T}\bigr{)},$

where $\Psi(\cdot)$ is a terminal cost. Recall that the nested expectiles
accumulate costs and risk so that it is the objective to minimize the value
function

$V^{u}(t,x)\coloneqq\tilde{e}_{\beta(\cdot)}^{\sigma(X)}\left(\left.\int_{t}^{T}\bigl{(}s,X_{s}^{u},u(s,X_{s}^{u})\bigr{)}\mathrm{d}s+\Psi\bigl{(}X_{T}^{u}\bigr{)}\right|X_{t}^{u}=x\right)$

among all policies $u\in\mathcal{U}$ chosen in a suitable set, where
$X_{t}^{u}$ solves the stochastic differential equation (5.1) for the policy
$u$.

######  Proposition 5.1.

The value function

$V(t,x)\coloneqq\inf_{u(\cdot)\in\mathcal{U}}V^{u}(t,x),$

solves the differential equation

$\frac{\partial V}{\partial
t}(t,x)=\mathcal{H}_{\beta}\left(t,x,\frac{\partial V}{\partial
x},\frac{\partial^{2}V}{\partial x^{2}}\right)$ (5.2)

with terminal condition $V(T,x)=\Psi(x)$, where

$\mathcal{H}_{\beta}(t,x,g,A)\coloneqq\sup_{u\in
U}\left\\{-c(t,x,u)-g\cdot\mu(t,x,u)-\frac{1}{2}A\,\sigma(t,x,u)^{2}-\sqrt{\frac{2}{\pi}\beta(t,x)}\cdot\bigl{|}g\cdot\sigma(t,x,u)\bigr{|}\right\\}$
(5.3)

is the Hamiltonian, cf. Fleming and Soner (2006, Section IV, (3.2)).

To accept the assertion recall that

$\displaystyle\frac{1}{h}\tilde{e}_{\beta(\cdot)}^{\sigma(X)}$
$\displaystyle\left(\left.\int_{t}^{t+h}c\bigl{(}s,X_{s}^{u},u(s,X_{s}^{u})\bigr{)}\mathrm{d}s+V(t+h,X_{t+h})-V(t,x)\right|X_{t}^{u}=x\right)$
$\displaystyle\xrightarrow[h\to 0]{}c(t,x,u)+\mathcal{G}_{\beta}V(t,x)$

by the definition of the risk generator. While the left-hand side vanishes by
the dynamic programming principle for the optimal policy, it follows for the
right-hand side that

$0=\inf_{u\in U}c(t,x,u)+\mathcal{G}_{\beta}V(t,x).$

With Proposition 4.7, this leads to the equation (5.2) with Hamiltonian (5.3).

The fundamental equation (5.2) is the Hamilton–Jacobi–Bellman (HJB) partial
differential equation. It is essential to observe that the HJB equation has
the additional term

$\sqrt{\frac{2}{\pi}\beta(t,x)}\cdot\left|\sigma(t,x,u)\frac{\partial
V}{\partial x}\right|$

involving the gradient; the total gradient in the Hamiltonian (5.2) thus comes
with the coefficient

$\mu(t,x,u)+\sqrt{\frac{2}{\pi}\beta(t,x)}\cdot\sigma(t,x,u)\cdot\operatorname{{sign}}\left(\sigma(t,x,u)\frac{\partial
V}{\partial x}\right).$

That is, risk aversion increases the trend $\mu$ by the amount
$+\sqrt{\frac{2}{\pi}\beta(t,x)}\cdot\sigma(t,x,u)$, while letting the
volatility $\sigma$ of the process unaffected.

In typical situations, $\frac{\partial V}{\partial x}$ does not change its
sign. For this reason, the classical theory on viscosity solutions on
existence of solutions of (5.2) applies directly, without modifications. As
well, explicit solutions of specific equations are known. In these situations,
the explicit results can be adapted to the risk averse situation, cf. Pichler
and Schlotter (2021) for applications from financial mathematics.

##  6 Summary

This paper exploits the unique properties of expectiles in stochastic and in
dynamic optimization. We start by giving tight comparisons with common risk
measures first. Next, we define the conditional expectile. The conditional
expectile can be nested to extend the scope of risk functionals (risk
measures) to stochastic processes in discrete and in continuous time. For the
random walk process or stochastic processes driven by a stochastic
differential equation, explicit evaluations of the nested risk functional are
available.

The risk generator is defined in analogy to the generator for stochastic
processes. The risk generator involves an additional term which is caused by
risk. With that, the risk generator is a non-linear differential operator. The
aspect of risk augments the Hamiltonian via an additional term, which is
responsible for risk only and the risk averse Hamilton–Jacobi–Bellman
equations thus derive accordingly.

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##  7 Appendix

The function
$x\mapsto(1-\alpha)\operatorname{{\mathds{E}}}(x-X)_{+}-\alpha\operatorname{{\mathds{E}}}(X-x)_{+}$
has slope

$\displaystyle(1-\alpha)\,P(X\leq x)+\alpha\,P(X\geq x)$
$\displaystyle=\alpha+(1-2\alpha)P(X\leq x)$ $\displaystyle\geq\alpha P(X\leq
x)+(1-2\alpha)P(X\leq x)$ $\displaystyle=(1-\alpha)P(X\leq x)$
$\displaystyle\geq 0$

and is therefore strictly increasing for every $\alpha\in(0,1)$ in the support
of $X$ so that the expectile is unique. Further, the slope is so that the
function is convex for $\alpha\leq\nicefrac{{1}}{{2}}$ and concave for
$\alpha\geq\nicefrac{{1}}{{2}}$.

Denote by $x_{\alpha}$ ($y_{\alpha}$, resp.) the expectile for $X$ ($Y$,
resp.), i.e.,

$\displaystyle\alpha\operatorname{{\mathds{E}}}(X-x_{\alpha})_{+}$
$\displaystyle=(1-\alpha)\operatorname{{\mathds{E}}}(x_{\alpha}-X)_{+}\text{
and}$ $\displaystyle\alpha\operatorname{{\mathds{E}}}(Y-y_{\alpha})_{+}$
$\displaystyle=(1-\alpha)\operatorname{{\mathds{E}}}(y_{\alpha}-Y)_{+}.$

With $x_{+}-(-x)_{+}=x$ we have further

$\displaystyle\big{(}\alpha-\frac{1}{2}\big{)}\operatorname{{\mathds{E}}}(X-x_{\alpha})_{+}$
$\displaystyle-\big{(}\frac{1}{2}-\alpha\Big{)}\operatorname{{\mathds{E}}}(x_{\alpha}-X)_{+}=\frac{1}{2}\operatorname{{\mathds{E}}}(X-x_{\alpha})\text{
and}$
$\displaystyle\Big{(}\alpha-\frac{1}{2}\Big{)}\operatorname{{\mathds{E}}}(Y-y_{\alpha})_{+}$
$\displaystyle-(\frac{1}{2}-\alpha)\operatorname{{\mathds{E}}}(y_{\alpha}-Y)_{+}=\frac{1}{2}\operatorname{{\mathds{E}}}(Y-y_{\alpha}).$

For $\alpha\geq\nicefrac{{1}}{{2}}$ we obtain by convexity of the function
$x\mapsto x_{+}$ that

$\displaystyle\big{(}\alpha-\frac{1}{2}\big{)}\operatorname{{\mathds{E}}}(X+Y-x_{\alpha}-y_{\alpha})_{+}$
$\displaystyle\leq\big{(}\alpha-\frac{1}{2}\big{)}\operatorname{{\mathds{E}}}(X-x_{\alpha})+\big{(}\alpha-\frac{1}{2}\big{)}\operatorname{{\mathds{E}}}(Y-y_{\alpha})$
$\displaystyle=\frac{1}{2}\operatorname{{\mathds{E}}}(X-x_{\alpha})+\big{(}\frac{1}{2}-\alpha\Big{)}\operatorname{{\mathds{E}}}(x_{\alpha}-X)_{+}+\frac{1}{2}\operatorname{{\mathds{E}}}(Y-y_{\alpha})+(\frac{1}{2}-\alpha)\operatorname{{\mathds{E}}}(y_{\alpha}-Y)_{+}$
$\displaystyle\leq\frac{1}{2}\operatorname{{\mathds{E}}}(X-x_{\alpha})+\frac{1}{2}\operatorname{{\mathds{E}}}(Y-y_{\alpha})+\big{(}\frac{1}{2}-\alpha\Big{)}\operatorname{{\mathds{E}}}(x_{\alpha}+y_{\alpha}-X-Y)_{+}.$

It follows that

$\alpha\operatorname{{\mathds{E}}}(X+Y-x_{\alpha}-y_{\alpha})_{+}\leq(1-\alpha)\operatorname{{\mathds{E}}}(x_{\alpha}+y_{\alpha}-X-Y)_{+}.$

The assertion follows by monotonicity again.