Monetary value measures in a category of probability spaces

TL;DR

This paper extends the concept of monetary value measures from a categorical framework to the broader setting of probability spaces, enhancing the theoretical foundation for financial risk assessment.

Contribution

It generalizes monetary value measures by shifting from a specific category to the category of probability spaces, broadening their applicability.

Findings

Extended monetary value measures to probability spaces

Provided a categorical framework for financial risk measures

Enhanced theoretical understanding of monetary value in probabilistic contexts

Abstract

We generalize the notion of monetary value measures developed with category theory in [Adachi, 2014] by extending their base category from the category \c{hi} to the category of probability spaces Prob introduced in [Adachi and Ryu, 2016].

Equations

$$\mathbf{Prob}(\bar{X},$$

$$\text{ is a measurable function satisfying }{\mathbb{P}}_Y\circ f^{-1}\ll {\mathbb{P}}_X\},$$

$$X$$

$${\int }_{A}ud{\mathbb{P}}_X={\int }_{f^{-1}(A)}vd{\mathbb{P}}_Y.$$

$${\int }_A{E}^{f^{-}}(v)d{\mathbb{P}}_X={\int }_{f^{-1}(A)}vd{\mathbb{P}}_Y.$$

$$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bar{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 11.74306pt\raise 6.54166pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.18056pt\hbox{$\scriptstyle{f^{-}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bar{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 47.03838pt\raise 5.61806pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.25696pt\hbox{$\scriptstyle{g^{-}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 64.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 64.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bar{Z}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$$

$$X$$

$$E^{f^{-}}((w\circ f)\cdot u){\sim }_{{\mathbb{P}}_X}w\cdot E^{f^{-}}(u).$$

$$\Phi :{\mathbf{Prob}}^{op}\to \mathbf{Set}$$

$$X$$

$$\Phi I{d}_X^{-}=I{d}_{L^1(\bar{X})}\text{and}\Phi f^{-}\circ \Phi g^{-}=\Phi (g^{-}\circ f^{-}).$$

$$0{\lesssim }_{{\mathbb{P}}_X}{E}^{f^{-}}(M-v){\sim }_{{\mathbb{P}}_X}{E}^{f^{-}}(M)-E^{f^{-}}(v).$$

$$E^{f^{-}}(M){\sim }_{{\mathbb{P}}_X}M{E}^{f^{-}}(1_Y){\sim }_{{\mathbb{P}}_X}M$$

$${\phi }^{f^{-}}(v):={\lambda }^{-1}\log E^{f^{-}}(e^{\lambda v}),(\forall v\in L^1(\bar{Y})).$$

$${\phi }^{f^{-}}(v+u\circ f)$$

$$\displaystyle=\lambda^{-1}\log E^{f^{-}}\big{(}e^{\lambda v}\cdot((e^{\lambda u})\circ f)\big{)}$$

$$\displaystyle\sim_{\mathbb{P}_{X}}\lambda^{-1}\log\Big{(}e^{\lambda u}\cdot E^{f^{-}}(e^{\lambda v})\Big{)}$$

$$=u+{\phi }^{f^{-}}(v).$$

$${\phi }^{f^{-}}(0_Y)={\lambda }^{-1}\log E^{f^{-}}(e^{\lambda 0_Y})={\lambda }^{-1}\log E^{f^{-}}(1_Y){\sim }_{{\mathbb{P}}_X}{\lambda }^{-1}\log 1_X=0_X.$$

$$1_A{\phi }^{f^{-}}(v){\sim }_{{\mathbb{P}}_X}{1}_A{\phi }^{f^{-}}(1_{f^{-1}(A)}v).$$

$$1_{f^{-1}(A)}v-1_{f^{-1}(A^c)}\Vert v{\Vert }_{{\mathcal{L}}^{\infty }(\bar{Y})}{\lesssim }_{{\mathbb{P}}_Y}{1}_{f^{-1}(A)}v+1_{f^{-1}(A^c)}v{\lesssim }_{{\mathbb{P}}_Y}{1}_{f^{-1}(A)}v+1_{f^{-1}(A^c)}\Vert v{\Vert }_{{\mathcal{L}}^{\infty }(\bar{Y})}.$$

$${\phi }^{f^{-}}(1_{f^{-1}(A)}v)-\Vert v{\Vert }_{{\mathcal{L}}^{\infty }(\bar{Y})}{1}_{A^c}$$

$$={\phi }^{f^{-}}(1_{f^{-1}(A)}v-1_{f^{-1}(A^c)}\Vert v{\Vert }_{{\mathcal{L}}^{\infty }(\bar{Y})})$$

$${\lesssim }_{{\mathbb{P}}_X}{\phi }^{f^{-}}(v)$$

$${\lesssim }_{{\mathbb{P}}_X}{\phi }^{f^{-}}(1_{f^{-1}(A)}v+1_{f^{-1}(A^c)}\Vert v{\Vert }_{{\mathcal{L}}^{\infty }(\bar{Y})})$$

$$={\phi }^{f^{-}}(1_{f^{-1}(A)}v+(\Vert v{\Vert }_{{\mathcal{L}}^{\infty }(\bar{Y})}{1}_{A^c})\circ f)$$

$${\sim }_{{\mathbb{P}}_X}{\phi }^{f^{-}}(1_{f^{-1}(A)}v)+\Vert v{\Vert }_{{\mathcal{L}}^{\infty }(\bar{Y})}{1}_{A^c}.$$

$${\phi }^{f^{-}}(1_{f^{-1}(A)}v)-1_{A^c}\Vert v{\Vert }_{{\mathcal{L}}^{\infty }(\bar{Y})}{\lesssim }_{{\mathbb{P}}_X}{\phi }^{f^{-}}(v){\lesssim }_{{\mathbb{P}}_X}{\phi }^{f^{-}}(1_{f^{-1}(A)}v)+1_{A^c}\Vert v{\Vert }_{{\mathcal{L}}^{\infty }(\bar{Y})}.$$

$$1_A{\phi }^{f^{-}}(1_{f^{-1}(A)}v)-{\lesssim }_{{\mathbb{P}}_X}{1}_A{\phi }^{f^{-}}(v){\lesssim }_{{\mathbb{P}}_X}{1}_A{\phi }^{f^{-}}(1_{f^{-1}(A)}v).$$

$${\phi }^{f^{-}}(1_{f^{-1}(A)}{v}_1+1_{f^{-1}(A^c)}{v}_2)$$

$$=1_{f^{-1}(A)}{\phi }^{f^{-}}(1_{f^{-1}(A)}{v}_1+1_{f^{-1}(A^c)}{v}_2)+1_{f^{-1}(A^c)}{\phi }^{f^{-}}(1_{f^{-1}(A)}{v}_1+1_{f^{-1}(A^c)}{v}_2)$$

$${\sim }_{{\mathbb{P}}_X}{1}_{f^{-1}(A)}{\phi }^{f^{-}}(1_{f^{-1}(A)}(1_{f^{-1}(A)}{v}_1+1_{f^{-1}(A^c)}{v}_2))+1_{f^{-1}(A^c)}{\phi }^{f^{-}}(1_{f^{-1}(A^c)}(1_{f^{-1}(A)}{v}_1+1_{f^{-1}(A^c)}{v}_2))$$

$$=1_{f^{-1}(A)}{\phi }^{f^{-}}(1_{f^{-1}(A)}{v}_1)+1_{f^{-1}(A^c)}{\phi }^{f^{-}}(1_{f^{-1}(A^c)}{v}_2)$$

$${\sim }_{{\mathbb{P}}_X}{1}_{f^{-1}(A)}{\phi }^{f^{-}}(v_1)+1_{f^{-1}(A^c)}{\phi }^{f^{-}}(v_2).$$

$${\phi }^{g^{-}\circ f^{-}}(w)$$

$$=({\phi }^{f^{-}}\circ {\phi }^{g^{-}})({\phi }^{g^{-}}(w)\circ g){\sim }_{{\mathbb{P}}_X}{\phi }^{g^{-}\circ f^{-}}({\phi }^{g^{-}}(w)\circ g).$$

$${\phi }^{g^{-}\circ f^{-}}(w_1){\sim }_{{\mathbb{P}}_X}{\phi }^{f^{-}}({\phi }^{g^{-}}(w_1)){\lesssim }_{{\mathbb{P}}_X}{\phi }^{f^{-}}({\phi }^{g^{-}}(w_2)){\sim }_{{\mathbb{P}}_X}{\phi }^{g^{-}\circ f^{-}}(w_2).$$

Full text

This work was supported by JSPS KAKENHI Grant Number 26330026.

In this paper, we generalize the notion of monetary value measures developed in [Adachi, 2014] by extending their base category from the category $\chi$ to the category $\mathbf{Prob}$ introduced in [Adachi and Ryu, 2016].

For those who are not familiar with financial risk management and/or monetary value measures, please refer to Section 2 of [Adachi, 2014].

1. A Category of Probability Spaces

In this section we overview a theory of a category of probability spaces. Please refer to [Adachi and Ryu, 2016] for the full discussions and proofs about the contents of this section.

Let $\bar{X}:=(X,\Sigma_{X},\mathbb{P}_{X})$, $\bar{Y}:=(Y,\Sigma_{Y},\mathbb{P}_{Y})$ and $\bar{Z}:=(Z,\Sigma_{Z},\mathbb{P}_{Z})$ be probability spaces.

Definition 1.1**.**

[Category $\mathbf{Prob}$]

A category $\mathbf{Prob}$ is the category whose objects are all probability spaces and the set of arrows between them are defined by

[TABLE]

where $f^{-}$ is a symbol corresponding uniquely to a measurable function $f$.

We fix the state space be a measurable space $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ for a simplicity. $\mathcal{L}^{\infty}(\bar{X})$ is a vector space consisting of $\mathbb{R}$-valued random variables $v$ such that $\mathbb{P}_{X}\mathchar 45\relax\operatorname*{ess\,sup}_{x\in X}\lvert v(x)\rvert<\infty$, while $\mathcal{L}^{1}(\bar{X})$ is a vector space consisting of $\mathbb{R}$-valued random variables $v$ such that $\int_{X}\lvert v\rvert\,d\mathbb{P}_{X}$ has a finite value. For two random variables $u_{1}$ and $u_{2}$, we write $u_{1}\sim_{\mathbb{P}_{X}}u_{2}$ when $\mathbb{P}_{X}(u_{1}\neq u_{2})=0$, and write $u_{1}\lesssim_{\mathbb{P}_{X}}u_{2}$ when $\mathbb{P}_{X}(u_{1}>u_{2})=0$. Note that $u_{1}\lesssim_{\mathbb{P}_{X}}u_{2}$ and $u_{2}\lesssim_{\mathbb{P}_{X}}u_{1}$ iff $u_{1}\sim_{\mathbb{P}_{X}}u_{2}$. $L^{\infty}(\bar{X})$ and $L^{1}(\bar{X})$ are quotient spaces $\mathcal{L}^{\infty}(\bar{X})/\sim_{\mathbb{P}_{X}}$ and $\mathcal{L}^{1}(\bar{X})/\sim_{\mathbb{P}_{X}}$, respectively.

Definition 1.2**.**

[Functor $\mathbf{L}$]

A functor $\mathbf{L}:\mathbf{Prob}\to\mathbf{Set}$ is defined by:

[TABLE]

Theorem 1.3**.**

Let $f^{-}$ be an arrow in $\mathbf{Prob}(\bar{X},\bar{Y})$. Then, for any $v\in\mathcal{L}^{1}(\bar{Y})$ there exists a $u\in\mathcal{L}^{1}(\bar{X})$ such that for every $A\in\Sigma_{X}$

[TABLE]

Moreover, $u$ is determined uniquely up to $\mathbb{P}_{X}$-null sets. In other words, if there are two $u_{1},u_{2}\in\mathcal{L}^{1}(\bar{X})$ both satisfying (1.1), then $u_{1}\sim_{\mathbb{P}_{X}}u_{2}$.

We write a version of this $u$ by $E^{f^{-}}(v)$, and call it a conditional expectation of $v$ along $f^{-}$. Therefore,

[TABLE]

Proposition 1.4**.**

Let $f^{-}$ and $g^{-}$ be arrows in $\mathbf{Prob}$ like:

[TABLE]

1. (1)

For $u\in\mathcal{L}^{1}(\bar{X})$, $E^{Id_{X}^{-}}(u)\sim_{\mathbb{P}_{X}}u$. 2. (2)

For $v_{1},v_{2}\in\mathcal{L}^{1}(\bar{Y})$, $v_{1}\sim_{\mathbb{P}_{Y}}v_{2}$ implies $E^{f^{-}}(v_{1})\sim_{\mathbb{P}_{X}}E^{f^{-}}(v_{2})$. 3. (3)

For $w\in\mathcal{L}^{1}(\bar{Z})$, $E^{f^{-}}(E^{g^{-}}(w))\sim_{\mathbb{P}_{X}}E^{g^{-}\circ f^{-}}(w)$.

Definition 1.5**.**

[Functor $\mathcal{E}$]

A functor $\mathcal{E}:\mathbf{Prob}^{op}\to\mathbf{Set}$ is defined by:

[TABLE]

We call $\mathcal{E}$ a conditional expectation functor.

Proposition 1.6**.**

Let $f^{-}:\bar{X}\to\bar{Y}$ be a $\mathbf{Prob}$-arrow, $u,v\in\mathcal{L}^{1}(\bar{Y})$ and $\alpha,\beta\in\mathbf{R}$.

1. (1)

Linearity:  $E^{f^{-}}(\alpha u+\beta v)\sim_{\mathbb{P}_{X}}\alpha E^{f^{-}}(u)+\beta E^{f^{-}}(v).$ 2. (2)

Positivity:  $E^{f^{-}}(v)\gtrsim_{\mathbb{P}_{X}}0$ if $v\gtrsim_{\mathbb{P}_{Y}}0$.

Theorem 1.7**.**

Let $f^{-}:\bar{X}\to\bar{Y}$ be a $\mathbf{Prob}$-arrow , $u\in\mathcal{L}^{1}(\bar{Y})$ and $w\in\mathcal{L}^{\infty}(\bar{X})$. Then we have

[TABLE]

2. Monetary Value Measures

A monetary value measure is defined as a presheaf on $\mathbf{Prob}$.

Definition 2.1**.**

[Monetary Value Measures]

A ** monetary value measure ** is a contravariant functor

[TABLE]

defined by

[TABLE]

where $\varphi^{f^{-}}$ satisfies

1. (1)

Cash invariance:  $(\forall v\in\mathcal{L}^{\infty}(\bar{Y}))(\forall u\in\mathcal{L}^{\infty}(\bar{X}))$

$\varphi^{f^{-}}(v+u\circ f)\sim_{\mathbb{P}_{X}}\varphi^{f^{-}}(v)+u$, 2. (2)

Monotonicity:  $(\forall v_{1}\in\mathcal{L}^{\infty}(\bar{Y}))(\forall v_{2}\in\mathcal{L}^{\infty}(\bar{Y}))$

$v_{1}\lesssim_{\mathbb{P}_{Y}}v_{2}\;\Rightarrow\;\varphi^{f^{-}}(v_{1})\lesssim_{\mathbb{P}_{X}}\varphi^{f^{-}}(v_{2})$, 3. (3)

Normalization:  $\varphi^{f^{-}}(0_{Y})\sim_{\mathbb{P}_{X}}0_{X}$ if $f^{-}$ is measure-preserving, 4. (4)

$v\in\mathcal{L}^{\infty}(\bar{Y})$ implies $\varphi^{f^{-}}(v)\in\mathcal{L}^{\infty}(\bar{X})$ if $f^{-}$ is measure-preserving.

We sometimes write $\Phi[\varphi^{\cdot}]$ for $\Phi$ for explicitly noting that arrows mapped by $\Phi$ are determined by $\varphi^{\cdot}$.

At this point, we do not require the monetary value measures to satisfy familiar conditions such as concavity or positive homogeneity. Instead of doing so, we want to see what kind of properties are deduced from this minimal setting.

The most crucial point of Definition 2.1 is that $\varphi$ does not move only in the direction of time but also moves over several absolutely continuous probability measures internally. This means we have a possibility to develop risk measures including ambiguity within this formulation.

Another key point of Definition 2.1 is that $\varphi$ is a contravariant functor. So, for any pair of arrows

\textstyle{\bar{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{-}}$$\textstyle{\bar{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{-}}$$\textstyle{\bar{Z}}

in $\mathbf{Prob}$, we have

[TABLE]

As an example of monetary value measures, we will introduce a notion of entropic value measures that depend on conditional expectations $E^{f^{-}}(v)$ of $v$ along $f^{-}$.

Before introducing entropic value measures, we need the following lemma.

Lemma 2.2**.**

Let $f^{-}:\bar{X}\to\bar{Y}$ be a measure-preserving arrow in $\mathbf{Prob}$. Then, $v\in\mathcal{L}^{\infty}(\bar{Y})$ implies $E^{f^{-}}(v)\in\mathcal{L}^{\infty}(\bar{X})$.

Proof.

Since $v\in\mathcal{L}^{\infty}(\bar{Y})$, there exists a non-negative $M\geq 0$ such that $-M\lesssim_{\mathbb{P}_{Y}}v\lesssim_{\mathbb{P}_{Y}}M$. Then, by Proposition 1.6,

[TABLE]

On the other hand, we have

[TABLE]

since $f$ is measure preserving. Therefore, we obtain $E^{f^{-}}(v)\lesssim_{\mathbb{P}_{X}}M$. Similarly, we have $-M\lesssim_{\mathbb{P}_{X}}E^{f^{-}}(v)$. So we get $E^{f^{-}}(v)\in\mathcal{L}^{\infty}(\bar{X})$. ∎

Proposition 2.3**.**

*[Entropic Value Measures]

Let $f^{-}:\bar{X}\to\bar{Y}$ be a $\mathbf{Prob}$-arrow, and $\lambda$ be a positive real number. Define a function $\varphi^{f^{-}}:L^{1}(\bar{Y})\to L^{1}(\bar{X})$ by*

[TABLE]

Then, $\Phi:=\Phi[\varphi^{\cdot}]$ is a monetary value measure. We call this $\Phi$ an entropic value measure.

Proof.

Let

\textstyle{\bar{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{-}}$$\textstyle{\bar{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{-}}$$\textstyle{\bar{Z}}

be arrows in $\mathbf{Prob}$. In order to show that $\Phi$ becomes a contravariant functor, we need to check three points: $\varphi^{Id_{Z}^{-}}(w)\sim_{\mathbb{P}_{Z}}w$, $\varphi^{f-}(\varphi^{g^{-}}(w))\sim_{\mathbb{P}_{X}}\varphi^{g^{-}\circ f^{-}}(w)$, and that $w_{1}\sim_{\mathbb{P}_{Z}}w_{2}$ implies $\varphi^{g^{-}}(w_{1})\sim_{\mathbb{P}_{Y}}\varphi^{g^{-}}(w_{2})$ for every $w,w_{1},w_{2}\in\mathcal{L}^{1}(\bar{Z})$. But, they are straightforward consequences of Proposition 1.4. So, we forward to check if $\varphi^{f^{-}}$ satisfies the four conditions of Definition 2.1.

Firstly, we show that $\varphi^{f^{-}}(v+u\circ f)\sim_{\mathbb{P}_{X}}\varphi^{f^{-}}(v)+u$ for $v\in\mathcal{L}^{\infty}(\bar{Y})$ and $u\in\mathcal{L}^{\infty}(\bar{X})$. But by Theorem 1.7, we have

[TABLE]

Secondly, we show that $v_{1}\lesssim_{\mathbb{P}_{Y}}v_{2}$ implies $\varphi^{f^{-}}(v_{1})\lesssim_{\mathbb{P}_{X}}\varphi^{f^{-}}(v_{2})$ for $v_{1},v_{2}\in\mathcal{L}^{\infty}(\bar{Y})$. But this comes from Proposition 1.6.

Thirdly, we show that $\varphi^{f^{-}}(0_{Y})\sim_{\mathbb{P}_{X}}0_{X}$ if $f^{-}$ is measure-preserving. But this is straightforward like the following:

[TABLE]

Lastly, we need to show that $v\in\mathcal{L}^{\infty}(\bar{Y})$ implies $\varphi^{f^{-}}(v)\in\mathcal{L}^{\infty}(\bar{X})$ when $f^{-}$ is measure-preserving. But this comes from Lemma 2.2. ∎

Here are some properties of monetary value measures.

Theorem 2.4**.**

*Let $\Phi=\Phi[\varphi^{\cdot}]:\mathbf{Prob}^{op}\to\mathbf{Set}$ be a monetary value measure, and

\textstyle{\bar{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{-}}$$\textstyle{\bar{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{-}}$$\textstyle{\bar{Z}}

be arrows in $\mathbf{Prob}$.*

1. (1)

If $f^{-}$ is measure-preserving, we have   $\Phi f^{-}\circ Lf^{-}=Id_{L\bar{X}}$. 2. (2)

Idempotence:  If $f^{-}$ is measure-preserving, we have

$\Phi f^{-}\circ Lf^{-}\circ\Phi f^{-}=\Phi f^{-}$. 3. (3)

Local property:  $(\forall v_{1}\in\mathcal{L}^{\infty}(\bar{Y}))(\forall v_{2}\in\mathcal{L}^{\infty}(\bar{Y}))(\forall A\in\Sigma_{X})$ \Phi f^{-}\big{[}1_{f^{-1}(A)}v_{1}+1_{f^{-1}(A^{c})}v_{2}\big{]}_{\sim_{\mathbb{P}_{Y}}}=[1_{f^{-1}(A)}]_{\sim_{\mathbb{P}_{X}}}\Phi f[v_{1}]_{\sim_{\mathbb{P}_{Y}}}+[1_{f^{-1}(A^{c})}]_{\sim_{\mathbb{P}_{X}}}\Phi f[v_{2}]_{\sim_{\mathbb{P}_{Y}}}. 4. (4)

Dynamic programming principle:  If $g^{-}$ is measure-preserving,

$\varphi^{g^{-}\circ f^{-}}(w)=\varphi^{g^{-}\circ f^{-}}(\varphi^{g^{-}}(w)\circ g)$* for $w\in\mathcal{L}^{\infty}(\bar{Z})$.* 5. (5)

Time consistency:  $(\forall w_{1}\in\mathcal{L}^{\infty}(\bar{Z}))(\forall w_{2}\in\mathcal{L}^{\infty}(\bar{Z}))$

$\varphi^{g^{-}}(w_{1})\lesssim_{\mathbb{P}_{Y}}\varphi^{g^{-}}(w_{2})\;\Rightarrow\;\varphi^{g^{-}\circ f^{-}}(w_{1})\lesssim_{\mathbb{P}_{X}}\varphi^{g^{-}\circ f^{-}}(w_{2})$.

Proof.

1. (1)

For $u\in\mathcal{L}^{\infty}(\bar{X})$, \Phi f^{-}(Lf^{-}[u]_{\sim_{\mathbb{P}_{X}}})=\big{[}\varphi^{f^{-}}(u\circ f)\big{]}_{\sim_{\mathbb{P}_{X}}} But, by cash invariance and normalization, we have $\varphi^{f^{-}}(u\circ f)=\varphi^{f^{-}}(0_{Y}+(u\circ f))\sim_{\mathbb{P}_{X}}\varphi^{f^{-}}(0_{Y})+u\sim_{\mathbb{P}_{X}}0_{X}+u=u$. 2. (2)

Immediate by (1). 3. (3)

First, we show that for any $A\in\Sigma_{X}$ and $v\in\mathcal{L}^{\infty}(\bar{Y})$,

[TABLE]

Since $v\in\mathcal{L}^{\infty}(\bar{Y})$, for every $y\in Y$ we have $\lvert v(y)\rvert\lesssim_{\mathbb{P}_{Y}}\|v\|_{\mathcal{L}^{\infty}(\bar{Y})}$. Therefore,

[TABLE]

Then noting that $1_{A}\circ f=1_{f^{-1}(A)}$, we have the following sequence of equations by cash invariance and monotonicity.

[TABLE]

Hence

[TABLE]

By multiplying $1_{A}$, we obtain

[TABLE]

Therefore, we get (2.3).

Next by using (2.3) twice, we have

[TABLE] 4. (4)

By (2), we have $\varphi^{g^{-}}(\varphi^{g^{-}}(w)\circ g)\sim_{\mathbb{P}_{Y}}\varphi^{g^{-}}(w)$ for $w\in\mathcal{L}^{\infty}(\bar{Z})$. So by (2.1),

[TABLE] 5. (5)

Assume $\varphi^{g^{-}}(w_{1})\lesssim_{\mathbb{P}_{Y}}\varphi^{g^{-}}(w_{2})$. Then, by monotonicity and (2.1),

[TABLE]

∎

In Theorem 2.4, two properties, dynamic programming principle and time consistency are usually introduced as axioms ([Detlefsen and Scandolo, 2006]). But, we derive them naturally here from the fact that the monetary value measure is a contravariant functor.

Before ending this section, we mention an interpretation of the Yoneda lemma in our setting.

Theorem 2.5**.**

*[The Yoneda Lemma]

For any monetary value measure $\Phi:\mathbf{Prob}^{op}\to\mathbf{Set}$ and an object $\bar{X}$ in $\mathbf{Prob}$, there exists a bijective correspondence $y_{\Phi,\bar{X}}$ specified by the following diagram:*

[TABLE]

where $\tilde{\mathbf{u}}$ is a natural transformation defined by for any $f^{-}:\bar{Y}\to\bar{X}$ in $\mathbf{Prob}$, $\tilde{\mathbf{u}}_{\bar{Y}}(f^{-}):=\Phi f^{-}\mathbf{u}$. Moreover, the correspondence is natural in both $\Phi$ and $\bar{X}$.

It makes sense to consider the representable functor $\mathbf{Prob}(-,\bar{X})$ as a generalized time domain with time horizon $\bar{X}$. Then a natural transformation from $\mathbf{Prob}(-,\bar{X})$ to $\Phi$ can be seen as a stochastic process that is (in a sense) adapted to $\Phi$, and its corresponding $\Sigma_{X}$-measurable random variable represents a terminal value (payoff) at the horizon.

The Yoneda lemma says that we have a bijective correspondence between those stochastic processes and random variables.