#
A Portfolio Choice Problem in the Framework of Expected Utility Operators^{ †}

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Review of Possibilistic Expected Utility Theories and Introduction to D-Operators

#### 2.1. Possibilistic Expected Utility and Possibilistic Variance of a Fuzzy Number

- a utility function u representing an agent;
- a random variable X representing the risk.

- a utility function u representing an agent;
- a fuzzy number A representing the risk (with the level sets ${\left[A\right]}^{\gamma}=[{a}_{1}\left(\gamma \right),{a}_{2}\left(\gamma \right)]$, $\gamma \in [0,1]$);
- a weighting function $f:[0,1]\to \mathbb{R}$ (f is a non-negative and increasing function that satisfies ${\int}_{0}^{1}f\left(\gamma \right)d\gamma =1$).

- (1)
- Setting $u={1}_{\mathbb{R}}$ (the identity of $\mathbb{R}$) in these concepts, the two possibilistic expected utilities introduce the same notion of possibilistic expected value:$${E}_{f}\left(A\right)={E}_{1}(f,{1}_{\mathbb{R}}\left(A\right))={E}_{2}(f,{1}_{\mathbb{R}}\left(A\right))=\frac{1}{2}{\int}_{0}^{1}[{a}_{1}\left(\gamma \right)+{a}_{2}\left(\gamma \right)]f\left(\gamma \right)d\gamma .$$
- (2)
- Setting $u\left(x\right)={(x-{E}_{f}\left(A\right))}^{2}$ in these concepts, two possibilistic variances are obtained (these two types of possibilistic variance were studied in several papers (see, e.g., [17,18,19,20,21]) and were applied in the study of different possibilistic models [18,22,25,29,30,36,37,38]):$$Va{r}_{1}(f,A)=\frac{1}{2}{\int}_{0}^{1}[{({a}_{1}\left(\gamma \right)-{E}_{f}\left(A\right))}^{2}+{({a}_{2}\left(\gamma \right)-{E}_{f}\left(A\right))}^{2}]f\left(\gamma \right)d\gamma ,$$$$Va{r}_{2}(f,A)={\int}_{0}^{1}[\frac{1}{{a}_{2}\left(\gamma \right)-{a}_{1}\left(\gamma \right)}{\int}_{{a}_{1}\left(\gamma \right)}^{{a}_{2}\left(\gamma \right)}{(x-{E}_{f}\left(A\right))}^{2}dx]f\left(\gamma \right)d\gamma .$$

#### 2.2. Expected Utility Operators and D-Operators

**Definition**

**1**

**.**An (f-weighted) expected utility operator is a function $T:\mathcal{F}\times \mathcal{C}\left(\mathbb{R}\right)\to \mathbb{R}$ such that for any $a,b\in \mathbb{R}$, $g,h\in \mathcal{C}\left(\mathbb{R}\right)$ and $A\in \mathcal{F}$, the following conditions are fulfilled:

- (a)
- $T(A,{1}_{\mathbb{R}})={E}_{f}\left(A\right)$;
- (b)
- $T(A,\overline{a})=a$;
- (c)
- $T(A,ag+bh)=aT(A,g)+bT(A,h)$;
- (d)
- $g\le h$ implies $T(A,g)\le T(A,h)$.

- the ${k}^{\mathrm{th}}$-order T-moment of A: $T(A,g)$, where $g\left(x\right)={x}^{k}$ for any $x\in \mathbb{R}$;
- the ${k}^{\mathrm{th}}$-order central T-moment of A: $T(A,g)$, where $g\left(x\right)={(x-{E}_{f}\left(A\right))}^{k}$ for any $x\in \mathbb{R}$.

- the T-variance of A: $Va{r}_{T}\left(A\right)=T(A,{(x-{E}_{f}\left(A\right))}^{2})$;
- the T-skewness of A: $S{k}_{T}\left(A\right)=T(A,{(x-{E}_{f}\left(A\right))}^{3})$;
- the T-kurtosis of A: ${K}_{T}\left(A\right)=T(A,{(x-{E}_{f}\left(A\right))}^{4})$.

**Example**

**1**

**.**The expected utility operator ${S}_{1}:\mathcal{F}\times \mathcal{C}\left(\mathbb{R}\right)\to \mathbb{R}$ is defined by:

**Example**

**2**

**.**The expected utility operator ${S}_{2}:\mathcal{F}\times \mathcal{C}\left(\mathbb{R}\right)\to \mathbb{R}$ is defined by ${S}_{2}(A,g)={\int}_{0}^{1}\left[\frac{1}{{a}_{2}\left(\gamma \right)-{a}_{1}\left(\gamma \right)}{\int}_{{a}_{1}\left(\gamma \right)}^{{a}_{2}\left(\gamma \right)}g\left(x\right)dx\right]f\left(\gamma \right)d\gamma ,$ for any fuzzy number A whose level sets are ${\left[A\right]}^{\gamma}=[{a}_{1}\left(\gamma \right),{a}_{2}\left(\gamma \right)]$, $\gamma \in [0,1]$ and for any $g\in \mathcal{C}\left(\mathbb{R}\right).$

**Definition**

**2.**

- (D
_{1}) - The function $\lambda \u27fcT(A,g(.,\lambda \left)\right)$ is derivable (with respect to λ);
- (D
_{2}) - $T(A,\frac{\partial g(.,\lambda )}{\partial \lambda})=\frac{d}{d\lambda}T(A,g(.,\lambda ))$.

**Proposition**

**1.**

**Proposition**

**2.**

- (a)
- $u\in \mathcal{U}.$
- (b)
- For any fuzzy number $A,$ the following equality holds:$\frac{d}{d\lambda}T(A,u(.,\lambda ))={a}_{1}\frac{d}{d\lambda}T(A,g(.,\lambda ))+{a}_{2}\frac{d}{d\lambda}T(A,h(.,\lambda ))$.

**Proof.**

## 3. T-Standard Model in the Possibilistic Portfolio Problem

#### 3.1. Portfolio Choice Problem and the T-Standard Model

- (H
_{1}) - The return of the risky asset is a fuzzy number ${B}_{0}$;
- (H
_{2}) - The formulation of the optimization problem will use the notion of generalized possibilistic expected utility associated with the D-operator T (see the previous section).

#### 3.2. Optimal Allocation Based on Absolute Risk Aversion and Prudence

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Remark**

**1.**

- the Arrow–Pratt index ${r}_{u}\left(w\right)$ and the prudence index ${P}_{u}\left(w\right)$
- the T-variance $Va{r}_{T}\left(A\right)$ and the T-skewness $S{k}_{T}\left(A\right)$.

**Example**

**3.**

**Example**

**4.**

- the indicators of risk aversion, prudence, and temperance associated with the utility function u;
- T-variance, T-skewness, and T-kurtosis associated with the fuzzy number A.

#### 3.3. Optimal Allocation in Terms of Absolute Risk Aversion, Prudence, and Temperance

**Proposition**

**6.**

**Theorem**

**1.**

**Proof.**

- $\frac{1}{3\phantom{\rule{0.166667em}{0ex}}!}{k}^{3}6{\alpha}^{\prime}\left(0\right){\mu}^{2}=\frac{{\left(k\mu \right)}^{3}}{{r}_{u}\left(w\right)}\frac{1}{T(A,{x}^{2})},$
- $\frac{1}{3\phantom{\rule{0.166667em}{0ex}}!}{k}^{3}3{P}_{u}\left(w\right){\alpha}^{\prime}\left(0\right){\alpha}^{\u2033}\left(0\right)T(A,{x}^{3})=\frac{1}{2}{\left(k\mu \right)}^{3}\frac{{\left({P}_{u}\left(w\right)\right)}^{2}}{{\left({r}_{u}\left(w\right)\right)}^{3}}\frac{{\left(T(A,{x}^{3})\right)}^{2}}{T(A,{x}^{2}){)}^{4}},$
- $\frac{1}{3\phantom{\rule{0.166667em}{0ex}}!}{k}^{3}9{P}_{u}\left(w\right)\mu {\left({\alpha}^{\prime}\left(0\right)\right)}^{2}T(A,{x}^{2})=\frac{3}{2}{\left(k\mu \right)}^{3}\frac{{P}_{u}\left(w\right)}{{\left({r}_{u}\left(w\right)\right)}^{2}}\frac{1}{(T(A,{x}^{2})},$
- $\frac{1}{3\phantom{\rule{0.166667em}{0ex}}!}{k}^{3}{\left({\alpha}^{\prime}\left(0\right)\right)}^{3}T(A,{x}^{4})\frac{{T}_{u}\left(w\right)}{{P}_{u}\left(w\right)}=\frac{1}{6}{\left(k\mu \right)}^{3}\frac{{T}_{u}\left(w\right)}{{P}_{u}\left(w\right){\left({r}_{u}\left(w\right)\right)}^{3}}\frac{T(A,{x}^{4})}{{\left(T(A,{x}^{2})\right)}^{3}}.$

**Corollary**

**1.**

**Proof.**

**Remark**

**2.**

**Example**

**5.**

- the weighting function is $f\left(t\right)=2t$, $t\in [0,1]$
- the agent’s utility function is $u\left(w\right)={w}^{a}$, $a>0$.

## 4. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Proposition 6

**Proof.**

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Georgescu, I.; Fono, L.A.
A Portfolio Choice Problem in the Framework of Expected Utility Operators. *Mathematics* **2019**, *7*, 669.
https://doi.org/10.3390/math7080669

**AMA Style**

Georgescu I, Fono LA.
A Portfolio Choice Problem in the Framework of Expected Utility Operators. *Mathematics*. 2019; 7(8):669.
https://doi.org/10.3390/math7080669

**Chicago/Turabian Style**

Georgescu, Irina, and Louis Aimé Fono.
2019. "A Portfolio Choice Problem in the Framework of Expected Utility Operators" *Mathematics* 7, no. 8: 669.
https://doi.org/10.3390/math7080669