An investment risk model with bilateral jumps

1.   Introduction

In 1965, Zadeh ^[18] introduced the concept of fuzzy theory, which has since undergone extensive research and various applications, including Choquet integrals of set-valued functions ^{[5,6,20,21,22]}, fuzzy set-valued measures ^[9,10,16], fuzzy random variable applications ^[1,4,17], theory for general quantum systems interacting with linear dissipative systems ^[3], and more. The relationship between fuzzy theory and probability theory has been a subject of much discussion ^[1,12,14], as both frameworks aim to capture the concept of uncertainty using membership functions and probability density functions (PDFs) whose values lie within the interval [0, 1].

Fuzzy theory and probability theory are two distinct mathematical frameworks, each with their own approach to modeling uncertainty. Fuzzy theory represents imprecision and vagueness in human reasoning using fuzzy sets, which assign degrees of membership to elements of a universe of discourse. Probability theory deals with randomness and uncertainty using probability distributions, which assign probabilities to outcomes of a random event. Despite their differences, both frameworks allow the expression of uncertainty using values that lie within the range of [0, 1], and they provide a means for decision-making under uncertainty, incorporating expert knowledge and data.

Research has explored the relationships between fuzzy theory and probability theory, revealing similarities in terms of mathematical structure and some analytical tools. For instance, fuzzy measures can be viewed as a generalization of probability measures, and Choquet integrals of set-valued functions are analogous to probability integrals. While the majority of fuzzy probability measure theories ^[7,13,15,19] have traditionally considered probability as the expected value of the membership function of fuzzy events, however, by using fuzzifying a PDF we define the fuzzifying probability of crisp events.

In this study, we propose the concept of fuzzifying probability for continuous random variables in the context of crisp events, along with its properties and associations with conventional probability theories. Therefore, fuzzy theory can be seen as an extension or generalization of probability theory. The main objective of this study is to introduce fuzzifying probability density functions and to investigate related properties by applying the concepts of fuzzifying probability of crisp events. This approach enables investigation of the ambiguities of the PDF and their impact on probability theories. Relevant definitions in probability theory will be briefly recalled to facilitate this investigation.

Definition 1.1. ^[8] Let $S$ be a sample space and $X$ be a real-valued continuous random variable on $S$. Then, a function ${f_{{ X }}}:S\to \mathbb{R}^+$ is a PDF of $X$ if it satisfies the following criteria:

(i) ${f_{{ X }}}(x)$ is positive everywhere in the support $S$, i.e., ${f_{{ X }}}(x) > 0$ for all $x\in S$, and

(ii) $\int_S {f_{{ X }}}(x) dx = 1$.

If ${f_{{ X }}}(x)$ is a PDF of the random variable $X$, then the probability $P$ that $X$ belongs to an event $E$ is defined as

Definition 1.2. ^[8] Let $\mathcal{A}$ be the $\sigma$-algebra of a sample space $S$. A real-valued function $P$ on $\mathcal{A}$ is a probability if $P$ satisfies the following properties:

(i) $P(E)\geq 0$ for all $E\in \mathcal{A}$,

(ii) $P(S) = 1$ and $P(\emptyset) = 0$, and

(iii) For any sequence of events $\{E_1, E_2, \cdots\}$ with $E_i\cap E_j = \emptyset\; (i\neq j)$, it holds

We next recall some basic fuzzy theory notions and definitions. Let $U$ and $V$ be two universal sets and $g:U\rightarrow V$ be a crisp function between these sets. Then the fuzzifying function $\widetilde{g}:U\rightarrow \mathcal{F}(V)$ is a mapping from the same domain to a new range $\mathcal{F}(V)$ comprising the family of all fuzzy sets on $V$. The fuzzy set $\widetilde{A}\in \mathcal{F}(V)$ of $V$ can be expressed as

where ${ \mathfrak{m}}_{\widetilde{A}}:V\rightarrow [0, 1]$ is a membership function of $\widetilde{A}$ (For more details see ^[2,11]). Recall that a fuzzy set $\widetilde{A}$ is said to be normal if there exists $v_0\in V$ such that ${ \mathfrak{m}}_{\widetilde{A}}(v_0) = 1$.

Let $I([0, 1])$ be the set of all intervals in $[0, 1]$ whose elements are described as

In particular, we consider $a = [a, a]$ for any $a\in [0, 1]$. Then the interval operators in $I([0, 1])$ are defined as follows.

Definition 1.3. ^[5,6,12] For each $\bar{a} = [a^-, a^+], \bar{b} = [b^-, b^+]\in I([0, 1])$, the arithmetic, comparison, and inclusion operators can be expressed as follows.

(i) $\bar{a}+\bar{b} = [a^-+b^-, a^++b^+]$,

(ii) $k \bar{a} = [ka^-, ka^+]$ for all $k\in[0, 1]$,

(iii) $\bar{a}\bar{b} = [a^-b^-, a^+b^+]$,

(iv) $\bar{a} \wedge \bar{b} = [a^-\wedge b^-, a^+\wedge b^+]$,

(v) $\bar{a} \vee \bar{b} = [a^-\vee b^-, a^+\vee b^+]$,

(vi) $\bar{a}\leq \bar{b}$ if and only if $a^-\leq b^-$ and $a^+\leq b^+$,

(vii) $\bar{a} < \bar{b}$ if and only if $\bar{a} \leq \bar{b}$ and $\bar{a}\neq \bar{b}$, and

(viii) $\bar{a} \subseteq \bar{b}$ if and only if $b^-\leq a^-$ and $a^+\leq b^+$.

Also, algebraic operations of fuzzy sets are defined as follows.

Definition 1.4. ^[11] Let $X$ be a nonempty set and $\widetilde{A}$ and $\widetilde{B}$ be fuzzy sets of $X$.

(i) The $\alpha$-cut $\widetilde{A}_\alpha$ of a fuzzy set $\widetilde{A}$ is defined as

(ii) The algebraic sum $\widetilde{A}+\widetilde{B}$ of two fuzzy sets $\widetilde{A}$ and $\widetilde{B}$ of $X$ is defined as

provided $\widetilde{A}_\alpha +\widetilde{B}_\alpha \subseteq [0, 1]$.

(iii) The algebraic product $\widetilde{A}\widetilde{B}$ of two fuzzy sets $\widetilde{A}$ and $\widetilde{B}$ of $X$ is defined as

Let $A$ be a measurable subset of $U$ and $f$ be an integrable function on $U$. If $\widetilde{f}$ is a fuzzifying function, then the fuzzifying integral ^[11] of $\widetilde{f}$ over $A$ is defined as

where $f^{-}_{\alpha}$ and $f^{+}_{\alpha}$ are $\alpha$-cut functions of $\widetilde{f}(x)$, i.e.,

2.   Fuzzifying PDF and fuzzifying probability

Let $S$ be a sample space with continuous random variable $X:S\rightarrow \mathbb{R}$ and $\mathcal{F}(\mathbb{R}^+)$ be the family of all fuzzy sets on $[0, \infty)$. Using the concepts ^[2] of fuzzifying functions to a PDF ${f_{{ X }}}:S\rightarrow [0, \infty)$, we define a fuzzifying PDF $\widetilde{f}_{ X }$ as follows. In order to facilitate theoretical development throughout the remainder of the paper, it is assumed that the fuzzifying PDF $\widetilde{f}_{ X }$ is integrable for all $\alpha$-cuts.

Definition 2.1. Let $X$ be a continuous random variable and ${f_{{ X }}}$ be a PDF of $X$. Then we define the fuzzifying PDF $\widetilde{f}_{ X }:S\rightarrow \mathcal{F}(\mathbb{R}^+)$ by fuzzifying ${f_{{ X }}}$ that satisfies the following conditions:

(i) ${\widetilde{f}_{ X }}(x) > 0$ for all $x\in S$, i.e.,

where ${ \mathfrak{m}}_{ {\widetilde{f}_{ X }}(x)} > 0$ means that there exists $u\in \mathbb{R}^+$ such that ${ \mathfrak{m}}_{ {\widetilde{f}_{ X }}(x)}(u) > 0$.

(ii) The fuzzifying integration (1.1) of $\widetilde{f}_{ X }$ satisfies

where $\widetilde{1}$ is a convex fuzzy set ^[11] of $1$ with ${ \mathfrak{m}}_{\widetilde{1}}(1) = 1$.

Note that from (1.1), the fuzzy set $\widetilde{1}$ in Definition 2.1 (ii) has its $\alpha$-cuts

If ${\widetilde{f}_{ X }}$ is a fuzzifying PDF of $X$, then the fuzzifying probability $\widetilde{P}$ that $X$ belongs to some event $E$ is given by the fuzzifying integral of ${\widetilde{f}_{ X }}$ over $E$, i.e.,

We consider the fuzzifying probability using the concept of fuzzifying functions in a similar way.

Definition 2.2. Let $\mathcal{A}$ be a $\sigma$-algebra of a sample space $S$ and $P:\mathcal{A}\to [0, 1]$ be a probability. Then the fuzzifying function $\widetilde{P}:\mathcal{A} \to F([0, 1])$ is called the fuzzifying probability if the following conditions are satisfied:

(i) $0\leq \widetilde{P}(E)\leq 1$ for each event $E$ of $S$.

(ii) $\widetilde{P}(S) = \widetilde{1}$, where $\widetilde{1}$ is a convex fuzzy set satisfying ${ \mathfrak{m}}_{\widetilde{1}}(1) = 1.$

(iii) For any sequence of events $\{E_1, E_2, \cdots\}$ with $E_i\cap E_j = \emptyset\; (i\neq j)$, it holds

The following theorem follows from Definitions 2.1 and 2.2.

Theorem 2.3. Let $\widetilde{f}_{ X}$ be a fuzzifying PDF for a continuous random variable $X$ and $\widetilde{P}$ be the fuzzifying probability with the density function $\widetilde{f}_{ X}$ given by (2.1). Then $\widetilde{P}$ is a fuzzifying probability.

Proof. We need only show that $\widetilde{P}$ satisfies the three conditions in Definition 2.2.

(i) Let $E$ be an element of $S$. Then from (1.1) and (2.1),

Since $0\leq \int_E f^-_{ { X }_\alpha} (x)dx \leq \int_E f^+_{ { X }_\alpha} (x)dx\leq 1$ for all $\alpha\in[0, 1]$, it implies $0\leq \widetilde{P}(X\in E)\leq 1$. Thus the first condition holds.

(ii) Since $\widetilde{f}_{ { X }_1}(x) = {f_{{ X }}} (x)$ for all $x\in S$, the $\alpha$-cut of $\widetilde{P}(X \in S)$ at $\alpha = 1$ can be expressed as

Hence the second condition is satisfied.

(iii) Let $\{E_1, E_2, \cdots\}$ be a sequence of disjoint events. Then

Thus, third condition is satisfied, which completes the proof.

□

Remark 2.4. Theorem 2.3 confirms the fuzzifying probability is a fuzzifying probability. Thus, we consider the fuzzifying probability to be $\widetilde{P}(E) = \widetilde{P}(X\in E)$.

Recall the negative-scalar product ^[11]: for $k\in \mathbb{R}^{-} = (-\infty, 0)$ and some interval $[a, b]$ in $\mathbb{R} = (-\infty, \infty)$ with $a\leq b$, the product $[a, b]$ by $k$ can be expressed as

Consider a fuzzy set $\widetilde{P}^* (E)$ for $E\subseteq S$ whose $\alpha$-cuts are defined by

Then the fuzzifying probability establishes the following property.

Theorem 2.5. Let $X$ be a continuous random variable on a sample space $S$ and $\widetilde{P}$ be a fuzzifying probability. Then

(i) $\widetilde{P}(E^c) = \widetilde{1}-\widetilde{P}^*(E)$ for $E\subseteq S$,

(ii) $\widetilde{P}(\emptyset) = 0$,

(iii) If $E_1\subseteq E_2$ in $S$, then $\widetilde{P}(E_1)\leq \widetilde{P}(E_2).$

Proof. We need only show that $\widetilde{P}$ satisfies the conditions.

(i) From (2.2) with $E^c = S-E$,

where the fuzzy set $\widetilde{P}^* (E)$ is given by (2.3).

(ii) The second condition is trivially satisfied by the definition

(iii) Since $\int_{E_2} f^{-}_{ { X }_\alpha} (x)dx \leq \int_{E_1} f^{-}_{ { X }_\alpha} (x)dx$ and $\int_{E_1} f^{+}_{ { X }_\alpha} (x)dx \leq \int_{E_2} f^{+}_{ { X }_\alpha} (x)dx$ for all $\alpha\in [0, 1],$

□

We present an example of the fuzzifying probability obtained from a fuzzifying PDF.

Example 2.6. Let $X$ be a continuous random variable with PDF ${f_{{ X }}}(x) = 3x^2, \; 0\leq x \leq 1$. Then, we consider the fuzzifying PDF $\widetilde{f}_{ X }(x) = \widetilde{3}x^2, \; 0\leq x \leq 1$ of ${f_{{ X }}}$, where a fuzzy set $\widetilde{3}$ of the constant $3$ is given by

Note that the membership function of the fuzzifying function is given by

From Definition 2.1 (iii), the corresponding fuzzifying probability can be expressed as

where

Thus, from (2.4),

and hence,

Therefore, the membership of the fuzzifying probability $\widetilde{P}$ for $0 < X < \frac{1}{3}$ is given by

Note that the probability $P$ over $0 < X < \frac{1}{3}$ is given by

Therefore, as observed in the graph of the membership function ${ \mathfrak{m}}_{\widetilde{P}(0 < X < \frac{1}{3})}$ in Figure 2, we see that $\widetilde{P}(0 < X < \frac{1}{3})$ establishes a normal fuzzy set of ${\frac{1}{3^3}}$ since ${ \mathfrak{m}}_{\widetilde{P}(0 < X < \frac{1}{3})}(\frac{1}{3^3}) = 1$.

3.   Fuzzifying expected values

We define the fuzzifying expected value of a random variable $X$ with the fuzzifying PDF ${\widetilde{f}_{ X }}$ as

and the fuzzifying expected value for a measurable function $g(X)$ of $X$ for ${\widetilde{f}_{ X }}$ as

Thus, we can derive the fuzzifying $n$-th moment of a random variable as follows.

Theorem 3.1. Let $X$ be a continuous random variable with PDF ${f_{{\mathit{X} }}}$ and $\mu_n = E(X^n)$ be the $n$-th moment about the origin for $X$. If $\widetilde{f}_{ X}$ is a fuzzifying PDF, then $\widetilde{E}(X^n) = \widetilde{\mu}_n$ is a fuzzy set of $\mu_n$ and $(\widetilde{\mu}_n)_1 = \mu_n$ for each $n\in \mathbb{N}$.

Proof. The definition of $\widetilde{E}$ directly provides that

hence $\widetilde{E}(X^n) = \widetilde{\mu}_n$ is a fuzzy set of $\mu_n$. The $\alpha$-cut of $\widetilde{E}(X^n)$ at $\alpha = 1$ in (3.1) can be expressed as

thus $(\widetilde{\mu}_n)_1 = \mu_n.$ □

We now proceed to introduce the concept of the fuzzifying variance of a random variable with a fuzzifying PDF, expressed in terms of the fuzzifying expected value.

Theorem 3.2. If $X$ is a random variable with a fuzzifying PDF ${f_{{\mathit{X} }}}$ and $\mu = E(X)$ is the expected value of $X$, then fuzzifying variance $\widetilde{Var}(X)$ of $X$ can be expressed as

where $\widetilde{\mu} = \widetilde{E}(X)$ and $\widetilde{1} = \left\{ \left.\left(\left[\int f^-_{ {\mathit{X} }_\alpha} (x)dx, \int f^+_{ {\mathit{X} }_\alpha} (x) dx\right], \alpha\right) \right| \alpha\in[0, 1] \right\}.$

Proof. From the definition of the fuzzifying variance,

□

Remark 3.3. Theorem 3.2 shows the fuzzy set

is a generalization of the constant $1$. Since $f^-_{ { X }_1} = f^+_{ { X }_1} = {f_{{ X }}}$ when $\alpha = 1$, $(\widetilde{1})_1$ is a PDF of $X$, hence

We extend Example 2.6 to introduce the concept of fuzzifying expected value and fuzzifying variance, and establish their relationship with the corresponding crisp measures.

Example 3.4. Consider $\widetilde{f}_{ X }(x) = \widetilde{3}x^2$ in Example 2.6. Then, the fuzzifying expected value of $X$ when $n = 1$ in Theorem 3.1 is

where $\widetilde{f}^{-}_{ { X }_\alpha} (x) = (\alpha+2) x^2$ and $\widetilde{f}^{+}_{ { X }_\alpha} (x) = (5-2\alpha) x^2$ for all $\alpha\in [0, 1].$

Therefore,

Thus, $\left(\widetilde{E}(X)\right)_\alpha = \left[\frac{\alpha+2}{4}, \frac{5-2\alpha}{4} \right]$, and hence

Since $E(X) = \int_0^1 x^3 dx = \frac{3}{4}$, $\widetilde{E}(X)$ can be represented by a fuzzy set $\widetilde{\frac{3}{4}}$ (see Figure 3).

We can express $\widetilde{E}(X^2)$ and $E(X^2)$ as

hence $\left(\widetilde{E}(X)\right)_\alpha = \left[\frac{\alpha+2}{5}, \frac{5-2\alpha}{5} \right]$ for all $\alpha\in[0, 1]$ and $E(X^2) = \int_0^1 3x^4 dx = \frac{3}{5}$. Thus $\widetilde{E}(X^2)$ comprises a fuzzy set $\widetilde{\frac{3}{5}}$ (Figure 3). From Theorem 3.2,

where $\widetilde{1}$ satisfies

and hence the membership function of $\widetilde{1}$ is

From (2.2),

for each $\alpha \in[0, 1]$. Thus, the fuzzifying variance $\widetilde{Var}(X)$ is

and the membership function of the fuzzifying variance is

Since $Var(X) = E(X^2)-(E(X))^2 = \frac{3}{5}-\left(\frac{3}{4}\right)^2 = \frac{3}{80}$, $\widetilde{Var}(X)$ is a fuzzy set of ${\frac{3}{80}}$ (see Figure 4).

In conclusion, the expression for the linearity of expectations for a random variable with a fuzzifying PDF is as follows.

Theorem 3.5. Let $g_j$ be integrable functions of a random variable $X$ and $k_j$ be positive integers for $j = 1, 2, \cdots, m$. Then,

Proof. Since $g_j(X)\widetilde{f}_{ X }(x) = g_j(X)[f^-_{ { X }_\alpha}(x), f^+_{ { X }_\alpha}(x)] = [g_j(X)f^-_{ { X }_\alpha}(x), g_j(X)f^+_{ { X }_\alpha}(x)]$ for all $x$,

which confirms linearity of fuzzifying expectations $\widetilde{E}\left(g_j (X)\right)$. □

4.   Conclusions

In this study, the concept of fuzzifying functions has been introduced to probability theory as a means of developing a fuzzifying PDF and a fuzzifying probability. Through this approach, we aim to investigate the ambiguities inherent in probability theories that are affected by uncertainties in the PDF. The validity of the fuzzifying probability was established through Theorem 2.3, while Theorems 3.1 and 3.2 provided the fuzzifying $n$-th moment about the origin of a random variable and the fuzzifying variance, respectively. To demonstrate the utility of our approach, we presented modeled examples in which the fuzzifying functions were shown to generalize crisp functions in probability theory. Examples 2.6 and 3.4 illustrated the fuzzifying probability and the fuzzifying expected value, respectively. Furthermore, we extended the concept of fuzzifying functions to Bernoulli, Poisson, and geometric random variables, among others, thus enabling us to investigate the uncertainties in probability theories arising from the ambiguities in PDFs. In summary, our approach of employing fuzzifying functions allows for the investigation of the impact of uncertainties in PDFs on probability theories, and our findings suggest that the concept of fuzzifying functions has the potential to enhance our understanding of probability theory.

Acknowledgements

The authors received no financial support for the research, authorship, and/or publication of this article.

Conflict of interest

The authors declare no conflict of interest.