This study aimed to investigate the risk-return relationship, provided volatility feedback was taken into account, in the South African market. Volatility feedback, a stronger measure of volatility, was treated as an important source of asymmetry in the investigation of the risk-return relationship. This study analyzed the JSE ALSI excess returns and realized variance for the sample period from 15 October 2009 to 15 October 2019. This study modelled the novel and robust Bayesian approach in a parametric and nonparametric framework. A parametric model has modelling assumptions, such as normality, and a finite sample space. A nonparametric approach relaxes modelling assumptions and allows for an infinite sample space; thus, taking into account every possible asymmetric risk-return relationship. Given that South Africa is an emerging market, which is subject to higher levels of volatility, the presence of volatility feedback was expected to be more pronounced. However, contrary to expectations, the test results from both the parametric and nonparametric Bayesian model showed that volatility feedback had an insignificant effect in the South African market. The risk-return relationship was then investigated free from empirical distortions that resulted from volatility feedback. The parametric Bayesian model found a positive risk-return relationship, in line with traditional theoretical expectations. However, the nonparametric Bayesian model found no relationship between risk and return, in line with early South African studies. Since the nonparametric Bayesian approach is more robust than the parametric Bayesian approach, this study concluded that there is no risk-return relationship. Therefore, investors can include South Africa in their investment portfolio with higher risk countries in order to spread their risk and derive diversification benefits. In addition, risk averse investors can find a safe environment within the South African market and earn a return in accordance to their risk tolerance.
Citation: Nitesha Dwarika. The risk-return relationship and volatility feedback in South Africa: a comparative analysis of the parametric and nonparametric Bayesian approach[J]. Quantitative Finance and Economics, 2023, 7(1): 119-146. doi: 10.3934/QFE.2023007
[1] | Xueqi Sun, Yongqiang Fu, Sihua Liang . Normalized solutions for pseudo-relativistic Schrödinger equations. Communications in Analysis and Mechanics, 2024, 16(1): 217-236. doi: 10.3934/cam.2024010 |
[2] | Wang Xiao, Xuehua Yang, Ziyi Zhou . Pointwise-in-time α-robust error estimate of the ADI difference scheme for three-dimensional fractional subdiffusion equations with variable coefficients. Communications in Analysis and Mechanics, 2024, 16(1): 53-70. doi: 10.3934/cam.2024003 |
[3] | Yining Yang, Cao Wen, Yang Liu, Hong Li, Jinfeng Wang . Optimal time two-mesh mixed finite element method for a nonlinear fractional hyperbolic wave model. Communications in Analysis and Mechanics, 2024, 16(1): 24-52. doi: 10.3934/cam.2024002 |
[4] | Shengbing Deng, Qiaoran Wu . Existence of normalized solutions for the Schrödinger equation. Communications in Analysis and Mechanics, 2023, 15(3): 575-585. doi: 10.3934/cam.2023028 |
[5] | Enzo Vitillaro . Nontrivial solutions for the Laplace equation with a nonlinear Goldstein-Wentzell boundary condition. Communications in Analysis and Mechanics, 2023, 15(4): 811-830. doi: 10.3934/cam.2023039 |
[6] | Zhen Wang, Luhan Sun . The Allen-Cahn equation with a time Caputo-Hadamard derivative: Mathematical and Numerical Analysis. Communications in Analysis and Mechanics, 2023, 15(4): 611-637. doi: 10.3934/cam.2023031 |
[7] | Luhan Sun, Zhen Wang, Yabing Wei . A second–order approximation scheme for Caputo–Hadamard derivative and its application in fractional Allen–Cahn equation. Communications in Analysis and Mechanics, 2025, 17(2): 630-661. doi: 10.3934/cam.2025025 |
[8] | Ho-Sik Lee, Youchan Kim . Boundary Riesz potential estimates for parabolic equations with measurable nonlinearities. Communications in Analysis and Mechanics, 2025, 17(1): 61-99. doi: 10.3934/cam.2025004 |
[9] | Mohamed Karim Hamdani, Lamine Mbarki, Mostafa Allaoui . A new class of multiple nonlocal problems with two parameters and variable-order fractional p(⋅)-Laplacian. Communications in Analysis and Mechanics, 2023, 15(3): 551-574. doi: 10.3934/cam.2023027 |
[10] | Yan Guo, Lei Wu . L2 diffusive expansion for neutron transport equation. Communications in Analysis and Mechanics, 2025, 17(2): 365-386. doi: 10.3934/cam.2025015 |
This study aimed to investigate the risk-return relationship, provided volatility feedback was taken into account, in the South African market. Volatility feedback, a stronger measure of volatility, was treated as an important source of asymmetry in the investigation of the risk-return relationship. This study analyzed the JSE ALSI excess returns and realized variance for the sample period from 15 October 2009 to 15 October 2019. This study modelled the novel and robust Bayesian approach in a parametric and nonparametric framework. A parametric model has modelling assumptions, such as normality, and a finite sample space. A nonparametric approach relaxes modelling assumptions and allows for an infinite sample space; thus, taking into account every possible asymmetric risk-return relationship. Given that South Africa is an emerging market, which is subject to higher levels of volatility, the presence of volatility feedback was expected to be more pronounced. However, contrary to expectations, the test results from both the parametric and nonparametric Bayesian model showed that volatility feedback had an insignificant effect in the South African market. The risk-return relationship was then investigated free from empirical distortions that resulted from volatility feedback. The parametric Bayesian model found a positive risk-return relationship, in line with traditional theoretical expectations. However, the nonparametric Bayesian model found no relationship between risk and return, in line with early South African studies. Since the nonparametric Bayesian approach is more robust than the parametric Bayesian approach, this study concluded that there is no risk-return relationship. Therefore, investors can include South Africa in their investment portfolio with higher risk countries in order to spread their risk and derive diversification benefits. In addition, risk averse investors can find a safe environment within the South African market and earn a return in accordance to their risk tolerance.
Ostrowski proved the following interesting and useful integral inequality in 1938, see [18] and [15, page:468].
Theorem 1.1. Let f:I→R, where I⊆R is an interval, be a mapping differentiable in the interior I∘ of I and let a,b∈I∘ with a<b. If |f′(x)|≤M for all x∈[a,b], then the following inequality holds:
|f(x)−1b−a∫baf(t)dt|≤M(b−a)[14+(x−a+b2)2(b−a)2] | (1.1) |
for all x∈[a,b]. The constant 14 is the best possible in sense that it cannot be replaced by a smaller one.
This inequality gives an upper bound for the approximation of the integral average 1b−a∫baf(t)dt by the value of f(x) at point x∈[a,b]. In recent years, such inequalities were studied extensively by many researchers and numerous generalizations, extensions and variants of them appeared in a number of papers, see [1,2,10,11,19,20,21,22,23].
A function f:I⊆R→R is said to be convex (AA−convex) if the inequality
f(tx+(1−t)y)≤tf(x)+(1−t)f(y) |
holds for all x,y∈I and t∈[0,1].
In [4], Anderson et al. also defined generalized convexity as follows:
Definition 1.1. Let f:I→(0,∞) be continuous, where I is subinterval of (0,∞). Let M and N be any two Mean functions. We say f is MN-convex (concave) if
f(M(x,y))≤(≥)N(f(x),f(y)) |
for all x,y∈I.
Recall the definitions of AG−convex functions, GG−convex functions and GA− functions that are given in [16] by Niculescu:
The AG−convex functions (usually known as log−convex functions) are those functions f:I→(0,∞) for which
x,y∈I and λ∈[0,1]⟹f(λx+(1−λ)y)≤f(x)1−λf(y)λ, | (1.2) |
i.e., for which logf is convex.
The GG−convex functions (called in what follows multiplicatively convex functions) are those functions f:I→J (acting on subintervals of (0,∞)) such that
x,y∈I and λ∈[0,1]⟹f(x1−λyλ)≤f(x)1−λf(y)λ. | (1.3) |
The class of all GA−convex functions is constituted by all functions f:I→R (defined on subintervals of (0,∞)) for which
x,y∈I and λ∈[0,1]⟹f(x1−λyλ)≤(1−λ)f(x)+λf(y). | (1.4) |
The article organized three sections as follows: In the first section, some definitions an preliminaries for Riemann-Liouville and new fractional conformable integral operators are given. Also, some Ostrowski type results involving Riemann-Liouville fractional integrals are in this section. In the second section, an identity involving new fractional conformable integral operator is proved. Further, new Ostrowski type results involving fractional conformable integral operator are obtained by using some inequalities on established lemma and some well-known inequalities such that triangle inequality, Hölder inequality and power mean inequality. After the proof of theorems, it is pointed out that, in special cases, the results reduce the some results involving Riemann-Liouville fractional integrals given by Set in [27]. Finally, in the last chapter, some new results for AG-convex functions has obtained involving new fractional conformable integrals.
Let [a,b] (−∞<a<b<∞) be a finite interval on the real axis R. The Riemann-Liouville fractional integrals Jαa+f and Jαb−f of order α∈C (ℜ(α)>0) with a≥0 and b>0 are defined, respectively, by
Jαa+f(x):=1Γ(α)∫xa(x−t)α−1f(t)dt(x>a;ℜ(α)>0) | (1.5) |
and
Jαb−f(x):=1Γ(α)∫bx(t−x)α−1f(t)dt(x<b;ℜ(α)>0) | (1.6) |
where Γ(t)=∫∞0e−xxt−1dx is an Euler Gamma function.
We recall Beta function (see, e.g., [28, Section 1.1])
B(α,β)={∫10tα−1(1−t)β−1dt(ℜ(α)>0;ℜ(β)>0)Γ(α)Γ(β)Γ(α+β) (α,β∈C∖Z−0). | (1.7) |
and the incomplete gamma function, defined for real numbers a>0 and x≥0 by
Γ(a,x)=∫∞xe−tta−1dt. |
For more details and properties concerning the fractional integral operators (1.5) and (1.6), we refer the reader, for example, to the works [3,5,6,7,8,9,14,17] and the references therein. Also, several new and recent results of fractional derivatives can be found in the papers [29,30,31,32,33,34,35,36,37,38,39,40,41,42].
In [27], Set gave some Ostrowski type results involving Riemann-Liouville fractional integrals, as follows:
Lemma 1.1. Let f:[a,b]→R be a differentiable mapping on (a,b) with a<b. If f′∈L[a,b], then for all x∈[a,b] and α>0 we have:
(x−a)α+(b−x)αb−af(x)−Γ(α+1)b−a[Jαx−f(a)+Jαx+f(b)]=(x−a)α+1b−a∫10tαf′(tx+(1−t)a)dt−(b−x)α+1b−a∫10tαf′(tx+(1−t)b)dt |
where Γ(α) is Euler gamma function.
By using the above lemma, he obtained some new Ostrowski type results involving Riemann-Liouville fractional integral operators, which will generalized via new fractional integral operators in this paper.
Theorem 1.2. Let f:[a,b]⊂[0,∞)→R be a differentiable mapping on (a,b) with a<b such that f′∈L[a,b]. If |f′| is s−convex in the second sense on [a,b] for some fixed s∈(0,1] and |f′(x)|≤M, x∈[a,b], then the following inequality for fractional integrals with α>0 holds:
|(x−a)α+(b−x)αb−af(x)−Γ(α+1)b−a[Jαx−f(a)+Jαx+f(b)]|≤Mb−a(1+Γ(α+1)Γ(s+1)Γ(α+s+1))[(x−a)α+1+(b−x)α+1α+s+1] |
where Γ is Euler gamma function.
Theorem 1.3. Let f:[a,b]⊂[0,∞)→R be a differentiable mapping on (a,b) with a<b such that f′∈L[a,b]. If |f′|q is s−convex in the second sense on [a,b] for some fixed s∈(0,1],p,q>1 and |f′(x)|≤M, x∈[a,b], then the following inequality for fractional integrals holds:
|(x−a)α+(b−x)αb−af(x)−Γ(α+1)b−a[Jαx−f(a)+Jαx+f(b)]|≤M(1+pα)1p(2s+1)1q[(x−a)α+1+(b−x)α+1b−a] |
where 1p+1q=1, α>0 and Γ is Euler gamma function.
Theorem 1.4. Let f:[a,b]⊂[0,∞)→R be a differentiable mapping on (a,b) with a<b such that f′∈L[a,b]. If |f′|q is s−convex in the second sense on [a,b] for some fixed s∈(0,1],q≥1 and |f′(x)|≤M, x∈[a,b], then the following inequality for fractional integrals holds:
|(x−a)α+(b−x)αb−af(x)−Γ(α+1)b−a[Jαx−f(a)+Jαx+f(b)]|≤M(1+α)1−1q(1+Γ(α+1)Γ(s+1)Γ(α+s+1))1q[(x−a)α+1+(b−x)α+1b−a] |
where α>0 and Γ is Euler gamma function.
Theorem 1.5. Let f:[a,b]⊂[0,∞)→R be a differentiable mapping on (a,b) with a<b such that f′∈L[a,b]. If |f′|q is s−concave in the second sense on [a,b] for some fixed s∈(0,1],p,q>1, x∈[a,b], then the following inequality for fractional integrals holds:
|(x−a)α+(b−x)αb−af(x)−Γ(α+1)b−a[Jαx−f(a)+Jαx+f(b)]|≤2s−1q(1+pα)1p(b−a)[(x−a)α+1|f′(x+a2)|+(b−x)α+1|f′(b+x2)|] |
where 1p+1q=1, α>0 and Γ is Euler gamma function.
Some fractional integral operators generalize the some other fractional integrals, in special cases, as in the following integral operator. Jarad et. al. [13] has defined a new fractional integral operator. Also, they gave some properties and relations between the some other fractional integral operators, as Riemann-Liouville fractional integral, Hadamard fractional integrals, generalized fractional integral operators etc., with this operator.
Let β∈C,Re(β)>0, then the left and right sided fractional conformable integral operators has defined respectively, as follows;
βaJαf(x)=1Γ(β)∫xa((x−a)α−(t−a)αα)β−1f(t)(t−a)1−αdt; | (1.8) |
βJαbf(x)=1Γ(β)∫bx((b−x)α−(b−t)αα)β−1f(t)(b−t)1−αdt. | (1.9) |
The results presented here, being general, can be reduced to yield many relatively simple inequalities and identities for functions associated with certain fractional integral operators. For example, the case α=1 in the obtained results are found to yield the same results involving Riemann-Liouville fractional integrals, given before, in literatures. Further, getting more knowledge, see the paper given in [12]. Recently, some studies on this integral operators appeared in literature. Gözpınar [13] obtained Hermite-Hadamard type results for differentiable convex functions. Also, Set et. al. obtained some new results for quasi−convex, some different type convex functions and differentiable convex functions involving this new operator, see [24,25,26]. Motivating the new definition of fractional conformable integral operator and the studies given above, first aim of this study is obtaining new generalizations.
Lemma 2.1. Let f:[a,b]→R be a differentiable function on (a,b) with a<b and f′∈L[a,b]. Then the following equality for fractional conformable integrals holds:
(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]=(x−a)αβ+1b−a∫10(1−(1−t)αα)βf′(tx+(1−t)a)dt+(b−x)αβ+1b−a∫10(1−(1−t)αα)βf′(tx+(1−t)b)dt. |
where α,β>0 and Γ is Euler Gamma function.
Proof. Using the definition as in (1.8) and (1.9), integrating by parts and and changing variables with u=tx+(1−t)a and u=tx+(1−t)b in
I1=∫10(1−(1−t)αα)βf′(tx+(1−t)a)dt,I2=∫10(1−(1−t)αα)βf′(tx+(1−t)b)dt |
respectively, then we have
I1=∫10(1−(1−t)αα)βf′(tx+(1−t)a)dt=(1−(1−t)αα)βf(tx+(1−t)a)x−a|10−β∫10(1−(1−t)αα)β−1(1−t)α−1f(tx+(1−t)a)x−adt=f(x)αβ(x−a)−β∫xa(1−(x−ux−a)αα)β−1(x−ux−a)α−1f(u)x−adux−a=f(x)αβ(x−a)−β(x−a)αβ+1∫xa((x−a)α−(x−u)αα)β−1(x−u)α−1f(u)du=f(x)αβ(x−a)−Γ(β+1)(x−a)αβ+1βJαxf(a), |
similarly
I2=∫10(1−(1−t)αα)βf′(tx+(1−t)b)dt=−f(x)αβ(b−x)+Γ(β+1)(b−x)αβ+1βxJαf(b) |
By multiplying I1 with (x−a)αβ+1b−a and I2 with (b−x)αβ+1b−a we get desired result.
Remark 2.1. Taking α=1 in Lemma 2.1 is found to yield the same result as Lemma 1.1.
Theorem 2.1. Let f:[a,b]→R be a differentiable function on (a,b) with a<b and f′∈L[a,b]. If |f′| is convex on [a,b] and |f′(x)|≤M with x∈[a,b], then the following inequality for fractional conformable integrals holds:
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤Mαβ+1B(1α,β+1)[(x−a)αβ+1b−a+(b−x)αβ+1b−a] | (2.1) |
where α,β>0, B(x,y) and Γ are Euler beta and Euler gamma functions respectively.
Proof. From Lemma 2.1 we can write
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤(x−a)αβ+1b−a∫10(1−(1−t)αα)β|f′(tx+(1−t)a)|dt+(b−x)αβ+1b−a∫10(1−(1−t)αα)β|f′(tx+(1−t)b)|dt≤(x−a)αβ+1b−a[∫10(1−(1−t)αα)βt|f′(x)|dt+∫10(1−(1−t)αα)β(1−t)|f′(a)|dt]+(b−x)αβ+1b−a[∫10(1−(1−t)αα)βt|f′(x)|dt+∫10(1−(1−t)αα)β(1−t)|f′(b)|dt]. | (2.2) |
Notice that
∫10(1−(1−t)αα)βtdt=1αβ+1[B(1α,β+1)−B(2α,β+1)],∫10(1−(1−t)αα)β(1−t)dt=B(2α,β+1)αβ+1. | (2.3) |
Using the fact that, |f′(x)|≤M for x∈[a,b] and combining (2.3) with (2.2), we get desired result.
Remark 2.2. Taking α=1 in Theorem 3.1 and s=1 in Theorem 1.2 are found to yield the same results.
Theorem 2.2. Let f:[a,b]→R be a differentiable function on (a,b) with a<b and f′∈L[a,b]. If |f′|q is convex on [a,b], p,q>1 and |f′(x)|≤M with x∈[a,b], then the following inequality for fractional conformable integrals holds:
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤M[B(βp+1,1α)αβ+1]1p[(x−a)αβ+1b−a+(b−x)αβ+1b−a] | (2.4) |
where 1p+1q=1, α,β>0, B(x,y) and Γ are Euler beta and Euler gamma functions respectively.
Proof. By using Lemma 2.1, convexity of |f′|q and well-known Hölder's inequality, we have
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤(x−a)αβ+1b−a[(∫10(1−(1−t)αα)βp)1p(∫10|f′(tx+(1−t)a)|qdt)1q]+(b−x)αβ+1b−a[(∫10(1−(1−t)αα)βp)1p(∫10|f′(tx+(1−t)b)|qdt)1q]. | (2.5) |
Notice that, changing variables with x=1−(1−t)α, we get
∫10(1−(1−t)αα)βp=B(βp+1,1α)αβ+1. | (2.6) |
Since |f′|q is convex on [a,b] and |f′|q≤Mq, we can easily observe that,
∫10|f′(tx+(1−t)a)|qdt≤∫10t|f′(x)|qdt+∫10(1−t)|f′(a)|qdt≤Mq. | (2.7) |
As a consequence, combining the equality (2.6) and inequality (2.7) with the inequality (2.5), the desired result is obtained.
Remark 2.3. Taking α=1 in Theorem 3.2 and s=1 in Theorem 1.3 are found to yield the same results.
Theorem 2.3. Let f:[a,b]→R be a differentiable function on (a,b) with a<b and f′∈L[a,b]. If |f′|q is convex on [a,b], q≥1 and |f′(x)|≤M with x∈[a,b], then the following inequality for fractional conformable integrals holds:
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤Mαβ+1B(1α,β+1)[(x−a)αβ+1b−a+(b−x)αβ+1b−a] | (2.8) |
where α,β>0, B(x,y) and Γ are Euler Beta and Euler Gamma functions respectively.
Proof. By using Lemma 2.1, convexity of |f″|q and well-known power-mean inequality, we have
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤(x−a)αβ+1b−a(∫10(1−(1−t)αα)βdt)1−1q(∫10(1−(1−t)αα)β|f′(tx+(1−t)a)|qdt)1q+(b−x)αβ+1b−a(∫10(1−(1−t)αα)βdt)1−1q(∫10(1−(1−t)αα)β|f′(tx+(1−t)b)|qdt)1q. | (2.9) |
Since |f′|q is convex and |f′|q≤Mq, by using (2.3) we can easily observe that,
∫10(1−(1−t)αα)β|f′(tx+(1−t)a)|qdt≤∫10(1−(1−t)αα)β[t|f′(x)|q+(1−t)|f′(a)|q]dt≤Mqαβ+1B(1α,β+1). | (2.10) |
As a consequence,
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤(x−a)αβ+1b−a(1αβ+1B(1α,β+1))1−1q(Mqαβ+1B(1α,β+1))1q+(b−x)αβ+1b−a(1αβ+1B(1α,β+1))1−1q(Mqαβ+1B(1α,β+1))1q=Mαβ+1B(1α,β+1)[(x−a)αβ+1b−a+(b−x)αβ+1b−a]. | (2.11) |
This means that, the desired result is obtained.
Remark 2.4. Taking α=1 in Theorem 3.2 and s=1 in Theorem 1.4 are found to yield the same results.
Theorem 2.4. Let f:[a,b]→R be a differentiable function on (a,b) with a<b and f′∈L[a,b]. If |f′|q is concave on [a,b], p,q>1 and |f′(x)|≤M with x∈[a,b], then the following inequality for fractional conformable integrals holds:
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤[B(βp+1,1α)αβ+1]1p[(x−a)αβ+1b−a|f′(x+a2)|+(b−x)αβ+1b−a|f′(x+b2)|] | (2.12) |
where 1p+1q=1, α,β>0, B(x,y) and Γ are Euler Beta and Gamma functions respectively.
Proof. By using Lemma 2.1 and well-known Hölder's inequality, we have
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤(x−a)αβ+1b−a[(∫10(1−(1−t)αα)βp)1p(∫10|f′(tx+(1−t)a)|qdt)1q]+(b−x)αβ+1b−a[(∫10(1−(1−t)αα)βp)1p(∫10|f′(tx+(1−t)b)|qdt)1q]. | (2.13) |
Since |f″|q is concave, it can be easily observe that,
|f′(tx+(1−t)a)|qdt≤|f′(x+a2)|,|f′(tx+(1−t)b)|qdt≤|f′(b+x2)|. | (2.14) |
Notice that, changing variables with x=1−(1−t)α, as in (2.6), we get,
∫10(1−(1−t)αα)βp=B(βp+1,1α)αβ+1. | (2.15) |
As a consequence, substituting (2.14) and (2.15) in (2.13), the desired result is obtained.
Remark 2.5. Taking α=1 in Theorem 3.2 and s=1 in Theorem 1.5 are found to yield the same results.
Some new inequalities for AG-convex functions has obtained in this chapter. For the simplicity, we will denote |f′(x)||f′(a)|=ω and |f′(x)||f′(b)|=ψ.
Theorem 3.1. Let f:[a,b]→R be a differentiable function on (a,b) with a<b and f′∈L[a,b]. If |f′| is AG−convex on [a,b], then the following inequality for fractional conformable integrals holds:
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤|f′(a)|(x−a)αβ+1αβ(b−a)[ω−1lnω−(ωln−αβ−1(ω)(Γ(αβ+1)−Γ(αβ+1,lnω)))]+|f′(b)|(b−x)αβ+1αβ(b−a)[ψ−1lnψ−(ψln−αβ−1(ψ)(Γ(αβ+1)−Γ(αβ+1,lnψ)))] |
where α>0,β>1, Re(lnω)<0∧Re(lnψ)<0∧Re(αβ)>−1,B(x,y),Γ(x,y) and Γ are Euler Beta, Euler incomplete Gamma and Euler Gamma functions respectively.
Proof. From Lemma 2.1 and definition of AG−convexity, we have
(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]≤(x−a)αβ+1b−a∫10(1−(1−t)αα)β|f′(tx+(1−t)a)|dt+(b−x)αβ+1b−a∫10(1−(1−t)αα)β|f′(tx+(1−t)b)|dt≤(x−a)αβ+1b−a[∫10(1−(1−t)αα)β|f′(a)|(|f′(x)||f′(a)|)tdt]+(b−x)αβ+1b−a[∫10(1−(1−t)αα)β|f′(b)|(|f′(x)||f′(b)|)tdt]. | (3.1) |
By using the fact that |1−(1−t)α|β≤1−|1−t|αβ for α>0,β>1, we can write
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤(x−a)αβ+1αβ(b−a)[∫10(1−|1−t|αβ)|f′(a)|(|f′(x)||f′(a)|)tdt]+(b−x)αβ+1αβ(b−a)[∫10(1−|1−t|αβ)|f′(b)|(|f′(x)||f′(b)|)tdt]. |
By computing the above integrals, we get the desired result.
Theorem 3.2. Let f:[a,b]→R be a differentiable function on (a,b) with a<b and f′∈L[a,b]. If |f′|q is AG−convex on [a,b] and p,q>1, then the following inequality for fractional conformable integrals holds:
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤(B(βp+1,1α)αβ+1)1p[|f′(a)|(x−a)αβ+1b−a(ωq−1qlnω)1q+|f′(b)|(b−x)αβ+1b−a(ψq−1qlnψ)1q]. |
where 1p+1q=1, α,β>0, B(x,y) and Γ are Euler beta and Euler gamma functions respectively.
Proof. By using Lemma 2.1, AG−convexity of |f′|q and well-known Hölder's inequality, we can write
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤(x−a)αβ+1b−a[(∫10(1−(1−t)αα)βp)1p(|f′(a)|q∫10(|f′(x)||f′(a)|)qtdt)1q]+(b−x)αβ+1b−a[(∫10(1−(1−t)αα)βp)1p(|f′(b)|q∫10(|f′(x)||f′(b)|)qtdt)1q]. |
By a simple computation, one can obtain
|(x−a)αβ+(b−x)αβ(b−a)αβf(x)−Γ(β+1)b−a[βxJαf(b)+βJαxf(a)]|≤(B(βp+1,1α)αβ+1)1p×[|f′(a)|(x−a)αβ+1b−a(ωq−1qlnω)1q+|f′(b)|(b−x)αβ+1b−a(ψq−1qlnψ)1q]. |
This completes the proof.
Corollary 3.1. In our results, some new Ostrowski type inequalities can be derived by choosing |f′|≤M. We omit the details.
The authors declare that no conflicts of interest in this paper.
[1] |
Adu G, Alagidede P, Karimu A (2015) Stock return distribution in the BRICS. Rev Dev Financ 5: 98–109. https://doi.org/10.1016/j.rdf.2015.09.002 doi: 10.1016/j.rdf.2015.09.002
![]() |
[2] | Bayes T (1763) An Essay towards Solving a Problem in the Doctrine of Chances by the Late Rev. Mr. Bayes, F. R. S. Communicated by Mr. Price, in a Letter to John Canton, A. M. F. R. S. Philos T (1683–1775) 53: 370–418. |
[3] |
Bekiros S, Jlassi M, Naoui K et al. (2017) The asymmetric relationship between returns and implied volatility: Evidence from global stock markets. J Financial Stab 30: 156–174. https://doi.org/10.1016/j.jfs.2017.05.006 doi: 10.1016/j.jfs.2017.05.006
![]() |
[4] | Beyhaghi P, Alimo SR, Bewley TR (2018) A multiscale, asymptotically unbiased approach to uncertainty quantification in the numerical approximation of infinite time-averaged statistics. Available from: https://arXiv.org/pdf/1802.01056.pdf. |
[5] | Brooks C (2014) Introductory Econometrics for Finance. New York: Cambridge University Press. |
[6] |
Chakraborty S, Lozano AC (2019) A graph Laplacian prior for Bayesian variable selection and grouping. Comput Stat Data Anal 136: 72–91. https://doi.org/10.1016/j.csda.2019.01.003 doi: 10.1016/j.csda.2019.01.003
![]() |
[7] |
Chang Y, Choi Y, Park JY (2017) A new approach to model regime switching. J Econom 196: 127–143. http://dx.doi.org/10.1016/j.jeconom.2016.09.005 doi: 10.1016/j.jeconom.2016.09.005
![]() |
[8] |
Demirer R, Gupta R, Lv Z, et al. (2019) Equity Return Dispersion and Stock Market Volatility: Evidence from Multivariate Linear and Nonlinear Causality Tests. Sustainability 11: 1–15. https://doi.org/10.3390/su11020351 doi: 10.3390/su11020351
![]() |
[9] |
Ferguson T (1973) A Bayesian analysis of some nonparametric problems. Ann Stat 1: 209–30. https://doi.org/10.1214/aos/1176342360 doi: 10.1214/aos/1176342360
![]() |
[10] | Ghahramani Z (2009) A Brief Overview of Nonparametric Bayesian Models. NIPS 2009 Workshop. Department of Engineering. University of Cambridge. United Kingdom. Available from: https://mlg.eng.cam.ac.uk/zoubin/talks/nips09npb.pdf. |
[11] |
Hansen PR, Lunde A (2006) Realized variance and market microstructure noise. J Bus Econ Stat 24: 127–61. https://doi.org/10.1198/073500106000000071 doi: 10.1198/073500106000000071
![]() |
[12] |
Harris RDF, Nguyen LH, Stoja E (2019) Extreme downside risk and market turbulence. Quant Finance 19: 1875–1892. https://doi.org/10.1080/14697688.2019.1614652 doi: 10.1080/14697688.2019.1614652
![]() |
[13] |
Jensen MJ, Maheu JM (2018) Risk, Return and Volatility Feedback: A Bayesian Nonparametric Analysis. J Risk Financ Manag 11: 1–29. https://doi.org/10.3390/jrfm11030052 doi: 10.3390/jrfm11030052
![]() |
[14] |
Jin X (2017) Time-varying return-volatility relation in international stock markets. Int Rev Econ Finance 51: 157–173. https://doi.org/10.1016/j.iref.2017.05.015 doi: 10.1016/j.iref.2017.05.015
![]() |
[15] |
Kalli M, Griffin J, Walker S (2011) Slice sampling mixture models. Stat Comput 21: 93–105. http://dx.doi.org/10.1007/s11222-009-9150-y doi: 10.1007/s11222-009-9150-y
![]() |
[16] |
Karabatsos G (2017) A menu-driven software package of Bayesian nonparametric (and parametric) mixed models for regression analysis and density estimation. Behav Res Methods 49: 335–362. https://doi.org/10.3758/s13428-016-0711-7 doi: 10.3758/s13428-016-0711-7
![]() |
[17] |
Kim C, Kim Y (2018) A unified framework jointly explaining business conditions, stock returns, volatility and "volatility feedback news" effects. Stud Nonlinear Dyn Economet 23: 1–12. https://doi.org/10.1515/snde-2016-0151 doi: 10.1515/snde-2016-0151
![]() |
[18] |
Mancino ME, Sanfelici S (2020) Identifying financial instability conditions using high frequency data. J Econ Interact Coord 15: 221–242. https://doi.org/10.1007/s11403-019-00253-6 doi: 10.1007/s11403-019-00253-6
![]() |
[19] |
Mandimika NZ, Chinzara Z (2012) Risk–return trade-off and behaviour of volatility on the South African stock market: evidence from both aggregate and disaggregate data. S Afr J Econ 80: 345–365. https://doi.org/10.1111/J.1813-6982.2012.01328.X doi: 10.1111/J.1813-6982.2012.01328.X
![]() |
[20] | Mangani R (2008) Modelling return volatility on the JSE Securities Exchange of South Africa. Afr Financ J 10: 55–71. |
[21] |
Markowitz H (1952) Portfolio selection. J Finance 7: 77–91. https://doi.org/10.1111/j.1540-6261.1952.tb01525.x doi: 10.1111/j.1540-6261.1952.tb01525.x
![]() |
[22] | National Treasury (2018) Budget Review, In: Economic Overview 2018, 13–24. Available from: http://www.treasury.gov.za/documents/national%20budget/2018/review/FullBR.pdf. |
[23] | Pindyck R (1984) Risk, Inflation, and the Stock Market. Am Econ Rev 74: 335–351. |
[24] | Sethuram J (1994) A constructive definition of Dirichlet priors. Stat Sinica 4: 639–650. |
[25] |
Steyn JP, Theart L (2019). Are South African equity investors rewarded for taking on more risk? J Econ Financ Sci 12: 1–10. https://doi.org/10.4102/jef.v12i1.448 doi: 10.4102/jef.v12i1.448
![]() |
[26] | Sultan I (2018) Asymmetric Covariance, Volatility and Time-Varying Risk Premium: Evidence from the Finnish Stock Market. Available from: https://lutpub.lut.fi/bitstream/handle/10024/158442/Final_Thesis.pdf; jsessionid = F051516CECB9A057A27074664D2D63EB?sequence = 1. |
[27] |
Umutlu M (2019) Does Idiosyncratic Volatility Matter at the Global Level? North Am J Econ Finance 47: 252–268. https://doi.org/10.1016/j.najef.2018.12.015 doi: 10.1016/j.najef.2018.12.015
![]() |