Volume 2008, Article ID 343024,14pages doi:10.1155/2008/343024
Research Article
The Weighted Square Integral
Inequalities for the First Derivative of the Function of a Real Variable
S. Hussain,1J. Peˇcari ´c,1, 2and M. Shashiashvili1, 3
1Abdus-Salam School of Mathematical Sciences, GC University, 35-C-2 Gulberg-III, Lahore 54660, Pakistan
2Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia
3Department of Probability Theory and Mathematical Statistics Chair, Tbilisi State University, 2 University Street, Tblisi 0143, Georgia
Correspondence should be addressed to S. Hussain,[email protected] Received 20 May 2008; Accepted 30 July 2008
Recommended by Wing-Sum Cheung
We generalize the square integral estimate for the derivative of the convex function by Shashiashvili2005to the case of the family of the weight functions, satisfying certain conditions.
This kind of generalization is especially valuable in the problems of mathematical finance for construction of the discrete time hedging strategies.
Copyrightq2008 S. Hussain et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The role of mathematical inequalities within the mathematical branches as well as in its enormous applications should not be underestimated. The appearance of the new mathematical inequality often puts on firm foundation for the heuristic algorithms and procedures used in applied sciences.
The present paper considers new type of weighted square integral inequalities for the first derivative of the convex function, the particular case which has been originally established by K. Shashiashvili and M. Shashiashvili 1 and subsequently applied to the hedging problems of mathematical finance by Hussain and Shashiashvili2.
The convexity property of the value functions of the various problems in finance leads to deep and unexpected results of great practical importance for the traders and practitioners dealing with the real-world financial markets. For example, it is shown in the article 3 by El Karoui et al.see also Hobson4 that the value functions of the European as well as American options are convex with respect to the underlying stock price; and the latter
property gives us the following remarkable robustness result. Even if the writer of the option uses incorrect mathematical model to describe the dynamics of stock prices, he is able to carry out his liabilities if only the incorrectly chosen volatility dominates the true volatility function.
The present paper is organized as follows. In Section 2 we prove the weighted square integral estimates for the first derivative of a function that is assumed to be twice continuously differentiable. Afterwards in Section 3 we consider more general case of the arbitrary convex functions which are not supposed even one time continuously differentiable.
We emphasize the fact that the latter case can be directly applied to problems of mathematical finance, especially to discrete time hedging of the European as well as American call options.
2. The weighted square integral estimates for the first derivative of a twice continuously differentiable function
In this section, we consider the pair of twice continuously differential functionsfxandgx defined on the closed bounded intervala, b.We assume that the functiongxis convex that isgx≥0and the following requirement is satisfied:
|fx| ≤gx, a≤x≤b. 2.1
Introduce the family of nonnegative twice continuously differentiable weight functions Hx, a≤x≤b,which satisfy the condition
Ha Hb 0, Ha Hb 0. 2.2
Theorem 2.1. Letfxandgxbe two twice continuously differentiable functions defined on the bounded intervala, b, which satisfy the requirement2.1and letHx, a ≤ x ≤ bbe arbitrary nonnegative weight function such that condition 2.2 is fulfilled. Then the following inequality is valid:
b
a
fx2Hxdx≤ b
a
f2x
2
sup
a≤x≤b|fx|
gx
Hxdx. 2.3
Proof. Using the integration-by-parts formula in the integral below, we have b
a
fx2HxdxfxfxHx b
a
− b
a
fHxfxdx −
b
a
fxfxHxdx− b
a
fxfxHxdx −1
2 b
a
f2xHxdx− b
a
fxfxHxdx
2.4
asHa Hb 0.
Let us transform the integral 1
2 b
a
f2xHxdx 1
2f2xHx b
a
−1 2
b
a
f2xHxdx −1
2 b
a
f2xHxdx
2.5
using the conditionHa Hb 0.
Inserting the latter expression in equality2.4, we obtain the estimate b
a
fx2Hxdx 1 2
b
a
f2xHxdx− b
a
fxfxHxdx
≤ 1 2
b
a
f2xHxdx b
a
|fx| |fx|Hxdx
≤ 1 2
b
a
f2xHxdx sup
a≤x≤b|fx|
b
a
|fx|Hxdx.
2.6
Taking into account requirement2.1, we get from the latter inequality2.6 b
a
fx2Hxdx≤ 1 2
b
a
f2xHxdx sup
a≤x≤b|fx|
b
a
gxHxdx. 2.7 Now we use twice the integration-by-parts formula and obtain
b
a
gxHxdxgxHx b
a
− b
a
gxHxdx −gxHx
b
a
b
a
gxHxdx
b
a
gxHxdx.
2.8
The latter equality together with the previous estimate2.7give us the required inequality 2.3.
Applying H ¨older inequality to the right-hand side of estimate 2.3, we get the following.
Corollary 2.2. For the functions fx, gx, and the weight functionHx, satisfying the same conditions as inTheorem 2.1, the following bound is valid
b
a
fx2Hxdx≤ f∞
1
2fpgp
Hq, 2.9
where 1≤p≤ ∞, andpandqare conjugate exponents, and
fp b
a
|fx|pdx 1/p
, 1≤p <∞, f∞ sup
a≤x≤b|fx|. 2.10
Remark 2.3. Let us notice that assumption2.1,|fx| ≤gx,is equivalent to the existence of the decomposition of the function fx as the difference of two twice continuously differentiable convex functionsfx f1x−f2x, a≤x≤bsuch thatf1x f2x gx.
Indeed the inequality|fx| ≤gxis the same as−gx≤fx≤gx, that is,
fx gx≥0, gx−fx≥0. 2.11
The latter means that the functions f1x 1
2fx gx, f2x 1
2gx−fx 2.12
are two convex functions such that
fx f1x−f2x, gx f1x f2x. 2.13 This remark suggests to write inequality2.9in a different form. Take two arbitrary twice continuously differentiable convex functionsf1xandf2xand define
fx f1x−f2x, gx f1x f2x, 2.14 fx f1x−f2x,andgx f1x f2x, it is obvious that|fx| ≤gx, then the inequality inCorollary 2.2will take the following form:
b
a
f1x−f2x2Hxdx≤ f1−f2∞
1
2f1−f2pf1f2p
Hq, 2.15 where 1≤p≤ ∞.
Consider the special case of the latter inequality when p ∞ and Hx is of the particular form
Hx x−a2b−x2, a≤x≤b. 2.16 Corollary 2.4. Letf1xandf2xbe two twice continuously differentiable convex functions defined on a closed bounded intervala, band let the weight functionHxbe equal to
Hx x−a2b−x2, a≤x≤b. 2.17 Then the following estimate holds
b
a
f1x−f2x2Hxdx≤ f1−f2∞ 4√
3
9 f1f2∞2√ 3
9 f1−f2∞
b−a3. 2.18 Proof. We have
Hx 12x2−12abx2a24abb2. 2.19 Calculate the integral
b
a
|Hx|dx2 b
a
|6x2−6abxa24abb2|dx. 2.20 Let us introduce the change of variablexaub−a, 0≤u≤1; from the above expression, we obtain
b
a
|Hx|dx2b−a3 1
0
|6u2−6u1|du 4√ 3
9 b−a3. 2.21
Taking into account the latter expression in estimate2.9, we come to the desired inequality 2.18.
Remark 2.5. Comparing the result stated in Corollary 2.4 with Theorem 2.1 from K.
Shashiashvili and M. Shashiashvili1, we come to the conclusion that the multiplier 4√ 3/9 is twice less than obtained in the latter paper.
3. The weighted square integral estimates for the difference of derivatives of two convex functions
In this section, we consider two arbitrary finite convex functionsfxandgxon an infinite interval0,∞. It is well known that they are continuous and have finite left- and right-hand derivativesfx−, fxandgx−, gxinside the open interval0,∞.
We will assume that there exists a positive numberAsuch that ifx≥A,we have
|fx−| ≤C, |gx−| ≤C, 3.1 whereCis certain positive constant.
Let us assume also that the difference of the functionsfxandgxis bounded on the infinite interval0,∞:
sup
x≥0|fx−gx|<∞. 3.2
Introduce now the family of nonnegative twice continuously differentiable weight functions Hxdefined on the open interval0,∞, which satisfy the following conditions:
x→lim0Hx 0, lim
x→ ∞Hx 0, lim
x→0Hx 0, lim
x→∞x Hx 0, 3.3
∞
0
|fx||gx||Hx|dx <∞ 3.4
this integral is understood in the improper sense as the limit
δ→0lim
b→∞
b
δ
|fx||gx||Hx|dx. 3.5
Theorem 3.1. For arbitrary two finite convex functionsfxandgxdefined on0,∞satisfying conditions3.1and3.2and for any nonnegative twice continuously differentiable weight function Hx, 0< x <∞, which satisfy conditions3.3and3.4, the following energy estimate is valid:
∞
0
fx−−gx−2Hxdx≤ 3 2sup
x≥0|fx−gx|
∞
0
|fx||gx||Hx|dx, 3.6
where fx− and gx− denote the left derivatives of the convex functions fx and gx, respectively.
Proof. We will prove the theorem in two stages. On first stage, we verify the validity of the statement for twice continuously differentiable convex functions satisfying conditions3.1 and3.2and on second stage we approximate arbitrary convex functions satisfying the same conditions by smooth ones inside the interval0,∞in an appropriate manner and afterwards we pass onto limit in the previously established estimate.
Thus let us assume at first that fxand gxare two convex functions defined on the interval0,∞which are twice continuously differentiable in the open interval0,∞and satisfy conditions3.1and3.2.
Introduce new functionFxas follows:
Fx fx−gx, 0≤x <∞, 3.7
thenFxis twice continuously differentiable inside the infinite interval0,∞and at point zero, it has finite limitF0.
Consider the following integral on a finite intervalδ, band use in it the integration by parts formulahereδandbare arbitrary strictly positive numbers,
b
δ
FxFHxdxFxFxHx b
δ
− b
δ
FxFxHxdx FbFbHb−FδFδHδ−
b
δ
FxFxHxdx.
3.8
Now bound the absolute value of the last integral in3.8:
b
δ
FxFxHxdx ≤ sup
δ≤x≤b|Fx|
b
δ
|fx−gx|Hxdx
≤ sup
δ≤x≤b|Fx|
b
δ
fx gxHxdx,
3.9
asfx≥0, gx≥0, 0< x <∞.
Let us transform the integral on the right-hand side of inequality3.9:
b
δ
fx gxHxdx fx gxHx b
δ
− b
δ
fx gxHxdx fb gbHb−fδ gδHδ
−
fx gxHx b
δ
− b
δ
fx gxHxdx ,
3.10
which implies b
δ
fx gxHxdx fb gbHb−fδ gδHδ−fb gbHb fδ gδHδ
b
δ
fx gxHxdx.
3.11 Using the above expression in inequality3.9, we obtain the estimate
b
δ
FxFxHxdx ≤ sup
δ≤x≤b|Fx|
|fb gb|Hb |fδ gδ|Hδ |fb gb||Hb||fδ gδ||Hδ|
b
δ
|fx gx||Hx|dx .
3.12
Thus from equality3.8, we come to the following bound:
b
δ
FxFHxdx
≤ |FbFb|Hb |FδFδ|Hδ sup
δ≤x≤b|Fx|
|fb gb|Hb |fδ gδ|Hδ |fb gb||Hb||fδ gδ||Hδ|
b
δ
|fx gx||Hx|dx .
3.13
On the other hand, b
δ
FxFHxdx b
δ
Fx2Hxdx b
δ
FxFxHxdx. 3.14 From here we have the chain of equalities
b
δ
Fx2Hxdx b
δ
FxFHxdx−1 2
b
δ
F2xHxdx
b
δ
FxFHxdx−1 2
F2xHx b
δ
− b
δ
F2xHxdx
b
δ
FxFHxdx−1
2F2bHb 1
2F2δHδ 1
2 b
δ
F2xHxdx.
3.15
Using bound3.13in expression3.15, we arrive to the estimate b
δ
Fx2Hxdx
≤ 1
2F2b|Hb|1
2F2δ|Hδ||FbFb|Hb |FδFδ|Hδ sup
δ≤x≤b|Fx|
× 3
2 b
δ
|fx||gx||Hx|dx|fb gb|Hb |fδ gδ|Hδ |fb gb| |Hb||fδ gδ| |Hδ| .
3.16
It is well known that any convex function is locally absolutely continuous see, e.g., 5, Proposition 17 of Chapter 5, that is,
fx2−fx1
x2
x1
fu−du, 0< x1≤x2<∞, 3.17
wherefu−denotes the left-derivative of the convex functionfxat pointu. As the left- hand derivativefx−of the convex functionfxis nondecreasing function, we have
fx1−≤fu−≤fx2−, if 0< x1≤u≤x2<∞. 3.18 Therefore, from expression3.17, we find
fx1− x2−x1≤fx2−fx1≤fx2−x2−x1, 3.19 where 0< x1≤x2<∞.
Takingx22x1, we get
fx1−x1≤f2x1−fx1. 3.20 Asx1is arbitrary positive number, we have
fx−x≤f2x−fx forx >0. 3.21 On the other hand, lettingx1to zero in inequality3.19, we write
fx2−f0≤fx2−x2, 3.22 that is,
fx−f0≤fx−x, x >0. 3.23
Ultimately we obtain the two-sided inequality
fx−f0≤fx−x≤f2x−fx forx >0, 3.24 which gives
x→0limxfx− 0
similarly lim
x→0xgx− 0
. 3.25
By equality3.17and using condition3.1, we obtain the bound
|fb| ≤ |fA|Cb−A≤ |fA|Cb forA≤b. 3.26 But we have
|fA| |fA|
A A≤ |fA|
A b ifA≤b. 3.27
Therefore we can write
|fbHb| ≤ |fAHb|Cb|Hb| ≤
|fA|
A C
b|Hb| ifA≤b 3.28 and similar bound is valid for|gbHb|
|gbHb| ≤
|gA|
A C
b|Hb| forA≤b. 3.29
Using condition3.3and bounds3.28and3.29, we get
b→∞limF2b|Hb| ≤ sup
0≤x<∞|Fx|lim
b→∞|fb||gb||Hb|0, 3.30
b→∞lim|fb gb||Hb|0. 3.31
Moreover, from conditions3.1and3.3, we find
δ→0limF2δ|Hδ| f0−g02lim
δ→0|Hδ|0,
b→∞lim|FbFb−|Hb≤ sup
0≤x<∞|Fx|lim
b→∞|fb−||gb−|Hb
≤2C sup
0≤x<∞|Fx|lim
b→∞Hb 0,
b→∞lim|fb− gb−|Hb≤2Clim
b→∞Hb 0,
δ→0lim|fδ gδ| |Hδ||f0 g0|lim
δ→0|Hδ|0.
3.32
By the mean value theorem, we have Hδ
δ Hδ−H0
δ Hϑδ, where 0< ϑδ< δ, 3.33 therefore from condition3.3, we get
δ→0lim Hδ
δ 0, 3.34
using the limit relations above and3.25we find
δ→0lim|FδFδ−|Hδ≤ sup
0≤x<∞|Fx|lim
δ→0|fδ−−gδ−|Hδ
≤ sup
0≤x<∞|Fx|lim
δ→0
|δfδ−|Hδ
δ |δgδ−|Hδ δ
0,
3.35
and similarly
δ→0lim|fδ− gδ−|Hδ 0. 3.36 Now we have to pass onto limit whenb → ∞and δ → 0 in inequality 3.16. Obviously, the left-hand side of the inequality increases and the right-hand side is bounded, whenb→
∞, δ → 0, therefore the left-hand side also converges to finite limit. Passing onto limitδ → 0, b→ ∞in inequality3.16using assumption3.4and the limit relations3.30–3.36, we come to the required estimate3.6.
Next we move to the second stage of the proof. Consider two arbitrary convex functions fxand gxdefined on 0,∞, satisfying conditions 3.1 and 3.2. We have to construct the sequences of twice continuously differentiablein the open interval0,∞
convex functionsfnxandgnxapproximating, respectively, the functionsfxandgx inside the interval0,∞in an appropriate manner.
To construct such sequences, we will use the following smoothing function:
ρx
⎧⎨
⎩ c·exp
1 xx−2
; 0< x <2,
0; otherwise,
3.37
where the factorcis chosen to satisfy the equality 2
0
ρxdx1. 3.38
Define forx∈0,∞
fnx ∞
0
nρnx−yfydy, gnx
∞
0
nρnx−ygydy,
3.39
wheren1,2, . . . .
For arbitrary fixedδ >0 consider the restriction of functionsfnxandgnxon the intervalδ, band letn≥4/δ.Thennx≥4 forx∈δ, b.
Perform in3.39the change of variableznx−y, then we find fnx
nx
−∞ρzf
x− z n
dz, gnx
nx
−∞ρzg
x− z n
dz.
3.40
Since the functionρzis equal to zero outside the interval0,2,we can write fnx
2
0
ρzf
x−z n
dz,
gnx 2
0
ρzg
x−z n
dz,
3.41
ifx∈δ, b, n≥4/δ.
From definition3.39, it is obvious that the functionsfnxand gnxare infinitely differentiable, while their convexity follows from the expressions3.41.
Now we will show the uniform convergence of the sequence fnx to fx on the intervalδ, b similarly, the uniform convergence ofgnxtogx. For this purpose, we use the uniform continuity of the functionfxon the intervalδ/2, b. For fixε >0 there exists δ > 0 such that we have
|fx2−fx1| ≤ε if|x2−x1|<δ, x1, x2∈ δ
2, b
. 3.42
Taken≥max{4/δ,4/δ}. Then for 0≤z≤2 andx∈δ, b,we get z
n ≤min δ
2,δ
2 , x− z n≥ δ
2. 3.43
Hence
f
x−z n
−fx
≤ε forn≥max 4
δ,4
δ 3.44
and consequently
|fnx−fx|
2
0
ρz
f
x−z n
−fx
dz
≤ε 3.45 forx∈δ, bandn≥max{4/δ,4/δ}.
This shows the uniform convergence of the sequencefnxtofx andgnxtogx on the intervalδ, b.
Next we need to differentiate 3.41. For this purpose, we will use the following inequality5, page 114concerning convex functionfxand its left-derivativefx−
fx1−≤ fx2−fx1
x2−x1 ≤fx2−, if 0< x1< x2<∞. 3.46 Take therein
x1
x− z n
−h, x2x− z
n, 3.47
where 0< h < δ/4.
We have f
x−z
n−h
−
≤ fx−z/n−fx−z/n−h
h ≤f
x− z
n
−
3.48 forx∈δ, b, 0≤z≤2, 0< h < δ/4, andn≥4/δ.
It is well known that the left derivative of the convex function is nondecreasing and as x−z
n−h≥ δ
4, x−z
n ≤b, 3.49
we can write
f δ
4 −
≤ fx−z/n−fx−z/n−h
h ≤fb−, 3.50
which shows that the family of functions
Φn,xh z fx−z/n−fx−z/n−h
h 3.51
is uniformly bounded by the constantD |fb−||fδ/4−|if onlyx∈δ, b, 0 ≤ z ≤ 2, 0< h <δ/4, andn≥4/δ.
Using expression3.41, we can write fnx−fnx−h
h
2
0
ρzfx−z/n−fx−z/n−h
h dz. 3.52
Taking limit as h tends to zero and using bounded convergence theorem, we obtain the formula
fnx 2
0
ρzf
x− z n
−
dz 3.53
forx∈δ, bandn≥4/δ.
Using3.53let us show that for fixedx∈δ, b,the sequencefnxconverges to the left-derivativefx−.
We have
fnx−fx−
2
0
ρz
f
x− z n
−
−fx−
dz, 3.54
wheren≥4/δ. Choose arbitraryε >0. Since the left-derivativefx−is left continuous, we can findNεsuch thatfor 0≤z≤2:
f
x−z n
−
−fx−
≤ε if onlyn≥Nε. 3.55
Hence we get
|fnx−fx−| ≤ 2
0
ρzε dzε ifx∈δ, b, n≥max 4
δ, Nε , 3.56 that is,
n→∞limfnx fx− for fixed x∈δ, b 3.57 and similarly we have
n→∞limgnx gx− ifx∈δ, b. 3.58 Let us write estimate3.16for the functionFnx fnx−gnxrestricted to the interval δ, b,
b
δ
Fnx2Hxdx≤ 1
2Fn2b|Hb|1
2F2nδ|Hδ||FnbFnb|Hb |FnδFnδ|Hδ sup
δ≤x≤b|Fnx|
× 3
2 b
δ
|fnx||gnx||Hx|dx
|fnb gnb|Hb |fnδ gnδ|Hδ |fnb gnb| |Hb||fnδ gnδ| |Hδ| .
3.59
Forx∈δ, b, 0≤z≤2, andn≥4/δ, we have f
δ 2 −
≤f
x− z n
−
≤fb−. 3.60
Multiplying this inequality byρzand integrating byzover0,2from expression3.53, we obtain
f δ
2 −
≤fnx≤fb−, 3.61
from which it follows that
|fnx| ≤ |fb−|
f δ
2 −
, if x∈δ, b, n≥ 4
δ. 3.62
Similarly for the functionsgnx,we can write
|gnx| ≤ |gb−|
g δ
2 −
. 3.63
From the latter bounds, we obtain
|Fnx| ≤ |fb−||gb−|
f δ
2 −
g δ
2 −
, 3.64 ifx∈δ, bandn≥4/δ.
Hence the sequence of the functionsFnxis uniformly bounded on the intervalδ, b forn ≥ 4/δ.Thus we can apply the bounded convergence theorem in the left-hand side of inequality3.59tendingnto infinity, we will have
b
δ
Fx−2Hxdx
≤ 1
2F2b|Hb|1
2F2δ|Hδ||FbFb−|Hb |FδFδ−|Hδ sup
δ≤x≤b|Fx|
× 3
2 b
δ
|fx||gx||Hx|dx|fb− gb−|Hb |fδ− gδ−|Hδ |fb gb||Hb||fδ gδ||Hδ| .
3.65 Finally, it remains to pass onto limit whenb → ∞andδ → 0 in the latter inequality. The left-hand side of inequality3.65obviously increases whenb→ ∞andδ→0 and the right- hand side is bounded by the assumption3.4and the limit relations3.30–3.36. Therefore passing onto limitb → ∞andδ → 0 in inequality3.65, we arrive to the desired estimate 3.6.
References
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2 S. Hussain and M. Shashiashvili, “Discrete time hedging of the American option,” accepted in Mathematical Finance.
3 N. El Karoui, M. Jeanblanc-Picqu'e, and S. E. Shreve, “Robustness of the Black and Scholes formula,” Mathematical Finance, vol. 8, no. 2, pp. 93–126, 1998.
4 D. G. Hobson, “Volatility misspecification, option pricing and superreplication via coupling,” The Annals of Applied Probability, vol. 8, no. 1, pp. 193–205, 1998.
5 H. L. Royden, Real Analysis, Prentice-Hall, New Delhi, India, 1997.
