Processing math: 93%
Research article Special Issues

Insider information and its relation with the arbitrage condition and the utility maximization problem

  • Within the well-known framework of financial portfolio optimization, we analyze the existing relationships between the condition of arbitrage and the utility maximization in presence of insider information. We assume that, since the initial time, the information flow is altered by adding the knowledge of an additional random variable including future information. In this context we study the utility maximization problem under the logarithmic and the Constant Relative Risk Aversion (CRRA) utilities, with and without the restriction of no temporary-bankruptcy.

    In particular, we show that the value of the insider information may be bounded while the arbitrage condition holds and we prove that the insider information does not always imply arbitrage for the insider by providing an explicit example.

    Citation: Bernardo D'Auria, José Antonio Salmerón. Insider information and its relation with the arbitrage condition and the utility maximization problem[J]. Mathematical Biosciences and Engineering, 2020, 17(2): 998-1019. doi: 10.3934/mbe.2020053

    Related Papers:

    [1] Ehsan Movahednia, Choonkil Park, Dong Yun Shin . Approximation of involution in multi-Banach algebras: Fixed point technique. AIMS Mathematics, 2021, 6(6): 5851-5868. doi: 10.3934/math.2021346
    [2] Songkran Pleumpreedaporn, Chanidaporn Pleumpreedaporn, Weerawat Sudsutad, Jutarat Kongson, Chatthai Thaiprayoon, Jehad Alzabut . On a novel impulsive boundary value pantograph problem under Caputo proportional fractional derivative operator with respect to another function. AIMS Mathematics, 2022, 7(5): 7817-7846. doi: 10.3934/math.2022438
    [3] Ehsan Movahednia, Young Cho, Choonkil Park, Siriluk Paokanta . On approximate solution of lattice functional equations in Banach f-algebras. AIMS Mathematics, 2020, 5(6): 5458-5469. doi: 10.3934/math.2020350
    [4] Weerawat Sudsutad, Chatthai Thaiprayoon, Sotiris K. Ntouyas . Existence and stability results for ψ-Hilfer fractional integro-differential equation with mixed nonlocal boundary conditions. AIMS Mathematics, 2021, 6(4): 4119-4141. doi: 10.3934/math.2021244
    [5] Tae Hun Kim, Ha Nuel Ju, Hong Nyeong Kim, Seong Yoon Jo, Choonkil Park . Bihomomorphisms and biderivations in Lie Banach algebras. AIMS Mathematics, 2020, 5(3): 2196-2210. doi: 10.3934/math.2020145
    [6] Araya Kheawborisut, Siriluk Paokanta, Jedsada Senasukh, Choonkil Park . Ulam stability of hom-ders in fuzzy Banach algebras. AIMS Mathematics, 2022, 7(9): 16556-16568. doi: 10.3934/math.2022907
    [7] Francisco Martínez, Inmaculada Martínez, Mohammed K. A. Kaabar, Silvestre Paredes . New results on complex conformable integral. AIMS Mathematics, 2020, 5(6): 7695-7710. doi: 10.3934/math.2020492
    [8] Weerawat Sudsutad, Wicharn Lewkeeratiyutkul, Chatthai Thaiprayoon, Jutarat Kongson . Existence and stability results for impulsive (k,ψ)-Hilfer fractional double integro-differential equation with mixed nonlocal conditions. AIMS Mathematics, 2023, 8(9): 20437-20476. doi: 10.3934/math.20231042
    [9] Ugyen Samdrup Tshering, Ekkarath Thailert, Sotiris K. Ntouyas . Existence and stability results for a coupled system of Hilfer-Hadamard sequential fractional differential equations with multi-point fractional integral boundary conditions. AIMS Mathematics, 2024, 9(9): 25849-25878. doi: 10.3934/math.20241263
    [10] Sabri T. M. Thabet, Sa'ud Al-Sa'di, Imed Kedim, Ava Sh. Rafeeq, Shahram Rezapour . Analysis study on multi-order ϱ-Hilfer fractional pantograph implicit differential equation on unbounded domains. AIMS Mathematics, 2023, 8(8): 18455-18473. doi: 10.3934/math.2023938
  • Within the well-known framework of financial portfolio optimization, we analyze the existing relationships between the condition of arbitrage and the utility maximization in presence of insider information. We assume that, since the initial time, the information flow is altered by adding the knowledge of an additional random variable including future information. In this context we study the utility maximization problem under the logarithmic and the Constant Relative Risk Aversion (CRRA) utilities, with and without the restriction of no temporary-bankruptcy.

    In particular, we show that the value of the insider information may be bounded while the arbitrage condition holds and we prove that the insider information does not always imply arbitrage for the insider by providing an explicit example.



    Hyers [1] made a response to the question of Ulam in the context of Banach spaces in relation to additive mappings and was a considerable step towards further solutions in this area. Note the concept of stability is a major property in the qualitative theory of differential equations. Over the last few years, results have been presented on numerous types of differential equations. The approach proposed by Hyers [1] which provides the additive function is named a direct technique. This technique is a significant and helpful tool used to investigate the stability of different functional equations. In recent years, a number of research monographs and articles have been studied on diverse applications and generalizations of the HUS, like k-additive mappings, differential equations, Navier–Stokes equations, ODEs, and PDEs (see [2,3,4]). Also in recent years, the stability of different (integral and differential, others functional) equations and other subjects (such as C-ternary algebras, groups, flows and Banach algebras) have been investigated. Fixed–point methods are useful when examining stability and fixed point theory proposes vital tools for solving problems arising in different fields of functional analysis, like equilibrium problems, differential equations, and dynamical systems.

    Assume Banach algebras Q and Q. Suppose (Q,Δ) is a probability measure space and suppose (Q,BQ) and (Q,BQ) are Borel measurable spaces. Then a map f:Q×QQ is a operator if {:f(,α)ν}Δ for each α in Q and νBQ. Assume =(1,,m) and Ω=(Ω1,,Ωm),mN. Then we have

    ΩıΩı,ı=1,,m;

    and also

    0ı0,ı=1,,m.

    Definition 1.1 ([5]). Let is a set and d:2[0,+]m,mN, is a given mapping. If the following conditions are satisfied, then we say d is a generalized metric on :

    (1) For each (g,g)×, we get

    d(g,g)=(0,,0)mg=g;

    (2) For each (g,g)×, we get

    d(g,g)=d(g,g)g=g;

    (3) For each g,g,g, we get

    d(g,g)+d(g,g)d(g,g).

    Theorem 1.2 ([5]). Assume the following assumptions:

    (1) d:2[0,+]m,mN, and (,d) is a complete generalized metric space.

    (2) L: is a strictly contractive mappingwith Lipschitz constant Z<1.

    Then for each g, either

    d(Lng,Ln+1g)=m(+,,+)

    for each nN{0} or there is a n0N such that

    (1) d(Lng,Ln+1g)m(+,,+),nn0;

    (2) The sequence {Lng} converges to a fixed point (g) of L;

    (3) (g) is the unique fixed point of L in the set ={gd(Ln0g,g)m(+,,+)};

    (4) d(g,(g))11Zd(g,Lg) for each g.

    We use fixed-point way to study the multi-stability of antiderivations associated with the following inequality:

    diag[f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες)),,fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες))]n×ndiag[θ1(f1(Λ,εζ+ς)f1(Λ,ε+ς)f1(Λ,εζς)+f1(Λ,ες)),,θn(fn(Λ,εζ+ς)fn(Λ,ε+ς)fn(Λ,εζς)+fn(Λ,ες))]n×n (1.1)

    for each ε,ζ,ςQ, ΛQ with |θ1|,,|θn|<1.

    For this section we refer the reader [6,7]. Assume (ν) denotes the real part of ν if νC. Also, let

    (1) Z+ be the set of the positive integers;

    (2) Z be the negative integer numbers;

    (3) R be the negative real numbers;

    (4) R+ be the positive real numbers.

    We begin by defining various functions which will be needed later. The gamma function is given by

    Γ(X)=0eYYX1dY,(X)>0,XC.

    Euler's functional equation is given by

    Γ(X+1)=XΓ(X),(X)>0,XC.

    Theorem 2.1 ([6]).If XN{0}, then

    Γ(X+1)=X!.

    Theorem 2.2 ([6]). Γ(0.5)=π0.5.

    The Pochhammer symbol is

    ()ȷ=ȷı=1(+ı1)=Γ(+ȷ)Γ()={1ȷ=0(+1)(+ȷ1)ȷN{0}

    where C and ȷ,ıN.

    Note that

    Γ(+ȷ)=(+1)(+ȷ1)Γ()

    where ȷN{0}.

    The Gauss hypergeometric series [7] is given by

    φ1(X):=2F1(α,B;T;X)=1+αBTX+α(α+1)B(B+1)T(T+1)X22+=n=0(α)n(B)n(T)nXnn!, (2.1)

    where α,B,T,XC, nN{0}, and |X|<1.

    Consider the Gauss differential equation

    (XX2)d2ωdX2+(T(α+B+1)X)dωdXαBω=0, (2.2)

    where α,B,XC, TC(Z{0}), and |X|<1. The hypergeometric series is a solution of (2.2).

    Theorem 2.3 ([6]).Let α,B,T,XC and |X|<1. Then

    2F1(α,B;T;X)=Γ(T)Γ(B)Γ(TB)10YB1(1Y)TB1(1XY)αdY,

    where (T)>(B)>0.

    Theorem 2.4. If (T)>0,|X|<1, and |arg(X)|<π, then

    2F1(α,B;T;X)=Γ(T)Γ(α)Γ(B)12πi+iiΓ(α+Y)Γ(B+Y)Γ(Y)Γ(T+Y)(X)YdY,

    We now present the Clausen hypergeometric series [7] and its properties:

    φ2(X):=pFq((α);(T);X)=pFq(α,,αp;T1,,Tq;X)=pFq(α1,,αpT1,,Tq;X)=k=0(α1)k(αp)k(T1)k(Tq)kXnn!, (2.3)

    where p,n,qN{0} and αn,X,TnC.

    Now, (2.3) is a solution of the following differential equation

    (M(q,η,Tn)ω)(X)(N(p,η,αn)ω)(X)=0,

    where

    (M(q,η,Tn)ω)(X)=(XddX)qn=1((XddX)ω(X)+(Tn1)ω(X))=XddX(qn=1((XddX+(Tn1))ω)(X)),

    and

    (N(p,η,αn)ω)(X)=Xpn=1(Xdω(X)dX+αnω(X))=Xpn=1((XddX+αn)ω)(X)

    and αn,X,TnC, p,n,qN{0}, and |X|<1,

    Theorem 2.5 ([6]).Suppose αnC(Z{0}):

    (1) The series converges only for X=0, if p>q+1.

    (2) The series converges absolutely for XC, if p<q.

    (3) The series converges absolutely for |X|<1 and diverges for |X|=1 and for |X|>1 it converges absolutely for (qk=1Tkpk=1αk)>0, if p=q+1.

    Assume the following notation [7]:

    Ξ:=qk=1bk+pj=1aj, (2.4)
    σ:=qk=1|bk|bk+pj=1|aj|aj, (2.5)

    and

    χ:=pj=1κj+qk=1ϑk+pq2, (2.6)

    where κj,ϑkC,k,jN,p,qN{0}, and bk,ajR+.

    The Wright generalized hypergeometric series is given by

    φ3(X):=pWq(X)=pWq((κ1,a1),,(κp,ap)(ϑ1,b1),,(ϑq,bq);X)=pWq((κp,ap)1,p(ϑq,bq)1,q;X)=s=0{pj=1Γ(κj+ajs)}{qk=1Γ(ϑk+bks)}Xss!, (2.7)

    where j,s,kN,XC,Ξ>1,κj,ϑkC,p,qN{0}, and bk,ajR+.

    Theorem 2.6 ([6]).Suppose XC,ϑk,κjC,j,s,kN, bk,ajR+, then

    (1) (2.7) is absolutely convergent for each valueof |X|=σ and of |X|<σ, and (χ)>0.5, if Ξ+1=0.

    (2) (2.7) is absolutely convergent for XC, if Ξ+1>0.

    Now, the Wright function is given by

    φ4(X):=K(ϑ,b,X)=0W1((b,ϑ);X)=k=01Γ(ϑ+bk)Xkk!, (2.8)

    where X,ϑC, and bR.

    Theorem 2.7 ([6]).Now (2.8) for bC (bZ{0} if ϑ=0) and ϑ>1 is an entire function of type δ=(1+ϑ)|ϑ|ϑ1+ϑ, andfinite order p=11+ϑ.

    Theorem 2.8 ([6]).Now (2.8) is an entire functionof X for each bC and ϑ>1.

    The Wright generalized Bessel function (Bessel-Maitland function) is given by

    φ5(X):=J(κ,a,X)=k=01Γ(κ+1+ak)(X)kk!=0W1((κ+1,b);X),

    where κ,XC, and aR.

    Theorem 2.9 ([6]).Suppose XC,j,s,kN, aj,bkR+, and κj,ϑkC. Then (2.7) is an entire function of X.

    Theorem 2.10 ([6]).Suppose bR and ϑC.

    (1) (2.8) is absolutely convergent for all |X|<1 and of |X|=1, and (χ)>0.5, if b+1=0.

    (2) (2.8) is absolutely convergent for XC, if b+1>0.

    Theorem 2.11 ([6]).Suppose b>1,ϑC. Then(2.8) is anentire function of X.

    Theorem 2.12 ([6]).Suppose XC,j,k,sN,κj,ϑkC, and aj,bkR+. Then

    pWq((κ1,1),,(κp,1)(ϑ1,1),,(ϑq,1);X)=pj=1Γ(κj)pk=1Γ(ϑk)pFq(κ1,,κpϑ1,,ϑq;X),

    where Ξ+10.

    The shifted Wright generalized hypergeometric series [6] is given by

    φ6(X):=pBq(X)=pBq((κ1,a1;ϑ1,b1),,(κp,ap;ϑp,bp)(ˆκ1,c1;ˆϑ1,d1),,(ˆκp,cp;ˆϑp,dp);X)=pBq((κp,ap;ϑp,bp)1,p(ˆκp,cp;ˆϑp,dp)1,q;X)=k=0{pm=1b(κm+amk;ϑm+bmk)}{qn=1b(ˆκn+cnk;ˆϑn+dnk)}Xkk!=k=0pm=1(Γ(κm+amk)Γ(ϑm+bmk))qn=1Γ((ˆκn+ˆϑn)+(cn+dn)k)pm=1Γ((ϑm+κm)+(bm+am)k)qn=1(Γ(ˆκn+cnk)Γ(ˆϑn+dnk))Xkk!,

    where m,nN,kN{0},κm,ϑm,ˆκn,ˆϑn,XC,p,qN{0}, and am,bm,cn,dnR+.

    We have the following special cases:

    0B0=eX,1B0(X)=1B0((κ,a;ϑ,b);X)=k=0b(κ+ak;ϑ+bk)Xkk!=k=0Γ(κ+ak)Γ(ϑ+bk)Γ[(b+b)k+(ϑ+κ)]Xkk!=2W1((κ,b),(ϑ,b)(ϑ+κ,b+b);X),0B1(X)=0B1((κ,a;ϑ,b);X)=k=01b(κ+ak;ϑ+bk)Xkk!=k=0Γ[(b+b)k+(ϑ+κ)]Γ(ϑ+bk)Γ(κ+ak)Xkk!=1W2((ϑ+κ,b+b)(κ,b),(ϑ,b);X),1B1(X)=1B1((ˆκ,c;ˆϑ,d)(κ,a;ϑ,b);X)=k=0b(κ+ak;ϑ+bk)b(ˆκ+ck;ˆϑ+dk)Xkk!=k=0Γ(κ+ak)Γ(ϑ+bk)Γ[(ˆκ+ˆϑ)+(c+d)k]Γ[(b+b)k+(κ+ϑ)]Γ(ˆϑ+dk)Γ(ˆκ+ck)Xkk!=3W3((κ,b),(ϑ,b),(ˆκ+ˆϑ,c+d)(ˆκ,c),(ˆϑ,d),(ϑ+κ,b+b);X),

    where kN{0},κm,ϑm,ˆκn,ˆϑn,XC, and am,bm,cn,dnR+.

    Now, we define the Wright generalized hypergeometric series (see [6]) as follows

    φ7(X):=[pWq]n(X)=ns=0{pj=1Γ(κj+ajs)}{qk=1Γ(ϑk+bks)}Xss!,

    where X,κj,ϑkC,s,j,k,q,pN, and aj,bkR+.

    Let

    diag[ρ1,,ρn]n×n=[ρ1000ρ2000ρn]n×n.

    Note that ρ:=diag[ρ1,,ρn]ϱ:=diag[ϱ1,,ϱn] if ρiϱi for each 1in.

    We denote W[X] as

    diag[φ1(X),,φn(X)]n×n.

    A HUR-stability with control functions W[X], is called multi-stability.

    We now propose the notion of antiderivations in Banach algebras and introduce the super-multi-stability of antiderivations in algebras Banach, associated to (1.1).

    Throughout this section, let Q be a complex Banach algebra and that θ1,,θnC{0} with |θ1|,,|θn|<1.

    In this subsection, we study the multi stability of the additive (θ1,,θn)-functional inequality (1.1).

    Lemma 3.1. Suppose fi:Q×QQ(i=1,,nN) are mappings satisfying fi(Λ,0)=0 and (1.1) for each ε,ζ,ςQ, and ΛQ. Then the mappings fi:Q×QQ,(i=1,,nN) are additive (the usual definition is at the end of the proof).

    Proof. Assume that fi:Q×QQ(i=1,,nN) satisfies (1.1).

    Replacing ζ by ζ in (1.1), we get

    diag[f1(Λ,εζ+ς)f1(Λ,ε+ς)f1(Λ,εζς)+f1(Λ,ες),,fn(Λ,εζ+ς)fn(Λ,ε+ς)fn(Λ,εζς)+fn(Λ,ες)]diag[θ1(f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες)),,θn(fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες))] (3.1)

    for each ε,ζ,ςQ, and ΛQ. According to (1.1) and (3.1) we have

    diag[f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες),,fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες)]diag[θ21(f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες)),,θ2n(fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες))]

    and so

    fi(Λ,ε+ζ+ς)fi(Λ,ε+ς)fi(Λ,ε+ζς)+fi(Λ,ες)=0,i=1,,n (3.2)

    for each ε,ζ,ςQ,ΛQ, since |θi|<1(i=1,,n).

    Letting ς=ε in (3.2),

    fi(Λ,2ε+ζ)fi(Λ,2ε)fi(Λ,ζ)=0,i=1,,n

    for each ε,ζQ,ΛQ. Thus fi(i=1,,n) are additive.

    Throughout the paper, let φji:(Q)3i[0,)i, 1in,1jin, and nN. Notice that M:=diag[φj1,,φjn] is a matrix valued control function such that φj1(φjn) represents the element at the 1th(nth) row and 1th(nth) column of the matrix M and φji demonstrates the jith given control function.

    Theorem 3.2. Let (φj1,,φjn):(Q×Q×Q)n[0,)n(1j1,,jnn), be functions such that there exists an (T1,,Tn)<(1,,1)n with

    diag[φj1(ε2,ζ2,ς2),,φjn(ε2,ζ2,ς2)]diag[T12φj1(ε,ζ,ς),,Tn2φjn(ε,ζ,ς)], (3.3)

    for all ε,ζ,ςQ. Suppose fi:Q×QQ(i=1,,n) are mappings satisfying fi(Λ,0)=0 and

    diag[f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες)+|θ1|φj11j1n(ε,ζ,ς),,fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες)+|θn|φjn1jnn(ε,ζ,ς)]diag[θ1(f1(Λ,εζ+ς)f1(Λ,ε+ς)f1(Λ,εζς)+f1(Λ,ες))+φj11j1n(ε,ζ,ς),,θn(fn(Λ,εζ+ς)fn(Λ,ε+ς)fn(Λ,εζς)+fn(Λ,ες))+φjn1jnn(ε,ζ,ς)], (3.4)

    for each ε,ζ,ςQ and ΛQ. Then there exist unique additive mappings fi:Q×QQ such that

    diag[f1(Λ,ε)f1(Λ,ε),,fn(Λ,ε)fn(Λ,ε)]n×ndiag[T12(1T1)φj11j1n(ε2,ε,ε2),,Tn2(1Tn)φjn1jnn(ε2,ε,ε2)]n×n, (3.5)

    for each εQ, and ΛQ.

    Proof. Replacing ζ by ζ in (3.4), we get

    diag[f1(Λ,εζ+ς)f1(Λ,ε+ς)f1(Λ,εζς)+f1(Λ,ες)+|θ1|φj11j1n(ε,ζ,ς),,fn(Λ,εζ+ς)fn(Λ,ε+ς)fn(Λ,εζς)+fn(Λ,ες)+|θn|φjn1jnn(ε,ζ,ς)]diag[θ1(f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες))+φj11j1n(ε,ζ,ς),,θn(fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες))+φjn1jnn(ε,ζ,ς)], (3.6)

    for each ε,ζ,ςQ, and ΛQ. According to (3.4) and (3.6) we have

    diag[f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες),,fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες)]n×ndiag[φj11j1n(ε,ζ,ς),,φjn1jnn(ε,ζ,ς)]n×n, (3.7)

    for each ε,ζ,ςQ, and ΛQ.

    Letting ε=ς=σ2 and ζ=σ in (3.7), we get

    diag[f1(Λ,2σ)2f1(Λ,σ),,fn(Λ,2σ)2fn(Λ,σ)]n×ndiag[φj11j1n(σ2,σ,σ2),,φjn1jnn(σ2,σ,σ2)]n×n, (3.8)

    for each σQ, and ΛQ.

    Let =(1,,n) and =(1,,n).

    Now, consider the set

    :={:(Q×Q)nQn:(Λ,0)=n(0,,0)}

    and define the generalized metric on by

    d(,)=inf{(μ1,,μn)Rn+:diag[1(Λ,ε)1(Λ,ε),n(Λ,ε)n(Λ,ε)]diag[μ1φj11j1n(ε2,ε,ε2),,μnφjn1jnn(ε2,ε,ε2)],εQ,ΛQ},

    where inf=(+,,+)n.

    Now (,d) is complete (also, see [8]).

    Let L:=(L1,,Ln). Now, we consider the linear mapping L: s.t.

    Lii(Λ,ε):=2i(Λ,ε2),i=1,,n

    for each εQ, and ΛQ.

    Let , be given s.t. d(,)=(ε1,,εn). Then

    diag[1(Λ,ε)1(Λ,ε),,n(Λ,ε)n(Λ,ε)]diag[ε1φj11j1n(ε2,ε,ε2),,εnφjn1jnn(ε2,ε,ε2)],

    for each εQ, and ΛQ. Hence

    diag[L11(Λ,ε)L11(Λ,ε),,Lnn(Λ,ε)Lnn(Λ,ε)]=diag[21(Λ,ε2)21(Λ,ε2),,2n(Λ,ε2)2n(Λ,ε2)]diag[2ε1φj11j1n(ε4,ε2,ε4),,2εnφjn1jnn(ε4,ε2,ε4)]diag[T1ε1φj11j1n(ε2,ε,ε2),,Tnεnφjn1jnn(ε2,ε,ε2)],

    for each εQ, and ΛQ. Thus d(,)=(ε1,,εn)n implies that

    d(L(Λ,ε),L(Λ,ε))(T1ε1,,Tnεn).

    Hence

    d(L(ε),L(ε))(T1,,Tn)d(,),

    for each ,. According to (3.8), we get

    diag[f1(Λ,ε)2f1(Λ,ε2),,fn(Λ,ε)2fn(Λ,ε2)]n×ndiag[φj11j1n(ε4,ε2,ε4),,φjn1jnn(ε4,ε2,ε4)]n×ndiag[T12φj11j1n(ε2,ε,ε2),,Tn2φjn1jnn(ε2,ε,ε2)]n×n,

    for each εQ,ΛQ, so d(f,Lf)(T12,,Tn2).

    According to Theorem 1.2 there exist mappings fi:QQ(i=1,,n) satisfying the following:

    (1) f is a fixed point of L, i.e.

    f(Λ,ε)=2f(Λ,ε2), (3.9)

    for each εQ, and ΛQ. The mapping f is a unique fixed point of L in the set

    k={:d(f,)<}.

    This implies that f is a unique mapping satisfying (3.8) s.t. there exist μ1,,μn(0,) satisfying

    diag[f1(Λ,ε)f1(Λ,ε),,fn(Λ,ε)fn(Λ,ε)]diag[μ1φj10j1n(ε2,ε,ε2),,μnφjn0jnn(ε2,ε,ε2)],

    for each εQ, and ΛQ.

    (2) Since limnd(Lnf,f)=0,

    limn2nfi(Λ,ε2n)=fi(Λ,ε),i=1,,n (3.10)

    for each εQ, and ΛQ.

    (3) d(f,f)(11T1,,11Tn)d(f,Lf), which implies

    diag[f1(Λ,ε)f1(Λ,ε),,fn(Λ,ε)fn(Λ,ε)]n×ndiag[T12(1T1)φj11j1n(ε2,ε,ε2),,Tn2(1Tn)φjn1jnn(ε2,ε,ε2)]n×n,

    for each εQ, and ΛQ. According to (3.3) and (3.4) we have

    diag[f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες),,fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες)]=diag[limn2nf1(Λ,ε+ζ+ς2n)f1(Λ,ε+ς2n)f1(Λ,ε+ζς2n)+f1(Λ,ες2n),,limn2nfn(Λ,ε+ζ+ς2n)fn(Λ,ε+ς2n)fn(Λ,ε+ζς2n)+fn(Λ,ες2n)]diag[limn2n|θ1|f1(Λ,εζ+ς2n)f1(Λ,ε+ς2n)f1(Λ,εζς2n)+f1(Λ,ες2n)+limn2n(φj11j1n(ε2n,ζ2n,ς2n)θ1φj11j1n(ε2n,ζ2n,ς2n)),,limn2n|θn|fn(Λ,εζ+ς2n)fn(Λ,ε+ς2n)fn(Λ,εζς2n)+fn(Λ,ες2n)+limn2n(φjn1jnn(ε2n,ζ2n,ς2n)θnφjn1jnn(ε2n,ζ2n,ς2n))]diag[θn(f1(Λ,εζ+ς)f1(Λ,ε+ς)f1(Λ,εζς)+f1(Λ,ες)),,θn(fn(Λ,εζ+ς)fn(Λ,ε+ς)fn(Λ,εζς)+fn(Λ,ες))]

    for each ε,ζ,ςQ, and ΛQ. According to Lemma 3.1, the mapping fi(i=1,,n) is additive.

    Definition 3.3. Assume Q is a complex Banach algebra. A C-linear mapping G:Q×QQ is called an antiderivation if it satisfies

    G(Λ,ε)G(Λ,ζ)=G(Λ,G(Λ,ε)ζ)+G(Λ,εG(Λ,ζ))

    for each ε,ζQ and ΛQ.

    Example 3.4. Suppose Qm is the collection of all polynomials of degree m with complex coefficients and

    Q={qmQm|q(Λ,0)=0,mN}.

    Define G:Q×QQ by

    G(Λ,nk=1bkχk)=nk=1bkkχk

    and G(Λ,0)=0. Then G is an antiderivation.

    Example 3.5. Consider the collection of all continuous functions on R, represented by C(R).

    Define G:Q×C(R)C(R) by

    G(Λ,g(ε))=ε0g(τ)dτ

    for each τR. Then G is an antiderivation.

    Lemma 3.6. [9]Suppose Q is complex Banach algebra and suppose f:Q×QQ is an additive mapping s.t. f(Λ,Jε)=Jf(Λ,ε) for each JT1:={ηC:|η|=1} and each εQ. Then f is C-linear.

    Theorem 3.7. Suppose φj1,,φjn:Q3[0,), (1j1,,jnn), are functions.

    (i) If there exist (T1,,Tn)<(1,,1) satisfying

    diag[φj11j1n(εJ,ζJ,ςJ),,φjn1jnn(εJ,ζJ,ςJ)]diag[T12φj11j1n(2ε,2ζ,2ς),,Tn2φjn1jnn(2ε,2ζ,2ς)], (3.11)

    and if fi:Q×QQ, (i=1,,n), are mappings satisfying fi(Λ,0)=0 and

    diag[Jf1(Λ,ε+ζ+ς)f1(Λ,J(ε+ς))f1(Λ,J(ε+ζς))+Jf1(Λ,ες)+|θ1|φj11j1n(ε,ζ,ς),,Jfn(Λ,ε+ζ+ς)fn(Λ,J(ε+ς))fn(Λ,J(ε+ζς))+Jfn(Λ,ες)+|θn|φjn1jnn(ε,ζ,ς)]diag[θ1(Jf1(Λ,εζ+ς)f1(Λ,J(ε+ς))f1(Λ,J(εζς))+Jf1(Λ,ες))+φj11j1n(ε,ζ,ς),,θn(Jfn(Λ,εζ+ς)fn(Λ,J(ε+ς))fn(Λ,J(εζς))+Jfn(Λ,ες))+φjn1jnn(ε,ζ,ς)], (3.12)

    for each JT1 and all ε,ζ,ςQ,ΛQ, then there exist unique C-linear mappings Gi:Q×QQ,(i=1,,n), s.t.

    diag[f1(Λ,ε)G1(Λ,ε),,fn(Λ,ε)Gn(Λ,ε)]diag[T12(1T1)φj11j1n(ε2,ε,ε2),,Tn2(1Tn)φjn1jnn(ε2,ε,ε2)], (3.13)

    for each εQ,ΛQ.

    (ii) In addition, if (T1,,Tn)<(12,,12) and fi,(i=1,,n), are continuous and satisfy fi(Λ,2ε)=2fi(Λ,ε) and

    diag[f1(Λ,ε)f1(Λ,ζ)f1(Λ,f1(Λ,ε)ζ)f1(Λ,εf1(Λ,ζ)),,fn(Λ,ε)fn(Λ,ζ)fn(Λ,fn(Λ,ε)ζ)fn(Λ,εfn(Λ,ζ))]diag[φj11j1n(ε,ζ,ε),,φjn1jnn(ε,ζ,ε)], (3.14)

    for each ε,ζQ, then fi:Q×QQ are antiderivations.

    Proof. By a similar method used in Theorem 3.2 the proof of (i) is straightforward. Now, we prove (ii).

    (ii) Since Gi=fi,(i=1,,n), are continuous and C-linear, we conclude from (3.11) and (3.14) that

    diag[G1(Λ,ε)G1(Λ,ζ)G1(Λ,G1(Λ,ε)ζ)G1(Λ,εG1(Λ,ζ)),,Gn(Λ,ε)Gn(Λ,ζ)Gn(Λ,Gn(Λ,ε)ζ)Gn(Λ,εGn(Λ,ζ))]=diag[limm4mJm(f1(Λ,ε2mJm)f1(Λ,ζ2mJm)G1(Λ,f1(Λ,ε2mJm)ζ2mJm)G1(Λ,ε2mJmf1(Λ,ζ2mJm))),,limm4mJm(fn(Λ,ε2mJm)fn(Λ,ζ2mJm)Gn(Λ,fn(Λ,ε2mJm)ζ2mJm)Gn(Λ,ε2mλmfn(Λ,ζ2mJm)))]=diag[limm4mJm(f1(Λ,ε2mJm)f1(Λ,ζ2mJm)f1(Λ,f1(Λ,ε2mJm)ζ2mJm)f1(Λ,ε2mJmf1(Λ,ζ2mJm))),,limm4mJm(fn(Λ,ε2mJm)fn(Λ,ζ2mJm)fn(Λ,f1(Λ,ε2mJm)ζ2mJm)fn(Λ,ε2mJmfn(Λ,ζ2mJm)))]diag[limm22mφj11j1n(ε2mJm,ζ2mJm,ε2mJm),,limm22mφjn1jnn(ε2mJm,ζ2mJm,ε2mJm)]diag[limm(2T1)mφj11j1n(ε,ζ,ε),,limm(2Tn)mφjn1jnn(ε,ζ,ε)],

    for each JT1 and each ε,ζQ,ΛQ. Since (2T1,,2Tn)n<(1,,1)n, the C-linear mappings Gi,(i=1,,n), are antiderivations. Thus the mappings fi:Q×QQ,(i=1,,n), are antiderivations.

    In this subsection, we investigate the super-multi-stability of continuous antiderivations in Banach algebras.

    Theorem 3.8. Consider φj11j1n,,φjn1jnn:Q3[0,).

    (i) If there exist (T1,,Tn)n(1,,1) satisfying

    diag[φj11j1n(εJ,ζJ,ςJ),,φjn1jnn(εJ,ζJ,ςJ)]diag[T12φj11j1n(2ε,2ζ,2ς),,Tn2φjn1jnn(2ε,2ζ,2ς)] (3.15)

    and if fi:Q×QQ,(i=1,,n), are mappings satisfying fi(Λ,0)=0 and

    diag[Jf1(Λ,ε+ζ+ς)f1(Λ,J(ε+ς))f1(Λ,J(ε+ζς))+Jf1(Λ,ες)+|θ1|φj11j1n(ε,ζ,ς),,Jfn(Λ,ε+ζ+ς)fn(Λ,J(ε+ς))fn(Λ,J(ε+ζς))+Jfn(Λ,ες)+|θn|φjn1jnn(ε,ζ,ς)]diag[θ1(Jf1(Λ,εζ+ς)f1(Λ,J(ε+ς))f1(Λ,J(εζς))+Jf1(Λ,ες))+φj11j1n(ε,ζ,ς),,θn(Jfn(Λ,εζ+ς)fn(Λ,J(ε+ς))fn(Λ,J(εζς))+Jfn(Λ,ες))+φjn1jnn(ε,ζ,ς)], (3.16)

    for each JC¯T1 and each ε,ζ,ςQ,ΛQ, then there are unique C-linear mappings Gi:Q×QQ,(i=1,,n), s.t.

    diag[f1(Λ,ε)G1(Λ,ε),,fn(Λ,ε)Gn(Λ,ε)]diag[T12(1T1)φj11j1n(ε2,ε,ε2),,Tn2(1Tn)φjn1jnn(ε2,ε,ε2)], (3.17)

    for each εQ,ΛQ.

    (ii) Furthermore, if (T1,,Tn)n(12,,12), φj11j1n,,φjn1jnn are continuous functions and also fi,(i=1,,n), are continuous and satisfy fi(Λ,2ε)=2fi(Λ,ε) and

    diag[f1(Λ,ε)f1(Λ,ζ)f1(Λ,f1(Λ,ε)ζ)f1(Λ,εf1(Λ,ζ)),,fn(Λ,ε)fn(Λ,ζ)fn(Λ,fn(Λ,ε)ζ)fn(Λ,εfn(Λ,ζ))]diag[φj11j1n(ε,ζ,ε),,φjn1jnn(ε,ζ,ε)],

    for each ε,ζQ,ΛQ, then fi:Q×QQ are continuous antiderivations.

    Proof. Using the same reasoning as in the proof of Theorem 3.7, we obtain the desired result.

    Here, let n=7.

    Corollary 3.9. Suppose fi:Q×QQ(i=1,,n) are mappings satisfying fi(Λ,0)=0 and

    diag[f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες),,fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες)]diag[θ1(f1(Λ,εζ+ς)f1(Λ,ε+ς)f1(Λ,εζς)+f1(Λ,ες))+(1|θ1|)φj11j1n(ε2+ζ2+ς2),,θn(fn(Λ,εζ+ς)fn(Λ,ε+ς)fn(Λ,εζς)+fn(Λ,ες))+(1|θn|)φjn1jnn(ε2+ζ2+ς2)]

    for each ε,ζ,ςQ, and ΛQ. Then there are unique additive mappings fi:Q×QQ s.t.

    diag[f1(Λ,ε)f1(Λ,ε),,fn(Λ,ε)fn(Λ,ε)]n×ndiag[φj11j1n(ε2),,φjn1jnn(ε2)],

    for each εQ and ΛQ.

    Proof. The proof follows from Theorem 3.2 by letting

    diag[φj11j1n(ε,ζ,ς),,φjn1jnn(ε,ζ,ς)]:=diag[φj11j1n(ε2+ζ2+ς2),,φjn1jnn(ε2+ζ2+ς2)],

    for each ε,ζ,ςQ. Choosing (T1,,Tn)=(47,,47), we obtain the desired result.

    Corollary 3.10. Suppose fi:Q×QQ,(i=1,,n) are mappings satisfying fi(Λ,0)=0 and

    diag[f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες),,fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες)]diag[θ1(f1(Λ,εζ+ς)f1(Λ,ε+ς)f1(Λ,εζς)+f1(Λ,ες))+(1|θ1|)φj11j1n(εζς),,θn(fn(Λ,εζ+ς)fn(Λ,ε+ς)fn(Λ,εζς)+fn(Λ,ες))+(1|θn|)φjn1jnn(εζς)]

    for each ε,ζ,ςQ and ΛQ. Then there are unique additive mappings fi:Q×QQ,(i=1,,n), s.t.

    diag[f1(Λ,ε)f1(Λ,ε),,fn(Λ,ε)fn(Λ,ε)]diag[φj11j1n(ε3),,φjn1jnn(ε3)],

    for each εQ and ΛQ.

    Proof. The proof follows from Theorem 3.2 by letting

    diag[φj11j1n(ε,ζ,ς),,φjn1jnn(ε,ζ,ς)]:=diag[φj11j1n(εζς),,φjn1jnn(εζς)],

    for each ε,ζ,ςQ and ΛQ. Choosing (T1,,Tn)=(89,,89), we obtain the desired result.

    Corollary 3.11. Let fi:Q×QQ,(i=1,,n) be odd mappings satisfying

    diag[f1(Λ,ε+ζ+ς)f1(Λ,ε+ς)f1(Λ,ε+ζς)+f1(Λ,ες),,fn(Λ,ε+ζ+ς)fn(Λ,ε+ς)fn(Λ,ε+ζς)+fn(Λ,ες)]diag[θ1[f1(Λ,εζ+ς)f1(Λ,ε+ς)f1(Λ,εζς)+f1(Λ,ες)]+(1|θ1|)φj11j1n(εζς),,θn[fn(Λ,εζ+ς)fn(Λ,ε+ς)fn(Λ,εζς)+fn(Λ,ες)]+(1|θn|)φjn1jnn(εζς)], (3.18)

    for each ε,ζ,ςQ and ΛQ. Then fi(i=1,,n) are additive.

    Proof. Putting ε=0 in (3.18), we get

    diag[f1(Λ,ζ+ς)f1(Λ,ς)f1(Λ,ζς)+f1(Λ,ς),,fn(Λ,ζ+ς)fn(Λ,ς)fn(Λ,ζς)+fn(Λ,ς)]diag[θ1(f1(Λ,ζ+ς)f1(Λ,ς)f1(Λ,ζς)+f1(Λ,ς)),,θn(fn(Λ,ζ+ς)fn(Λ,ς)fn(Λ,ζς)+fn(Λ,ς))], (3.19)

    for each ζ,ςQ and ΛQ. Replacing ζ by ζ in (3.19), we have

    diag[f1(Λ,ζ+ς)f1(Λ,ς)f1(Λ,ζς)+f1(Λ,ς),,fn(Λ,ζ+ς)fn(Λ,ς)fn(Λ,ζς)+fn(Λ,ς)]diag[θ1[f1(Λ,ζ+ς)f1(Λ,ς)f1(Λ,ζς)+f1(Λ,ς)],,θn[fn(Λ,ζ+ς)fn(Λ,ς)fn(Λ,ζς)+fn(Λ,ς)]], (3.20)

    for each \zeta, \varsigma\in \mathscr{Q} and \Lambda\in\mathscr{Q}^{\prime} . From (3.19) and (3.20), it follows that

    \begin{equation*} f_{i}(\Lambda , \zeta+\varsigma)-f_{i}(\Lambda , \varsigma)-f_{i}(\Lambda , \zeta-\varsigma)+f_{i}(\Lambda , -\varsigma) = 0,\:\:\:\:\:i = 1,\ldots , n \end{equation*}

    for each \zeta, \varsigma\in \mathscr{Q} and \Lambda\in\mathscr{Q}^{\prime} . Since f_{i}, (i = 1, \ldots, n) , are odd mappings,

    \begin{equation*} f_{i}(\varsigma+\zeta)+f_{i}(\varsigma-\zeta)-2f_{i}(\varsigma) = 0,\:\:\:\:\:i = 1,\ldots , n \end{equation*}

    for each \zeta, \varsigma\in \mathscr{Q} and \Lambda\in\mathscr{Q}^{\prime} . Thus the mappings f_{i}, (i = 1, \ldots, n) , are additive.

    Corollary 3.12. Suppose f_{i}: \mathscr{Q}^{\prime}\times \mathscr{Q} \to \mathscr{Q}, (i = 1, \cdots, n), are mappings satisfying f_{i}(\Lambda, 0) = 0 and

    \begin{align*} &\mathit{\text{diag}}\bigg[\:\:\bigg\Vert f_{1}(\Lambda, \varepsilon+\zeta+\varsigma)- f_{1}(\Lambda, \varepsilon+\varsigma)- f_{1}(\Lambda, \varepsilon+\zeta-\varsigma)+ f_{1}(\Lambda, \varepsilon-\varsigma)\bigg\Vert , \cdots , \\ & \quad \bigg\Vert f_{n}(\Lambda, \varepsilon+\zeta+\varsigma)- f_{n}(\Lambda, \varepsilon+\varsigma)- f_{n}(\Lambda, \varepsilon+\zeta-\varsigma)+ f_{n}(\Lambda, \varepsilon-\varsigma)\bigg\Vert \:\:\bigg]\nonumber\\ &\preceq \mathit{\text{diag}}\bigg[\:\: \bigg\Vert \theta_{1} ( f_{1}(\Lambda, \varepsilon-\zeta+\varsigma)- f_{1}(\Lambda, \varepsilon+\varsigma)- f_{1}(\Lambda, \varepsilon-\zeta-\varsigma)+ f_{1}(\Lambda, \varepsilon-\varsigma))\bigg\Vert \\ & \quad +(1-\vert \theta_{1} \vert) {\underbrace{\varphi_{j_{1}}^{{\circledS}}}_{1\leq j_{1}\leq n}}\bigg(\Vert \varepsilon^{4}+\zeta^{4}+\varsigma^{4} \Vert\bigg)\:\: , \cdots , \\ & \quad \bigg\Vert \theta_{n} ( f_{n}(\Lambda, \varepsilon-\zeta+\varsigma)- f_{n}(\Lambda, \varepsilon+\varsigma)- f_{n}(\Lambda, \varepsilon-\zeta-\varsigma)+ f_{n}(\Lambda, \varepsilon-\varsigma))\bigg\Vert \\ & \quad +(1-\vert \theta_{n} \vert){\underbrace{\varphi_{j_{n}}^{{\circledS}}}_{1\leq j_{n}\leq n}}\bigg( \Vert \varepsilon^{4}+\zeta^{4}+\varsigma^{4} \Vert\bigg) \:\:\bigg] \end{align*}

    and

    \begin{align*} &\mathit{\text{diag}}\bigg[\:\:\bigg\Vert f_{1}(\Lambda, \varepsilon)f_{1}(\Lambda, \zeta)-f_{1}(\Lambda, f_{1}(\Lambda, \varepsilon)\zeta)-f_{1}(\Lambda, \varepsilon f_{1}(\Lambda, \zeta))\bigg\Vert \: , \cdots , \\ & \quad \bigg\Vert f_{n}(\Lambda, \varepsilon)f_{n}(\Lambda, \zeta)-f_{n}(\Lambda, f_{n}(\Lambda, \varepsilon)\zeta)-f_{n}(\Lambda, \varepsilon f_{n}(\Lambda, \zeta))\bigg\Vert \: \bigg] \\ & \preceq \mathit{\text{diag}}\bigg[\:\: {\underbrace{\varphi_{j_{1}}^{{\circledS}}}_{1\leq j_{1}\leq n}}\bigg(\Vert 2\varepsilon^{4}+\zeta^{4}\Vert\bigg) , \cdots , {\underbrace{\varphi_{j_{n}}^{{\circledS}}}_{1\leq j_{n}\leq n}}\bigg(\Vert 2\varepsilon^{4}+\zeta^{4}\Vert \bigg)\bigg]_{n\times n}, \end{align*}

    for each \varepsilon, \zeta, \varsigma\in \mathscr{Q}, \: \Lambda\in \mathscr{Q}^{\prime} . If f_{i}(\Lambda, 2\varepsilon) = 2 f_{i}(\Lambda, \varepsilon) foreach \varepsilon, \zeta, \varsigma\in \mathscr{Q}, \: \Lambda\in \mathscr{Q}^{\prime} , and f_{i}, (i = 1, \cdots, n), are continuous, then the mappings f_{i}: \mathscr{Q}^{\prime}\times \mathscr{Q}\to \mathscr{Q}, (i = 1, \cdots, n), are antiderivations.

    Proof. The proof follows from Theorem 3.7 by letting

    \begin{eqnarray*} &&\text{diag}\bigg[\:\: \underbrace{\varphi_{j_{1}}}_{1\leq j_{1}\leq n}(\varepsilon , \zeta , \varsigma) , \cdots , \underbrace{\varphi_{j_{n}}}_{1\leq j_{n}\leq n}(\varepsilon , \zeta , \varsigma) \bigg]\\ &&: = \text{diag}\bigg[\:\:{\underbrace{\varphi_{j_{1}}^{{\circledS}}}_{1\leq j_{1}\leq n}} \bigg(\Vert \varepsilon^{4}+\zeta^{4}+\varsigma^{4} \Vert\bigg) \: , \cdots , \:{\underbrace{\varphi_{j_{n}}^{{\circledS}}}_{1\leq j_{n}\leq n}}\bigg( \Vert \varepsilon^{4}+\zeta^{4}+\varsigma^{4} \Vert \bigg) \:\bigg]_{n\times n} \end{eqnarray*}

    for each \varepsilon, \zeta, \varsigma\in \mathscr{Q} . Choosing (\mathcal{T}_{1}, \cdots, \mathcal{T}_{n}) = \overbrace{(\frac{8}{17}, \cdots, \frac{8}{17})}^{n} , we obtain the desired result.

    In this study, we investigated the concept of antiderivations in Banach algebras and study multi-super-stability of antiderivations in Banach algebras, associated with functional inequalities.

    The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) for funding and supporting this work through Research Partnership Program no RP-21-09-08.

    The authors declare that they have no competing interests.



    [1] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, Rev. Econ. Stat., 51 (1969), 247-257.
    [2] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, J. Econ. Theory, 3 (1971), 373-413.
    [3] T. Jeulin and M. Yor, Nouveaux résultats sur le grossissement des tribus, Ann. Sci. Éc. Norm. Supér., 11 (1978), 429-443.
    [4] J. Jacod, Grossissement Initial, Hypothése (H') et Théoréme de Girsanov, Springer, 1985.
    [5] M. Chaleyat-Maurel and T. Jeulin, Grossissement gaussien de la filtration brownienne, in Grossissements de filtrations: Exemples et applications (eds. T. Jeulin and M. Yor), Springer Berlin Heidelberg, Berlin, Heidelberg, 1985, 59-109.
    [6] H. Föllmer and P. Imkeller, Anticipation cancelled by a girsanov transformation: A paradox on wiener space, Ann. Inst. Henri Poincaré Probab. Stat., 29 (1993), 569-586.
    [7] I. Pikovsky and I. Karatzas, Anticipative portfolio optimization, Adv. Appl. Probab., 28 (1996), 1095-1122.
    [8] A. Grorud and M. Pontier, Insider trading in a continuous time market model, Int. J. Theor. Appl. Finance, 01 (1998), 331-347.
    [9] J. Amendinger, P. Imkeller and M. Schweizer, Additional logarithmic utility of an insider, Stochastic Process Appl., 75 (1998), 263-286.
    [10] J. Amendinger, D. Becherer and M. Schweizer, A monetary value for initial information in portfolio optimization, Finance Stoch., 7 (2003), 29-46.
    [11] S. Ankirchner, S. Dereich and P. Imkeller, The shannon information of filtrations and the additional logarithmic utility of insiders, Ann. Probab., 34 (2006), 743-778.
    [12] F. Baudoin and L. Nguyen-Ngoc, The financial value of a weak information on a financial market, Finance Stoch., 8 (2004), 415-435.
    [13] F. Biagini and B. Øksendal, A general stochastic calculus approach to insider trading, Appl. Math. Optim., 52 (2005), 167-181.
    [14] A. Aksamit and M. Jeanblanc, Enlargement of Filtration with Finance in View, Springer International Publishing, 2017.
    [15] S. Ankirchner and P. Imkeller, Finite utility on financial markets with asymmetric information and structure properties of the price dynamics, Ann. Inst. Henri Poincare Probab. Stat., 41 (2005), 479-503.
    [16] B. Acciaio, C. Fontana and C. Kardaras, Arbitrage of the first kind and filtration enlargements in semimartingale financial models, Stochastic Process. Appl., 126 (2016), 1761-1784.
    [17] H. N. Chau, A. Cosso and C. Fontana, The value of informational arbitrage, preprint, URL https://arXiv.org/abs/1804.00442.
    [18] H. N. Chau, W. Runggaldier and P. Tankov, Arbitrage and utility maximization in market models with an insider, Math. Financ. Econ., 12 (2018), 589-614.
    [19] F. Delbaen, Representing martingale measures when asset prices are continuous and bounded, Math. Finance, 2 (1992), 107-130.
    [20] P. Imkeller, Random times at which insiders can have free lunches, Stoch. Stoch. Rep., 74 (2002), 465-487.
    [21] C. Fontana, M. Jeanblanc and S. Song, On arbitrages arising with honest times, Finance Stoch., 18 (2014), 515-543.
    [22] C. Kardaras, On the characterisation of honest times that avoid all stopping times, Stochastic Process. Appl., 124 (2014), 373-384.
    [23] F. Delbaen and W. Schachermayer, A general version of the fundamental theorem of asset pricing, Math. Ann., 300 (1994), 463-520.
    [24] F. Delbaen and W. Schachermayer, The Mathematics of Arbitrage, Springer Finance, SpringerVerlag, 2006.
    [25] F. Baudoin, Conditioned stochastic differential equations: Theory, examples and application to finance, Stochastic Process. Appl., 100 (2002), 109-145.
    [26] P. Imkeller, M. Pontier and F. Weisz, Free lunch and arbitrage possibilities in a financial market model with an insider, Stochastic Process. Appl., 92 (2001), 103-130.
    [27] N. Bauerle and U. Rieder, Portfolio optimization with markov-modulated stock prices and interest rates, IEEE Trans. Automat. Control, 15 (2004), 442-447.
    [28] K. Itô, Extension of stochastic integrals, in Proceedings of International Symposium on Stochastic Differential Equations (Kyoto University, 1976), Wiley, New York, USA, 1978, 95-109.
    [29] P. E. Protter, Stochastic Integration and Differential Equations, Springer Berlin Heidelberg, 2005.
    [30] S. Dragomir and R. Agarwa, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, App. Math. Lett., 11 (1997), 91-95.
  • This article has been cited by:

    1. Safoura Rezaei Aderyani, Reza Saadati, Chenkuan Li, Themistocles M. Rassias, Choonkil Park, Special functions and multi-stability of the Jensen type random operator equation in C^{*}-algebras via fixed point, 2023, 2023, 1029-242X, 10.1186/s13660-023-02942-0
    2. Safoura Rezaei Aderyani, Reza Saadati, Chenkuan Li, Tofigh Allahviranloo, 2024, Chapter 13, 978-3-031-55563-3, 337, 10.1007/978-3-031-55564-0_13
    3. Safoura Rezaei Aderyani, Reza Saadati, Chenkuan Li, Tofigh Allahviranloo, 2024, Chapter 14, 978-3-031-55563-3, 351, 10.1007/978-3-031-55564-0_14
    4. Safoura Rezaei Aderyani, Azam Ahadi, Reza Saadati, Hari M. Srivastava, Aggregate special functions to approximate permuting tri-homomorphisms and permuting tri-derivations associated with a tri-additive ψ-functional inequality in Banach algebras, 2024, 44, 0252-9602, 311, 10.1007/s10473-024-0117-z
    5. Safoura Rezaei Aderyani, Reza Saadati, Stability and controllability results by n–ary aggregation functions in matrix valued fuzzy n–normed spaces, 2023, 643, 00200255, 119265, 10.1016/j.ins.2023.119265
    6. Safoura Rezaei Aderyani, Reza Saadati, Chenkuan Li, Tofigh Allahviranloo, 2024, Chapter 11, 978-3-031-55563-3, 275, 10.1007/978-3-031-55564-0_11
    7. Safoura Rezaei Aderyani, Reza Saadati, Chenkuan Li, Tofigh Allahviranloo, 2024, Chapter 1, 978-3-031-55563-3, 1, 10.1007/978-3-031-55564-0_1
    8. Safoura Rezaei Aderyani, Reza Saadati, Chenkuan Li, Tofigh Allahviranloo, 2024, Chapter 10, 978-3-031-55563-3, 251, 10.1007/978-3-031-55564-0_10
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4062) PDF downloads(398) Cited by(3)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog