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Using Lyapunov's method for analysing of chaotic behaviour on financial time series data: a case study on Tehran stock exchange

  • In the last decade there is a constantly growing interest in application of mathematics methods and econophysics methods to solve various problems concerning finance,economics,etc. Chaos and its application are importance for most of the current financial and economic phenomena. Financial markets can potentially provide financial long-term series which can be used in analysing and forecasting. Most recent studies,shows the existence of long and short-range correlations in the financial market and economic phenomena. For testing the existence of chaotic behaviour,Lyapunov's method is one of the best methods. In the current study time-series tests of Lyapunov's method were applied,among listed companies on Tehran stock exchange over a period from 2005 to 2015. The obtained findings prove the existence multifractality process in the evolution of time series stock price.

    Citation: Mohammad Reza Abbaszadeh, Mehdi Jabbari Nooghabi, Mohammad Mahdi Rounaghi. Using Lyapunov's method for analysing of chaotic behaviour on financial time series data: a case study on Tehran stock exchange[J]. National Accounting Review, 2020, 2(3): 297-308. doi: 10.3934/NAR.2020017

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  • In the last decade there is a constantly growing interest in application of mathematics methods and econophysics methods to solve various problems concerning finance,economics,etc. Chaos and its application are importance for most of the current financial and economic phenomena. Financial markets can potentially provide financial long-term series which can be used in analysing and forecasting. Most recent studies,shows the existence of long and short-range correlations in the financial market and economic phenomena. For testing the existence of chaotic behaviour,Lyapunov's method is one of the best methods. In the current study time-series tests of Lyapunov's method were applied,among listed companies on Tehran stock exchange over a period from 2005 to 2015. The obtained findings prove the existence multifractality process in the evolution of time series stock price.



    1. Introduction

    The celebrated Cahn-Hilliard equation was proposed to describe phase separation phenomena in binary systems [9]. Its "standard" version has the form of a semilinear parabolic fourth order equation, namely

    utΔ(Δu+f(u))=0. (1.1)
    Here the unknown u stands for the relative concentration of one phase, or component, in a binary material, and f is the derivative of a non-convex potential F whose minima represent the energetically more favorable configurations usually attained in correspondance, or in proximity, of pure phases or concentrations. In view of the fact that u is an order parameter, often it is normalized in such a way that the pure states correspond to the values u=±1, whereas 1<u<1 denotes the (local) presence of a mixture. We will also adopt this convention. In such setting the values u[1,1] are generally interpreted as "nonphysical" and should be somehow excluded. In view of the fourth-order character of (1.1), no maximum principle is available for u. Hence, the constraint u[1,1] is generally enforced by assuming F to be defined only for u(1,1) (or for u[1,1]; both choices are admissible under proper structure conditions) and to be identically + outside the interval [1,1]. A relevant example is given by the so-called logarithmic potential
    F(u)=(1u)log(1u)+(1+u)log(1+u)λ2u2,λ0, (1.2)
    where the last term may induce nonconvexity. Such a kind of potential is generally termed as a singular one and its occurrence may give rise to mathematical difficulties in the analysis of the system. For this reason, singular potentials are often replaced by "smooth" approximations like the so-called double-well potential taking, after normalization, the form F(u)=(1u2)2. Of course, in the presence of a smooth double-well potential, solutions are no longer expected to satisfy the physical constraint u[1,1].

    The mathematical literature devoted to (1.1) is huge and the main properties of the solutions in terms of regularity, qualitative behavior, and asymptotics are now well-understood, also in presence of singular potentials like (1.2) (cf., e.g., [20,21] and the references therein). Actually, in recent years, the attention has moved to more sophisticated versions of (1.1) related to specific physical situations. Among these, we are interested here in the so-called hyperbolic relaxation of the equation. This can be written as

    αutt+utΔ(Δu+f(u))=0, (1.3)
    where α>0 is a (small) relaxation parameter and the new term accounts for the occurrence of "inertial" effects. Equation (1.3) may be used in order to describe strongly non-equilibrium decomposition generated by deep supercooling into the spinodal region occurring in certain materials (e.g., glasses), see [11,12]. From the mathematical point of view, equation (1.3) carries many similarities with the semilinear (damped) wave equation, but is, however, much more delicate to deal with. For instance, in space dimension N=3 the existence of global in time strong solutions is, up to our knowledge, an open issue also in the case when f is a globally Lipschitz (nonlinear) function [17], whereas for N=2 the occurrence of a critical exponent is observed in case f has a polynomial growth [16,18]. The situation is somehow more satisfactory in space dimension N=1 (cf., e.g., [26,27]) due to better Sobolev embeddings (in particular all solutions taking values in the "energy space" are also uniformly bounded). It is however worth noting that, in the case when f is singular, even the existence of (global) weak solutions is a mathematically very challenging problem. Indeed, at least up to our knowledge, this seems to be an open issue even in one space dimension.

    The picture is only partially more satisfactory when one considers a further relaxation of the equation containing a "strong damping" (or "viscosity") term, namely

    αutt+utΔ(δutΔu+f(u))=0, (1.4)
    with δ>0 (a physical justification for this equation is given, e.g., in [22]). The new term induces additional regularity and some parabolic smoothing effects, and, for this reason, (1.4) is mathematically more tractable in comparison to (1.3). Indeed, existence, regularity and large time behavior of solutions have been analyzed in a number of papers (cf., e.g., [6,7,13,15,19] and references therein). In all these contributions, however, f is taken as a smooth function of at most polynomial growth at infinity. Here, instead, we will consider (1.4) with the choice of a singular function f.

    To explain the related difficulties, the main point stands, of course, in the low number of available a priori estimates. This is a general feature of equations of the second order in time, and, as a consequence, approximating sequences satisfy very poor compactness properties. In particular, the second order term utt can be only controlled in a space like L1(0,T;X), where X is a Sobolev space of negative order. In view of the bad topological properties of L1, this implies that in the limit the term ut cannot be shown to be (and, in fact, is not expected to be) continuous in time, but only of bounded variation. In particular, it may present jumps with respect to the time variable. In turn, the occurrence of these jumps is strictly connected to the fact that it is no longer possible to compute the singular term f(u) in the "pointwise" sense.

    Indeed, in the weak formulation f(u) is suitably reinterpreted in the distributional sense, and, in particular, concentration phenomena may occur. This idea comes from the theory of convex integrals in Sobolev spaces introduced in the celebrated paper by Brezis [8] and later developed and adapted to cover a number of different situations (cf., e.g., [3,4,23] and references therein). In our former paper in collaboration with E.Bonetti and E.Rocca [5] we have shown that this method can be adapted to treat equations of the second order in time. Actually, using duality methods in Sobolev spaces of parabolic type (i.e., depending both on space and on time variables), we may provide the required relaxation of the term f(u) accounting for the possible occurrence of concentration phenomena with respect to time. The reader is referred to [5] for further considerations and extended comments and examples.

    Equation (1.4) will be considered here in the simplest mathematical setting. Namely, we will settle it in a smooth bounded domain ΩRN, N3 (we remark however that the results could be easily extended to any spatial dimension), in a fixed reference interval (0,T) of arbitrary length, and with homogeneous Dirichlet boundary conditions. Then, existence of weak solutions will be proved by suitably adapting the approach of [5]. It is worth observing that, as happens for the mentioned strongly damped wave equation and for other similar models, an alternative weak formulation could be given by restating the problem in the form of a variational inequality. However, as noted in [5], we believe the concept of solution provided here to be somehow more flexible. In particular, with this method we may provide an explicit characterization of the (relaxed) term f(u) (which may be thought of as a physical quantity representing the vincular reaction provided by the constraint) in terms of regularity (for instance, for equation (1.4) concentration phenomena are expected to occur only with respect to the time variable t). Moreover, we can prove that at least some weak solutions satisfy a suitable form of the energy inequality. This can be seen as a sort of selection principle for "physical" solutions (note, indeed, that uniqueness is not expected to hold).

    The plan of the paper is the following: in the next Section 2 we introduce our assumptions on coefficients and data and state our main result regarding existence of at least one solution to a suitable weak formulation of equation (1.4). The proof of this theorem is then carried out in Section 3 by means of an approximation - a priori estimates - compactness argument.

    2. Main result

    2.1. Preliminaries

    We consider the viscous Cahn-Hilliard equation with inertia:

    αutt+utΔw=0,w=δutΔu+β(u)λu. (2.1)
    Here the coefficients α and δ are strictly positive constants, whereas λ0. Moreover, β is a maximal monotone operator in R×R satisfying
    ¯D(β)=[1,1],0β(0). (2.2)
    Actually, β represents the monotone part of @f(u)@ (cf. (1.3)). The domain D(β) has been normalized just for mathematical convenience. Following [2], there exists a convex and lower semicontinuous function @j: \mathbb{R} \to [0, + \infty ]@ such that @\beta=\partial j@, @\overline{D(j)}=[-1, 1]@, and @j(0)=\min \;j=0@. For all @ϵ\in(0, 1)@ we denote by @j^ϵ:\mathbb{R} \to [0, +\infty)@ the Moreau-Yosida regularization of j, and by @\beta^ϵ:=\partial j^ϵ=(j^ϵ)'@ the corresponding Yosida approximation of @\beta=\partial j@.

    By a direct check (cf. also [20, Appendix A]), one may prove, based on (2.2), that there exist constants c1>0 and c20 independent of , such that

    β(r)rc1|β(r)|c2. (2.3)
    Let us also introduce some functional spaces: we set H:=L2(Ω) and V:=H10(Ω), so that V=H1(Ω). Moreover, we put
    V:=H1(0,T;H),
    and, for all t(0,T],
    Vt:=H1(0,t;H).
    We denote by (,) and ,Rangle the scalar product in H and the duality pairing between V and V, respectively. The scalar products on L2(0,T;H) and on L2(0,t;H), for t(0,T), are indicated respectively by
    ((,))andby((,))t.
    Correspondingly, the duality products between V and V and between Vt and Vt are noted as
    ,and,t,
    respectively.

    Next, we indicate by A:D(A)RightarrowH, with domain D(A):=H2(Ω)H10(Ω), the Laplace operator with homogeneous Dirichlet boundary condition seen as an unbounded linear operator on H. Hence, A is strictly positive and its powers As are well defined for all sR. In particular, D(A1/2)=H10(Ω)=V. Moreover, A may be extended to the space V and it turns out that A:VRightarrowV is an isomorphism. In particular, V is a Hilbert space when endowed with the scalar product

    (u,v):=v,A1uRangle=u,A1vRangleforu,vV.
    The associated norm is then given by u2V=(u,u) for uV. Correspondingly, the scalar products of the spaces L2(0,T;V) and L2(0,t;V) are denoted by
    ((,))((,)),t
    respectively. In particular, we have
    ((u,v))=T0v,A1udtforu,vL2(0,T;V),
    with a similar characterization holding for ((,)),t.

    2.2. Relaxation of the constraint

    We now provide a brief sketch of the relaxation of β mentioned in the introduction, referring to [5, Sec. 2] for additional details. First of all, we introduce the functional J:H[0,+], J(u):=Ωj(u)dx for all uH, whose value is intended to be + if j(u)L1(Ω). Moreover it is convenient to define

    J(u):=T0Ωj(u)dxdtuL2((0,T)×Ω), (2.4)
    and its counterpart on (0,t), namely
    Jt(u):=t0Ωj(u)dxdsuL2((0,t)×Ω). (2.5)
    Then, the relaxed version of β will be intended as a maximal monotone operator in the duality between V and V. Indeed, we first introduce JV:=JV, the restriction of J to V. Then, we consider its subdifferential JV with respect to the duality pairing between V and V. Namely, for ξV and uV, we say that
    ξJV(u)JV(z)ξ,zu+JV(u)zV. (2.6)
    In order to emphasize that JV consists in a relaxation of β, we will simply note JV=:βw (w standing for "weak"). Proceeding in a similar way for the functional Jt, we define the subdifferential Jt,Vt of the operator Jt,Vt:=JtVt. This will be indicated simply by βw,t.

    In this setting it is not true anymore that an element ξ of the set βw(u) (recall that β is a multivalued operator and, as a consequence, βw may be multivalued as well) admits a "pointwise" interpretation as "ξ(t,x)=β(u(t,x))". Indeed, ξ belongs to the negative order Sobolev space V and concentration phenomena are expected to occur. Nevertheless, the maps β still provide a suitable approximation of βw. Referring the reader to [5,24] for additional details and comments, we just mention here some basic facts. First of all, let us define J(u):=Ωj(u)dx and mathcalJ(u):=T0Ωj(u)dxdt. Then, one may prove that the functionals J converge to J in the sense of Mosco-convergence with respect to the topology of L2(0,T;H). Moreover, their restrictions to V Mosco-converge to JV in the topology of V. The analogue of these properties also holds for restrictions to time subintervals (0,t). Referring the reader to [1, Chap. 3] for the definition and basic properties of Mosco-convergence, here we just recall that this convergence notion for functionals implies (and is in fact equivalent to) a related notion of convergence for their subdifferentials, called graph-convergence (or G-convergence). Namely, noting that the function β represents the subdifferential of J both with respect to the topology of L2(0,T;H) and to that of V, it turns out that the operators β, if identified with their graphs, G-converge to β in the topology of L2(0,T;H)×L2(0,T;H) and G-converge to βw in the topology of V×V. As a consequence of the latter property, we may apply the so-called Mintystrick in the duality between V and V. This argument will be the main tool we will use in order to take the limit in the approximation of the problem and can be simply stated in this way: once one deals with a sequence {v}V satisfying vv weakly in V and β(v)ξ weakly in V, then the inequality

    limsup∈↘0ξ,vξ,v (2.7)
    implies that ξβw(v). In other words, ξ is identified as an element of the set βw(v)V.

    2.3. Statement of the main result

    We start with presenting our basic concept of weak solution, which can be seen as an adaptation of [5, Def. 2.2].

    Definition 2.1. A pair (u,η) is called a weak solution to the initial-boundary value problem for the viscous Cahn-Hilliard equation with inertia whenever the following conditions hold:

    (a) There hold the regularity properties

    utBV(0,T;H4(Ω))L(0,T;V)L2(0,T;H), (2.8)
    uL(0,T;V)L2(0,T;D(A)), (2.9)
    η. (2.10)

    (b) For any test function φV, there holds the following weak version of (2.1) :

    α(ut(T),φ(T))α(u1,φ(0))α((ut,φt))+((ut,φ))+δ((ut,φ))+((A1/2u,A1/2φ))+η,φλ((u,φ))=0. (2.11)

    Moreover, for all t[0,T] there exists η(t)V such that

    α(ut(t),φ(T))α(u1,φ(0))α((ut,φt)),t+((ut,φ)),t+δ((ut,φ))t+((A1/2u,A1/2φ))t+η(t),φtλ((u,φ))t=0, (2.12)
    for all φVt.

    (c) The functionals η and η(t) satisfy

    ηβw(u),η(t)βw,t(u(0,t)) for all t(0,T), (2.13)
    and the following compatibility condition holds true:
    η(t),φt=η,ˉφforallφt,0andallt[0,T), (2.14)
    where Vt,0:={φVt:φ(t)=0} and ˉφ is the trivial extension of φVt,0 to V, i.e., ˉφ(s)=φ(t)=0 for all s(t,T].

    (d) There holds the Cauchy condition

    u|t=0=u0a.e.inΩ. (2.15)

    Correspondingly, we conclude this section with our main result, stating existence of at least one weak solution.

    Theorem 2.2. Let T>0 and let the initial data satisfy

    u0V,j(u0)L1(Ω),u1H. (2.16)
    Then, there exists a solution (u,η) to the viscous Cahn-Hilliard equation with inertia in the sense of Def. 2.1. Moreover, u satisfies the energy inequality
    α2ut(t2)2V+12A1/2u(t2)2H+J(u(t2))λ2u(t2)2H+t2t1(δut2H+ut2V)dsα2ut(t1)2V+12A1/2u(t1)2H+J(u(t1))λ2u(t1)2H, (2.17)
    for almost every t1[0,T) (surely including t1=0) and every t2(t1,T].

    3. Proof of Theorem 2.2

    3.1. Approximation

    We consider a regularization of system (2.1), namely for ∈∈(0,1) we denote by (u,w) the solution to

    αutt+ut+Aw=0, (3.1)
    w=δut+Au+β(u)λu, (3.2)
    coupled with the initial conditions
    u|t=0=u0Rmandut|t=0=u1,Rma.e.inΩ. (3.3)
    Recall that β was defined in Subsec. 2.1. The following result provides existence of a unique smooth solution to (3.1)-(3.3) once the initial data are suitably regularized:

    Theorem 3.1. Let T>0, u0D(A)=H2(Ω)V, u1D(A1/2)=V. Then there exists a unique function u with

    uW1,(0,T;H)H1(0,T;V)L(0,T;D(A)), (3.4)
    utW1,(0,T;D(A1)), (3.5)
    satisfying (3.1)-(3.3). Moreover, for every t1,t2[0,T], there holds the approximate energy balance
    α2ut(t2)2V+12A1/2u(t2)2H+J(u(t2))λ2u(t2)2H+t2t1(δut2H+ut2V)ds=α2ut(t1)2V+12A1/2u(t1)2H+J(u(t1))λ2u(t1)2H. (3.6)

    The proof of the above result is standard (see, e.g., [14, Thm. 2.1]{GGMP2}). Actually, one can replicate the a priori estimates corresponding to the regularity properties (3.4)-(3.5) by multiplying (3.1) by ut, (3.2) by Aut, and using the Lipschitz continuity of β. The regularity of β is also essential for having uniqueness, as one can show via standard contractive methods. Then, to prove the energy equality it is sufficient to test (3.1) by A1ut, (3.2) by ut, and integrate the results with respect to the time and space variables. It is worth observing that these test functions are admissible thanks to the regularity properties (3.4)-(3.5). As a consequence of this fact, we can apply standard chain-rule formulas to obtain that (3.6) holds with the equal sign, which will no longer be the case in the limit.

    As a first step in the proof of Theorem 2.2, we need to specify the required regularization of the initial data:

    Lemma 3.2. Let (2.16) hold. Then there exist two families {u0}D(A)V and {u1}V, ∈∈(0,1), satisfying

    J(u0)J(u0)Rm∈>0Rmandu0u0RminV, (3.7)
    u1u1RminH. (3.8)

    Also the above lemma is standard. Indeed, one can construct u0, u1 by simple singular perturbation methods (see, e.g., [23, Sec. 3]). Let us then consider the solutions u to the regularized system (3.1)-(3.3) with the initial data provided by Lemma 3.2. Then, taking a test function φV, multiplying (3.1) by A1φ, (3.2) by φ, and performing standard manipulations, one can see that u also satisfies the weak formulation (compare with (2.11))

    α(ut(T),φ(T))α(u1,φ(0))α((ut,φt))+((ut,φ))+δ((ut,φ))+((A1/2u,A1/2φ))+((β(u),φ))λ((u,φ))=0. (3.9)

    Correspondingly, the analogue over subintervals (0,t) also holds. Namely, for φVt one has (compare with (2.12))

    α(ut(t),φ(t))α(u1,φ(0))α((ut,φt)),t+((ut,φ)),t+δ((ut,φ))t+((A1/2u,A1/2φ))t+((β(u),φ))tλ((u,φ))t=0. (3.10)

    3.2. A priori estimates

    We now establish some a priori estimates for u. The estimates will be uniform in and permit us to take ∈↘0 at the end. First of all, the energy balance (3.6) and the uniform bounded properties (3.7)-(3.8) of approximating initial data provide the existence of a constant M>0, independent of , such that the following bounds hold true:

    Rm||uRm|Rm|L(0,T;V)M, (3.11a)
    Rm||uRm|Rm|H1(0,T;V)M, (3.11b)
    δ1/2Rm||uRm|Rm|H1(0,T;H)M, (3.11c)
    α1/2Rm||uRm|Rm|W1,(0,T;V)M, (3.11d)
    Rm||j(u)Rm|Rm|L(0,T;L1(Ω))M, (3.11e)
    for all ∈∈(0,1). More precisely, thanks to the fact that, for every (fixed) ∈∈(0,1), ut lies in C0([0,T];V) by (1.1) - (1.1), we are allowed to evaluate ut pointwise in time. Hence, (3.11d) may be complemented by
    ut(t)VM for every t[0,T], (3.12)
    and in particular, for t=T. Analogously, thanks to uC0([0,T];V), in addition to (1.1) we also have
    u(t)VM for every t[0,T]. (3.13)

    Next, taking φ=u in (3.9) and rearranging terms, we infer

    T0Ωβ(u)udxdtαut(T)Vu(T)V+αu1Vu0V+αut2L2(0,T;V)+utL2(0,T;V)uL2(0,T;V)+δutL2(0,T;H)uL2(0,T;H)+A1/2u2L2(0,T;H)+λu2L2(0,T;H). (3.14)

    Then, thanks to estimates (3.11), (3.12) and (3.13), we may check that the right-hand side of (3.14) is bounded uniformly with respect to . Consequently, using also (2.3), we infer

    β(u)L1(0,T;L1(Ω))M. (3.15)

    Now, since we assumed N3, we know that L1(Ω)D(A1), the latter being a closed subspace of H2(Ω). Moreover, A can be extended to a bounded linear operator A:D(A1)RightarrowD(A2)H4(Ω). Then, letting X:=H4(Ω) (note that for N>3 the argument still works up to suitably modifying the choice of X) and rewriting (3.1)-(3.2) as a single equation, i.e., \

    αutt+ut+δAut+A2u+A(β(u))λAu=0, we may check by a comparison of terms that αutW1,1(0,T;X)M. (3.16)

    Actually, we used here the estimates (3.11) together with (3.15).

    Next, thanks to the last of (3.4), we are allowed to multiply (3.1) by u and (3.2) by Au. Using the monotonicity of β and the bounds (3.11), standard arguments lead us to the additional estimate

    uL2(0,T;D(A))M, (3.18)
    still holding for M>0 independent of .

    Finally, for all φVt we can compute from (3.10)

    t0β(u),φdsRight|αut(t)Vφ(t)V+αu1Vφ(0)V+αutL2(0,t;V)φtL2(0,t;V)+utL2(0,t;V)φL2(0,t;V)+δutL2(0,t;H)φL2(0,t;H)+uL2(0,t;D(A))φL2(0,t;H)+λuL2(0,t;H)φL2(0,t;H) (3.19)
    and the right-hand side, by (3.11), (3.12) and (3.18), is less or equal to CφVt, with C depending only on the (controlled) norms of u. Hence it follows that there exists a constant M>0 independent of such that
    β(u)VtM, (3.20)
    for every t(0,T]. In particular, β(u)VM.

    3.3. Passage to the limit

    Using the estimates obtained above, we now aim to pass to the limit as ∈↘0 in the weak formulation (3.9). Firstly, (3.11), (3.18) and (3.20) imply that there exist uW1,(0,T;V)H1(0,T;H)L(0,T;V)L2(0,T;D(A)) and ηV such that

    uRightharpoonupuRmweaklystarinW1,(0,T;V)RmandweaklyinL2(0,T;D(A)), (3.21a)
    uRightharpoonupuRmweaklystarinL(0,T;V)RmandweaklyinH1(0,T;H), (3.21b)
    utRightharpoonuputRmweaklystarinBV(0,T;X), (3.21c)
    β(u)RightharpoonupηRmweaklyin. (3.21d)

    Here and below all convergence relations are implicitly intended to hold up to extraction of a (non relabeled) subsequence of ∈↘0.

    Thanks to (3.21a)-(3.21b) and (3.21) we also infer

    uϵ(t)u(t)weaklyinVforallt[0,T]. (3.21e)
    Next, condition (1.1) implies, thanks to the Aubin-Lions lemma, that
    uu strongly in L2(0,T;V). (3.21f)

    A generalized version of the same lemma [25, Cor. 4, Sec. 8] implies, thanks to (3.21b) and (3.21c),

    utut strongly in L2(0,T;V). (3.21g)

    From (3.21c) and a proper version of the Helly selection principle [10, Lemma 7.2], we infer

    ut(t)Rightharpoonuput(t) weakly in X for all t[0,T]. (3.21h)

    Combining this with (3.21), we obtain more precisely

    ut(t)Rightarrowut(t) weakly in V and strongly in D(A1) for all t[0,T]. (3.21i)

    Hence, using (3.21), we can take ∈↘0 in (3.9) and get back (2.11). Indeed, it is not difficult to check that all terms pass to the limit. Notice however that, in view of (3.21d), the L^2scalarproduct ((β(u),φ)) is replaced by the V-V duality η,φ in the limit.

    Let us now consider the weak formulation on subintervals. Taking φVt, t[0,T], we may rearrange terms in (3.10) to get

    ((β(u),φ))t=α(ut(t),φ(t))+α(u1,φ(0))+α((ut,φt)),t((ut,φ)),tδ((ut,φ))t((A1/2u,A1/2φ))t+λ((u,φ))t=0. (3.22)

    Now, withoutextractingfurthersubsequences, it can be checked that, as a consequence of (3.21), the right-hand side tends to

    α(ut(t),φ(t))+α(u1,φ(0))+α((ut,φ))t,t((ut,φ)),tδ((ut,φ))t((A1/2u,A1/2φ))t+λ((u,φ))t=:η(t),φt. (3.23)

    Hence we have proved (2.11) and (2.12). The compatibility property (2.14) is also a straighforward consequence of this argument.

    Next, to prove (2.13), according to (2.7), we need to show

    lim sup∈↘0β(u),uη,u. (3.24)

    Thanks to (3.9) with φ=u, we have

    β(u),u=α(ut(T),u(T))+α(u1,u0)+α((ut,ut))((ut,u))δ((ut,u))A1/2u2L2(0,T;H)+λu2L2(0,T;H). (3.25)

    Then, we take the lim sup of the above expression as ∈↘0. Then, using relations (3.21) and standard lower semicontinuity arguments we infer that the lim sup of the above expression is less or equal to

    α(ut(T),u(T))+α(u1,u0)+α((ut,ut))((ut,u))δ((ut,u))A1/2u2L2(0,T;H)+λu2L2(0,T;H)=η,u, (3.26)
    the last equality following from (2.11) with the choice φ=u. Combining (3.25) with (3.26) we obtain (3.24), whence the first of (2.13). The same argument applied to the subinterval (0,t) entails η(t)βw(u(0,t)), for all t(0,T], as desired.

    Finally, we need to prove the energy inequality inequality (2.17). To this aim, we consider the approximate energy balance (3.6) and take its lim inf as ∈↘0.

    Then, by standard lower semicontinuity arguments, it is clear that the left-hand side of (2.17) is less or equal to the lim inf of the left-hand side of (3.6). The more delicate point stands, of course, in dealing with the right-hand sides. Indeed, we claim that there exists the limit

    lim∈↘0(α2ut(t1)2V+12A1/2u(t1)2H+J(u(t1))λ2u(t1)2H)=(α2ut(t1)2V+12A1/2u(t1)2H+J(u(t1))λ2u(t1)2H), (3.27)
    at least for almost every t1[0,t), surely including t1=0. We just sketch the proof of this fact, which follows closely the lines of the argument given in [5, Section 3] to which we refer the reader for more details.

    First, we observe that the last summand passes to the limit in view of (3.21e) and the compact embedding VH. Next, the convergence

    (α2ut(t1)2V+12A1/2u(t1)2H)(α2ut(t1)2V+12A1/2u(t1)2H)
    holds for almost every choice of t1 and up to extraction of a further subsequence of ∈↘0 in view of (3.21f) and (3.21g) (indeed, because these are just L2-bounds with respect to time, we cannot hope to get convergence for every t1[0,T)). Finally, we need to show
    Jϵ(uϵ(t1))J(u(t1)).

    This is the most delicate part, which proceeds exactly as in [5, Section 3], to which the reader is referred. Note, finally, that (3.27) for t1=0 can be easily proved as a direct consequence of Lemma 3.2 (again, we refer the reader to [5] for details). The proof is concluded.



    [1] Arnold L, Doyle MM, Sri Namachchivaya N (1997) Small noise expansion of moment Lyapunov exponents for two-dimensional systems. Dyn Stab Syst 12: 187-211.
    [2] Dai M, Hou J, Gao J, et al. (2016) Mixed multifractal analysis of China and US stock index series. Chaos Soliton Fract 87: 268-275.
    [3] De Luca G, Loperfido N (2004) A Skew-in-Mean GARCH Model for Financial Returns, In Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality, CRC Press, 205-222.
    [4] De Luca G, Loperfido N (2015) Modelling Multivariate Skewness in Financial Returns: a SGARCH Approach. Eur J Financ 21: 1113-1131.
    [5] Engle RF (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation. Econometrica 50: 987-1008.
    [6] Fernandez A, Swanson NR (2017) Further Evidence on the Usefulness of Real-Time Datasets for Economic Forecasting. Quant Financ Econ 1: 2-25.
    [7] Foley DK (2016) Can economics be a physical science? Eur Phys J-Spec Top 225: 3171-3178.
    [8] Ghadiri Moghadam A, Jabbari Nooghabi M, Rounaghi MM, et al. (2014) Chaos Process Testing (Time-Series in The Frequency Domain) in Predicting Stock Returns in Tehran Stock Exchange. Indian J Sci Res 4: 202-210.
    [9] Gomez IS (2017) Lyapunov exponents and poles in a non Hermitian dynamics. Chas Soliton Fract 99: 155-161.
    [10] Gupta P, Batra SS, Jayadeva (2017) Sparse Short-Term Time Series Forecasting Models via Minimum Model Complexity. Neurocomputing 243: 1-11.
    [11] Gyamerah SA (2019) Modelling the volatility of Bitcoin returns using GARCH models. Quant Financ Econ 3: 739-753.
    [12] Ibarra-Valdez C, Alvarez J, Alvarez-Ramirez J (2016) Randomness confidence bands of fractal scaling exponents for financial price returns. Chaos Soliton Fract 83: 119-124.
    [13] Jahanshahi H, Yousefpour A, Wei Z, et al. (2019) A financial hyperchaotic system with coexisting attractors: Dynamic investigation, entropy analysis, control and synchronization. Chaos Soliton Fract 126: 66-77.
    [14] Kaplan DT, Glass L (1992) Direct test for determinism in a time series. Phys Rev Lett 68: 427-430.
    [15] Lahmiri S (2017a) A study on chaos in crude oil markets before and after 2008 international financial crisis. Physica A 466: 389-395.
    [16] Lahmiri S (2017b) Investigating existence of chaos in short and long term dynamics of Moroccan exchange rates. Physica A 465: 655-661.
    [17] Lahmiri S, Uddin GS, Bekiros S (2017) Nonlinear dynamics of equity, currency and commodity markets in the aftermath of the global financial crisis. Chaos Soliton Fract 103: 342-346.
    [18] Li DY, Nishimura Y, Men M (2014) Fractal markets: Liquidity and investors on different time horizons. Physica A 407: 144-151.
    [19] Li R, Wang J (2017) Symbolic complexity of volatility duration and volatility difference component on voter financial dynamics. Digit Signal Process 63: 56-71.
    [20] Madaleno M, Vieira E (2018) Volatility analysis of returns and risk: Family versus nonfamily firms. Quant Financ Econ 2: 348-372.
    [21] Mantegna RN, Stanley HE (1996) Turbulence and financial markets. Nature 383: 587-588.
    [22] Mantegna RN, Palagyi Z, Stanley HE (1999) Applications of statistical mechanics to finance. Physica A 274: 216-221.
    [23] Mastroeni L, Vellucci P, Naldi M (2018) Co-existence of stochastic andchaotic behaviour in the copper price time series. Resour Policy 58:295-302.
    [24] Mastroeni L, Vellucci P, Naldi M (2019) A reappraisal of the chaoticparadigm for energy commodity prices. Energ Econ 82: 167-178.
    [25] Moradi M, Jabbari Nooghabi M, Rounaghi MM (2019) Investigation of fractal market hypothesis and forecasting time series stock returns for Tehran Stock Exchange and London Stock Exchange. Int J Financ Econ, 1-17.
    [26] Münnix M, Shimada T, Schä fer R (2012) Identifying states of a financial market. Sci Rep 2: 644-647.
    [27] Nair BB, Kumar PK, Sakthivel NR, et al. (2017) Clustering stock price time series data to generate stock trading recommendations: An empirical study. Expert Syst Appl 70: 20-36.
    [28] Niu H, Wang H (2013a) Complex dynamic behaviors of oriented percolation-based financial time series and Hang Seng index. Chaos Soliton Fract 52: 36-44.
    [29] Niu H, Wang H (2013b) Volatility clustering and long memory of financial time series and financial price model. Digit Signal Process 23: 489-498.
    [30] Ola MR, Jabbari Nooghabi M, Rounaghi MM (2014) Chaos Process Testing (Using Local Polynomial Approximation Model) in Predicting Stock Returns in Tehran Stock Exchange. Asian J Res Bank Financ 4: 100-109.
    [31] Peinke J, Parisi J, Parisi OE, et al. (1992) Encounter with Chaos, Springer-Verlag.
    [32] Podsiadlo M, Rybinski H (2016) Financial time series forecasting using rough sets with time-weighted rule voting. Expert Syst Appl 66: 219-233.
    [33] Rosini L, Shenai V (2020) Stock returns and calendar anomalies on the London Stock Exchange in the dynamic perspective of the Adaptive Market Hypothesis: A study of FTSE100 & FTSE250 indices over a ten year period. Quant Financ Econ 4: 121-147.
    [34] Rounaghi MM, Abbaszadeh MR, Arashi M (2015) Stock price forecasting for companies listed on Tehran stock exchange using multivariate adaptive regression splines model and semi-parametric splines technique. Physica A 438: 625-633.
    [35] Rounaghi MM, Nassir Zadeh F (2016) Investigation of market efficiency and Financial Stability between S & P 500 and London Stock Exchange: Monthly and yearly Forecasting of Time Series Stock Returns using ARMA model. Physica A 456: 10-21.
    [36] Rydber TH (2000) Realistic Statistical Modelling of Financial Data. Int Stat Rev 68: 233-258.
    [37] Salvino LW, Cawley R (1994) Smoothness implies determinism: a method to detect it in time series. Phys Rev Lett 73:1091-1094.
    [38] Shynkevich Y, McGinnity TM, Coleman SA, et al. (2017) Forecasting price movements using technical indicators: Investigating the impact of varying input window length. Neurocomputing 264: 71-88.
    [39] Su CH, Cheng CH (2016) A Hybrid Fuzzy Time Series Model Based on ANFIS and Integrated Nonlinear Feature Selection Method for Forecasting Stock. Neurocomputing 205: 264-273.
    [40] Sun Y, Wang X, Wu Q, et al. (2011) On stability analysis via Lyapunov exponents calculated based on radial basis function networks. Int J Control 84: 1326-1341.
    [41] Takaishi T (2017) Volatility estimation using a rational GARCH model. Quant Financ Econ 2: 127-136.
    [42] Wayland R, Bromley D, Pickett D, et al. (1993) Recog-nizing determinism in a time series. Phys Rev Lett 70: 580-582.
    [43] Wei LY (2016) A hybrid ANFIS model based on empirical mode decomposition for stock time series forecasting. Appl Soft Comput 42: 368-376.
    [44] Zahedi J, Rounaghi MM (2015) Application of artificial neural network models and principal component analysis method in predicting stock prices on Tehran Stock Exchange. Physica A 438: 178-187.
    [45] Zhong X, Enke D (2017) A Comprehensive Cluster and Classification Mining Procedure for Daily Stock Market Return Forecasting. Neurocomputing 267: 152-168.
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