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[Math.CA] 13 Aug 2001 K014 QUADRATIC VARIATION, p-VARIATION AND INTEGRATION WITH APPLICATIONS TO STOCK PRICE MODELLING Rimas Norvaiˇsa1 Institute of Mathematics and Informatics, Vilnius, Lithuania October 29, 2018 arXiv:math/0108090v1 [math.CA] 13 Aug 2001 1This research was partially supported by Lithuanian State Science and Studies Foundation Grant K014. Contents 1 Introduction and results 1 2 The p-variation and integrals of Stieltjes type 12 2.1 Additive and multiplicative functions . ........... 12 2.2 TheWienerclass .................................. 17 2.3 Extended Riemann-Stieltjes integrals . ........... 23 2.4 Chain rules for extended Riemann-Stieltjes integrals . ............... 30 2.5 TheLeftandRightYoungintegrals . ...... 33 3 Quadratic variation and restricted integrals of Stieltjes type 38 3.1 The quadratic λ-variation and λ-covariation . 38 3.2 The Left and Right λ-integrals ............................ 51 3.3 Vectorvaluedfunctions . ..... 64 3.4 The product λ-integral................................. 73 3.5 Extended Dol´eans exponentials . ........ 83 3.6 Augmented linear λ-integralequations . 90 3.7 The evolution representation . ....... 99 3.8 Non-existence of λ-integrals .............................. 108 4 Extension of the class of semimartingales 115 4.1 Stochastic processes and p-variation ......................... 115 4.2 Stochastic integral and the Left Young integral . ............ 121 4.3 (2 ǫ)-semimartingales and the stochastic ⊖-integral . 123 − 4.4 Quadratic covariation of (2 ǫ)-semimartingales . 126 − 4.5 The Itˆoformula and the linear stochastic ⊖-integral equation . 129 4.6 Concludingremarks ............................... 134 5 Stock price modelling in continuous time 135 5.1 Evolutionaryassetpricingmodel . ........ 135 5.2 Almost sure approximation by a discrete time model . ........... 140 5.3 Optionpricingandhedging . 144 5.4 Returns and the p-variation: examples . 148 5.5 Estimating the p-variationindex ........................... 152 A Convergence of directed functions 159 Bibliography 162 i Chapter 1 Introduction and results This long paper deals with several aspects of calculus for real-valued functions having a quadratic variation. Here the notion of a quadratic variation is a property of a “deterministic” function rather than the well-known property of a Brownian motion discovered by L´evy [63]. Let f be a regulated real-valued function on a closed interval [a, b], that is, for a s<t b, there exist ≤ ≤ the limits f(t ) := limu t f(u) and f(s+) := limu s f(u). Let λ = λm: m 1 be a nested − ↑ ↓ { ≥ } sequence of partitions λ = tm: i = 0,...,n(m) of [a, b] such that λ is dense in [a, b]; the m { i } ∪m m class of all such λ is denoted by Λ[a, b]. We say that f has the quadratic λ-variation on [a, b], if there is a regulated function V on [a, b] such that V (a) = 0 and for each a s<t b, letting ≤ ≤ xm := (tm t) s for i = 0,...,n(m), i i ∧ ∨ n(m) V (t) V (s) = lim [f(xm) f(xm )]2, (1.1) m i i 1 − →∞ − − Xi=1 V (t) V (t ) = [f(t) f(t )]2 and V (s+) V (s) = [f(s+) f(s)]2. (1.2) − − − − − − The function [f]λ := V , if exists, is nondecreasing and is called the bracket function of f. Actu- ally, existence of V with the stated properties is equivalent to the definition of the quadratic λ- variation (see Definition 3.3 and Proposition 3.4). The class of all functions having the quadratic λ-variation on [a, b] is denoted by Qλ[a, b]. F¨olmer [34] introduced a quadratic variation for a regulated and right-continuous function on [0, ) to be a Radon measure on [0, ) if exists as ∞ ∞ a limit of sums of point masses with respect to a sequence of partitions of a half line [0, ) with ∞ vanishing mesh. The bracket function [f]λ being a function rather than a measure, allows us to use a Stieltjes type integration for all classes of functions appearing in this paper. We notice that the quadratic variation of a function defined by Wiener [109], if exists, does not depend on partitions, and is closely related to the local 2-variation of a function defined by Love and Young [68]. To recall its definition let f be a function on [a, b], and let 0 <p< . For a partition ∞ κ = t : i = 0,...,n of [a, b], let { i } n p sp(f; κ) := f(ti) f(ti 1) . (1.3) | − − | Xi=1 We say that f has the local p-variation, 1 <p< , if the limit ∞ lim sp(f; κ) (1.4) κ,P 1 exists in the sense of refinements of partitions (see Appendix A for details). Wiener’s quadratic variation of f means that (1.4) holds for p = 2 and the limit in the sense of refinements is replaced by the limit as the mesh of partitions tends to zero. Notice that the limit (1.1) is taken along a fixed sequence of partitions. Both Wiener’s quadratic variation and the local 2-variation do not exist for almost all sample functions of a Brownian motion. Again let f be a function on [a, b]. For 0 <p< , the p-variation of f is defined by ∞ v (f; [a, b]) := sup s (f; κ): κ Ξ[a, b] , (1.5) p { p ∈ } where Ξ[a, b] denotes the class of all partitions of [a, b]. We say that f has bounded p-variation if v (f; [a, b]) < , and denote the class of all such functions by [a, b]. Usefulness of the p ∞ Wp p-variation property hinges on its relation to the Stieltjes type integrability theory established around the late of thirties by L. C. Young in connection to applications in the theory of Fourier series. Let g and f be a pair of functions on [a, b] such that 1 1 f [a, b], g [a, b] and p− + q− > 1 for p,q > 0. (1.6) ∈Wp ∈Wq Then by the L. C. Young Theorem on Stieltjes integrability [113] the integral b g df exists (a) ∫a in the sense of Riemann-Stieltjes if f and g have no common discontinuities, (b) in the sense of refinement Riemann-Stieltjes if f and g have no common discontinuities on the right and no common discontinuities on the left, (c) always in the sense defined by L. C. Young. The functional calculus developed by many authors later on has a full-fledged application to the class of functions having bounded p-variation for some 0 <p< 2 (see [25]). In particular, the theory applies path by path to L´evy processes without a Gaussian component, and to a fractional Brownian motion having the Hurst exponent H (1/2, 1) because their almost all ∈ sample functions have bounded p-variation for some 0 <p< 2. Let B = B(t): t 0 be a { ≥ } standard Brownian motion, and let X = X(t): t 0 be a stochastic process both defined on a { ≥ } complete probability space (Ω, , Pr). Then by L. C. Young’s Theorem on Stieltjes integrability, F for almost all ω Ω, the Riemann-Stieltjes integral ∈ T (RS) X(t,ω) dB(t,ω) (1.7) Z0 exists provided for some 0 <p< 2, X has almost all sample functions of bounded p-variation on [0, T ]. This fact was used by P. L´evy to solve suitable Riemann-Stieltjes integral equations with respect to a Brownian motion (see e.g. [64]). In (1.7), X may be an α-stable L´evy process with 0 <α< 2, or X may be a fractional Brownian motion with the Hurst index 1/2 <H< 1. However, one cannot replace X in (1.7) by B (see Section 3.8 for details). A function f on [a, b] having bounded p-variation for some 1 p< 2 also has the quadratic ≤ λ-variation for each sequence of partitions λ = λ : m 1 Λ[a, b] such that { m ≥ }∈ t (a, b): [f(t+) f(t)][f(t) f(t )] = 0 λ . (1.8) { ∈ − − − 6 }⊂∪m m In this case, the bracket function [f] at t [a, b] is a sum of squared jumps over [a, t] (see λ ∈ Corollary 3.15 below). The converse is not true: given λ Λ[0, 1], almost all sample functions ∈ of a Brownian motion have the quadratic λ-variation on [0, 1], and have the p-variation on [0, 1] unbounded for each 1 p 2. Another related example is provided by a subclass of ≤ ≤ self-affine functions introduced by Kˆono [57] (see Example 3.6 below). An extension of the p- variation calculus to Brownian motion path like functions require a more refined constructions than the Riemann-Stieltjes integral. The quadratic λ-variation and related integral constructions introduced below make the core of our attempts towards building a desirable calculus. 2 Evolution representation problem. The results in this paper are motivated by and applied to an evolution representation problem. To sketch the problem, let B = (B, ) be a Banach k · k algebra with unit 1I. A family U = U(t,s): a s t b B is called an evolution in B if it { ≤ ≤ ≤ }⊂ is multiplicative, that is, U(r, t)U(t,s)= U(r, s) for a s t r b, ≤ ≤ ≤ ≤ (1.9) U(t,t) = 1I for a t b. ( ≤ ≤ The notion of an evolution generalizes the concept of a one-parameter semigroup of bounded linear operators on a Banach space. The classical Hille-Yosida theorem describes any strongly continuous, contractive semigroup in terms of its generator. Analogous pairing results have been established for evolutions under various conditions on the function Ua defined by Ua(t) := U(t, a) for t [a, b]. For example, U is analytic, or almost differentiable, or of bounded variation is ∈ a one set of conditions.
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