Moderate deviations for Euler-Maruyama approximation of Hull-White stochastic volatility model

We consider the Euler-Maruyama discretization of stochastic volatility model dS_t = σ_tS_tdW_t, dσ_t = ωσ_tdZ_t, t ∈ [0, T], which has been widely used in financial practice, where W_t,Z_t, t ∈ [0, T], are two uncorrelated standard Brownian motions. Using asymptotic analysis techniques, the moderate deviation principles for log S_n (or log |S_n| in case S_n is negative) are obtained as n → ∞ under different discretization schemes for the asset price process S_t and the volatility process σ_t. Numerical simulations are presented to compare the convergence speeds in different schemes.     

Canonical ensemble of particles in a self-avoiding random walk

We consider an ensemble of particles not interacting with each other and randomly walking in the d-dimensional Euclidean space ? ^d . The individual moves of each particle are governed by the same distribution, but after the completion of each such move of a particle, its position in the medium is “marked” as a region in the form of a ball of diameter r ₀, which is not available for subsequent visits by this particle. As a result, we obtain the corresponding ensemble in ? ^d of marked trajectories in each of which the distance between the centers of any pair of these balls is greater than r ₀. We describe a method for computing the asymptotic form of the probability density W _n (r) of the distance r between the centers of the initial and final balls of a trajectory consisting of n individual moves of a particle of the ensemble. The number n, the trajectory modulus, is a random variable in this model in addition to the distance r. This makes it necessary to determine the distribution of n, for which we use the canonical distribution obtained from the most probable distribution of particles in the ensemble over the moduli of their trajectories. Averaging the density W _n (r) over the canonical distribution of the modulus n allows finding the asymptotic behavior of the probability density of the distance r between the ends of the paths of the canonical ensemble of particles in a self-avoiding random walk in ? ^d for 2 ≤ d < 4.     

Limit theorems for inverse process <Emphasis Type="Italic">T</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> of Hawkes process

We consider linear Hawkes process N _t and its inverse process T _n . The limit theorems for N _t are well known and studied by many authors. In this paper, we study the limit theorems for T _n . In particular, we investigate the law of large numbers, the central limit theorem and the large deviation principle for T _n . The main tool of the proof is based on immigration-birth representation and the observations on the relation between N _t and T _n .     

Limiting law results for a class of conditional mode estimates for functional stationary ergodic data

The main purpose of the present work is to establish the functional asymptotic normality of a class of kernel conditional mode estimates when functional stationary ergodic data are considered. More precisely, consider a random variable (X,Z) taking values in some semi-metric abstract space E × F. For a real function φ defined on F and for each x ∈ E, we consider the conditional mode, say ?_φ(x), of the real random variable φ(Z) given the event “X = x”. While estimating the conditional mode function by Θ?_φ,n(x), using the kernel-type estimator, we establish the limiting law of the family of processes {Θ?_φ(x) - Θ_φ(x)} (suitably normalized) over Vapnik–Chervonenkis class C of functions φ. Beyond ergodicity, no other assumption is imposed on the data. This paper extends the scope of some previous results established under mixing condition for a fixed function φ. From this result, the asymptotic normality of a class of predictors is derived and confidence bands are constructed. Finally, a general notion of bootstrapped conditional mode constructed by exchangeably weighting samples is presented. The usefulness of this result will be illustrated in the construction of confidence bands.     

Automorphisms of Coxeter groups of type <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

A Coxeter system (W, S) is said to be of type K _n if the associated Coxeter graph Γ_S is complete on n vertices and has only odd edge labels. If W satisfies either of: (1) n = 3; (2) W is rigid; then the automorphism group of W is generated by the inner automorphisms of W and any automorphisms induced by Γ_S. Indeed, Aut(W) is the semidirect product of Inn(W) and the group of diagram automorphisms, and furthermore W is strongly rigid. We also show that if W is a Coxeter group of type K _n then W has exactly one conjugacy class of involutions and hence Aut(W) = Spec(W).     

On Lipschitz extension from finite subsets

We prove that for every n ∈ ? there exists a metric space (X, d _X), an n-point subset S ? X, a Banach space (Z, \({\left\| \right\|_Z}\)) and a 1-Lipschitz function f: S → Z such that the Lipschitz constant of every function F: X → Z that extends f is at least a constant multiple of \(\sqrt {\log n} \). This improves a bound of Johnson and Lindenstrauss [JL84]. We also obtain the following quantitative counterpart to a classical extension theorem of Minty [Min70]. For every α ∈ (1/2, 1] and n ∈ ? there exists a metric space (X, d _X), an n-point subset S ? X and a function f: S → ?₂ that is α-Hölder with constant 1, yet the α-Hölder constant of any F: X → ?₂ that extends f satisfies \({\left\| F \right\|_{Lip\left( \alpha \right)}} > {\left( {\log n} \right)^{\frac{{2\alpha - 1}}{{4\alpha }}}} + {\left( {\frac{{\log n}}{{\log \log n}}} \right)^{{\alpha ^2} - \frac{1}{2}}}\). We formulate a conjecture whose positive solution would strengthen Ball’s nonlinear Maurey extension theorem [Bal92], serving as a far-reaching nonlinear version of a theorem of König, Retherford and Tomczak-Jaegermann [KRTJ80]. We explain how this conjecture would imply as special cases answers to longstanding open questions of Johnson and Lindenstrauss [JL84] and Kalton [Kal04].     

Estimation of the depth of reversible circuits consisting of NOT,CNOT and 2-CNOT gates

The paper discusses the asymptotic depth of a reversible circuits consisting of NOT, CNOT and 2-CNOT gates. The reversible circuit depth function D(n, q) is introduced for a circuit implementing a mapping f: Z₂ⁿ → Z₂ⁿ as a function of n and the number q of additional inputs. It is proved that for the case of implementation of a permutation from A(Z₂ⁿ) with a reversible circuit having no additional inputs the depth is bounded as D(n, 0) ? 2ⁿ/(3log₂n). It is also proved that for the case of transformation f: Z₂ⁿ → Z₂ⁿ with a reversible circuit having q₀ ~ 2ⁿ additional inputs the depth is bounded as D(n,q₀) ? 3n.     

Finite Groups Whose <Emphasis Type="Italic">n</Emphasis>-Maximal Subgroups Are Modular

Let G be a finite group. If M_n< M_n?1< · · · < M₁< M₀ = G with Mi a maximal subgroup of M_i?1 for all i = 1,..., n, then M_n (n > 0) is an n-maximal subgroup of G. A subgroup M of G is called modular provided that (i) 〈X,M ∩ Z〉 = 〈X,M〉 ∩ Z for all X ≤ G and Z ≤ G such that X ≤ Z, and (ii) 〈M,Y ∩ Z〉 = 〈M,Y 〉 ∩ Z for all Y ≤ G and Z ≤ G such that M ≤ Z. In this paper, we study finite groups whose n-maximal subgroups are modular.     

Words with minimal growth function

Words of slow growth, which are a natural generalization of Sturmian words, are studied in the paper. An infinite word W = (w _n ) _n∈? is called a word of slow growth if its complexity function satisfies the relation T _W (n + 1) ? T _W (n) = 1 for all sufficiently large n. The aim of this paper is to describe all words of slow growth in terms of rotations of a circle.     

Isometries between JB*-triples

Let Z and W be JB*-triples and let T be a linear isometry from Z into W. For any z ∈ Z with ||z||<1, we show that Open image in new window if the Möbius transform induced by T(z) preserves the unit ball of T(Z). We show further that T is, locally, a triple homomorphism via a tripotent: for any z ∈ Z, there is a tripotent u in W** such that Open image in new window for all a, b, c in the smallest subtriple Z _z of Z containing z, and also, {u,T(·),u}:Z _z →W** is an isometry.     

Harmonic analysis associated with a discrete Laplacian

It is well known that the fundamental solution of

$${u_t}\left( {n,t} \right) = u\left( {n + 1,t} \right) - 2u\left( {n,t} \right) + u\left( {n - 1,t} \right),n \in \mathbb{Z},$$

with u(n, 0) = δ _nm for every fixed m ∈ Z is given by u(n, t) = e ^?2t I _n?m (2t), where I _k (t) is the Bessel function of imaginary argument. In other words, the heat semigroup of the discrete Laplacian is described by the formal series W _t f(n) = Σ _m∈Z e ^?2t I _n?m (2t)f(m). This formula allows us to analyze some operators associated with the discrete Laplacian using semigroup theory. In particular, we obtain the maximum principle for the discrete fractional Laplacian, weighted ? ^p (Z)-boundedness of conjugate harmonic functions, Riesz transforms and square functions of Littlewood-Paley. We also show that the Riesz transforms essentially coincide with the so-called discrete Hilbert transform defined by D. Hilbert at the beginning of the twentieth century. We also see that these Riesz transforms are limits of the conjugate harmonic functions. The results rely on a careful use of several properties of Bessel functions.

     

On the summability factors of infinite series

The author has established that if [λ_n] is a convex sequence such that the series Σn ^-1λ_n is convergent and the sequence {K _n} satisfies the condition |K _n|=O[log(n+1)]^k(C, 1),k?0, whereK _n denotes the (R, logn, 1) mean of the sequence {n log (n+1)a _n}, then the series Σlog(n+1)^1-kλ_n a _n is summable |R, logn, 1|. The result obtained for the particular casek=0 generalises a previous result of the author [1].     

A Classification of Spectrum-determined Circulant Digraphs

Some researchers have proved that ádám’s conjecture is wrong. However, under special conditions, it is right. Let Z_n be a cyclic group of order n and C_n(S) be the circulant digraph of Z_n with respect to S ? Z_n\{0}. In the literature, some people have used a spectral method to solve the isomorphism for the circulants of prime-power order. In this paper, we also use the spectral method to characterize the circulants of order p^aq^bw^c(where p, q and w are all distinct primes), and to make ádám’s conjecture right.     

Rectangular <Emphasis Type="Italic">R</Emphasis>-Transform as the Limit of Rectangular Spherical Integrals

In this paper, we connect rectangular free probability theory and spherical integrals. We prove the analogue, for rectangular or square non-Hermitian matrices, of a result that Guionnet and Maïda proved for Hermitian matrices in (J. Funct. Anal. 222(2):435–490, 2005). More specifically, we study the limit, as n and m tend to infinity, of \(\frac{1}{n}\log\mathbb{E}\{\exp[\sqrt{nm}\theta X_{n}]\}\), where θ∈?, X _n is the real part of an entry of U _n M _n V _m and M _n   is a certain n×m deterministic matrix and U _n and V _m are independent Haar-distributed orthogonal or unitary matrices with respective sizes n×n and m×m. We prove that when the singular law of M _n converges to a probability measure μ, for θ small enough, this limit actually exists and can be expressed with the rectangular R-transform of μ. This gives an interpretation of this transform, which linearizes the rectangular free convolution, as the limit of a sequence of log-Laplace transforms.     

Primes in tuples II

We prove that

$ \mathop{ \lim \inf}\limits_{n \rightarrow \infty} \frac{p_{n+1}-p_{n}}{\sqrt{\log p_{n}} \left(\log \log p_{n}\right)^{2}}< \infty, $

where p _n denotes the nth prime. Since on average p _n+1?p _n is asymptotically log _n , this shows that we can always find pairs of primes much closer together than the average. We actually prove a more general result concerning the set of values taken on by the differences p?p′ between primes which includes the small gap result above.

     

On the total multiplicity of zeros of generalized polynomials related to nonoscillating trajectories

For a continuous curve L = {x: x = Z(t), t ∈ [a, b]} in R ⁿ , we study the number of zeros of the function l _h (t) = 〈h, Z(t)〉, where h ∈ R ⁿ . We introduce the notion of multiple zeros for such functions and study the possibility of estimating the total multiplicity of such zeros under the assumption that the system {z ₁(t), z ₂(t), …, z _n (t)} of coordinates of the function Z(t) is a Chebyshev system on [a, b].     

Relationships between generalized Wiener integrals and conditional analytic Feynman integrals over continuous paths

Let C[0, t] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0, t], and define a random vector Z _n: C[0, t] → R ⁿ⁺¹ by \({Z_n}\left( x \right) = \left( {x\left( 0 \right) + a\left( 0 \right),\int_o^{{t_1}} {h\left( s \right)dx\left( s \right) + x\left( 0 \right) + a\left( {{t_1}} \right),...,\int_0^{{t_n}} {h\left( s \right)dx\left( s \right) + x\left( 0 \right) + a\left( {{t_n}} \right)} } } \right)\), where a ∈ C[0, t], h ∈ L ₂[0, t], and 0 < t ₁ <... < t _n ≤ t is a partition of [0, t]. Using simple formulas for generalized conditional Wiener integrals, given Z _n we will evaluate the generalized analytic conditional Wiener and Feynman integrals of the functions F in a Banach algebra which corresponds to Cameron-Storvick’s Banach algebra S. Finally, we express the generalized analytic conditional Feynman integral of F as a limit of the non-conditional generalized Wiener integral of a polygonal function using a change of scale transformation for which a normal density is the kernel. This result extends the existing change of scale formulas on the classical Wiener space, abstract Wiener space and the analogue of the Wiener space C[0, t].     

On the boundedness of generalized solutions of higher-order nonlinear elliptic equations with data from an Orlicz–Zygmund class

In the present paper, a 2mth-order quasilinear divergence equation is considered under the condition that its coefficients satisfy the Carathéodory condition and the standard conditions of growth and coercivity in the Sobolev space W^m,p(Ω), Ω ? Rⁿ, p > 1. It is proved that an arbitrary generalized (in the sense of distributions) solution u ∈ W₀^m,p (Ω) of this equation is bounded if m ≥ 2, n = mp, and the right-hand side of this equation belongs to the Orlicz–Zygmund space L(log L)^n?1(Ω).     

Asymptotics for the Euler-Discretized Hull-White Stochastic Volatility Model

We consider the stochastic volatility model d S _t = σ _t S _t d W _t ,d σ _t = ω σ _t d Z _t , with (W _t ,Z _t ) uncorrelated standard Brownian motions. This is a special case of the Hull-White and the β=1 (log-normal) SABR model, which are widely used in financial practice. We study the properties of this model, discretized in time under several applications of the Euler-Maruyama scheme, and point out that the resulting model has certain properties which are different from those of the continuous time model. We study the asymptotics of the time-discretized model in the n→∞ limit of a very large number of time steps of size τ, at fixed \(\beta =\frac 12\omega ^{2}\tau n^{2}\) and \(\rho ={\sigma _{0}^{2}}\tau \), and derive three results: i) almost sure limits, ii) fluctuation results, and iii) explicit expressions for growth rates (Lyapunov exponents) of the positive integer moments of S _t . Under the Euler-Maruyama discretization for (S _t ,logσ _t ), the Lyapunov exponents have a phase transition, which appears in numerical simulations of the model as a numerical explosion of the asset price moments. We derive criteria for the appearance of these explosions.     

Theorems on Large Deviations for Randomly Indexed Sum of Weighted Random Variables

In this paper, we consider a random variable \(Z_{t}=\sum_{i=1}^{N_{t}}a_{i}X_{i}\), where \(X, X_{1}, X_{2}, \ldots\) are independent identically distributed random variables with mean E X=μ and variance D X=σ ²>0. It is assumed that Z ₀=0, 0≤a _i <∞, and N _t , t≥0 is a non-negative integer-valued random variable independent of X _i , i=1,2,…?. The paper is devoted to the analysis of accuracy of the standard normal approximation to the sum \(\tilde{Z}_{t}=(\mathbf{D}Z_{t})^{-1/2}(Z_{t}-\mathbf{E}Z_{t})\), large deviation theorems in the Cramer and power Linnik zones, and exponential inequalities for \(\mathbf{P}(\tilde{Z}_{t}\geq x)\).