Insider trading equilibrium in a market with memory

We consider the Kyle-Back model for insider trading, with the difference that the classical Brownian motion noise of the noise traders is replaced by the noise of a fractional Brownian motion B ^H with Hurst parameter ${H>\frac{1}{2}}$ (when ${H=\frac{1}{2}, B^H}$ coincides with the classical Brownian motion). Heuristically, for ${H>\frac{1}{2}}$ this means that the noise traders has some ??memory??, in the sense that any increment from time t on has a positive correlation with its value at t. (In other words, the noise trading is a persistent stochastic process). It also means that the paths of the noise trading process are more egular than in the classical Brownian motion case. We obtain an equation for the optimal (relative) trading intensity for the insider in this setting, and we show that when ${H\rightarrow\frac{1}{2}}$ the solution converges to the solution in the classical case. Finally, we discuss how the size of the Hurst coefficient H influences the optimal performance and portfolio of the insider.     

多重分数Stratonovich积分的另一种构造(英文)

本文研究了当Hurst参数日小于1/2时关于分数布朗运动的随机积分问题.利用分数布朗运动的性质和卷积逼近的方法,获得了多重分数Stratonovich积分的另一种构造.     

Smoothness of densities for area-like processes of fractional Brownian motion

We consider a process given by a n-dimensional fractional Brownian motion with Hurst parameter ${\frac{1}{4} < H < \frac{1}{2}}$ , along with an associated Lévy area-like process, and prove the smoothness of the density for this process with respect to Lebesgue measure.     

On the Weighted Local Time and the Tanaka Formula for the Multidimensional Fractional Brownian Motion

Abstract

We determine the weighted local time for the multidimensional fractional Brownian motion from the occupation time formula. We also discuss on the Itô and Tanaka formula for the multidimensional fractional Brownian motion. In these formulas the Skorohod integral is applicable if the Hurst parameter of fractional Brownian motion is greater than 1/2. If the Hurst parameter is less than 1/2, then we use the Skorohod type integral introduced by Nualart and Zakai for the stochastic integral and establish the Itô and Tanaka formulas.     

The \frac{4}{3}-Variation of the Derivative of the Self-intersection Brownian Local Time and Related Processes

In this paper, we compute the \(\frac{4}{3}\) -variation of the derivative of the self-intersection Brownian local time by applying techniques from the theory of fractional martingales (Hu et al. in Ann Probab 37:2404–2430, 2009).     

On the Expectations of Multiple Stratonovich Integrals

Abstract

In the construction of numerical methods for solving stochastic differential equations it becomes necessary to calculate the expectations of products of multiple stochastic integrals. In the Itô case, explicit formulae for the expectation of a multiple integral with integrand identically equal to 1 and for the product of two such integrals are known. In this paper formulae for the expectation of any multiple Stratonovich integral as well as for the product of a broad class of two Stratonovich integrals have been derived.     

Wiener–Itô Theorem in Terms of Wick Tensors

Wick tensors are used to describe homogeneous chaos and to define multiple Wiener integrals. The Wiener–Itô decomposition is expressed by the formula $$\varphi (x) = \sum\limits_{n = 0}^\infty {\frac{1}{{n!}}\int_B {[D^n (\mu \varphi )(0)]^ \sim}} (x + iy,...,x + iy)d\mu (y)$$ . We use this formula to interpret Hida's original idea of generalized multiple Wiener integrals as generalized functions acting on the space of test functions.     

多重分数斯特拉托诺维奇积分的一个注记

本文研究了多重分数斯特拉托诺维奇积分,通过卷积逼近技巧和分数布朗运动的随机积分的性质,构造了当Hurst参数小于二分之一时的多重随机积分.这种方法是新的不同于文[8]中的构造方法.     

Numerical Solution Based on Hat Functions for Solving Nonlinear Stochastic Itô Volterra Integral Equations Driven by Fractional Brownian Motion

This paper presents a numerical method for solving nonlinear stochastic Itô Volterra integral equations driven by fractional Brownian motion with Hurst parameter \( H \in (0,1)\) via of hat functions. Using properties of the generalized hat basis functions and fractional Brownian motion, new stochastic operational matrix of integration is achieved and the nonlinear stochastic equation is transformed into nonlinear system of algebraic equations which by solving it, an approximation solution with high accuracy is obtained. In addition, error analysis of the method is investigated, and by some examples, efficiency and accuracy of the suggested method are shown.     

On the Shape of the Ground State Eigenfunction for Stable Processes

We prove that the ground-state eigenfunction for symmetric stable processes of order α∈(0,2) killed upon leaving the interval (?1,1) is concave on $(-\frac{1}{2},\frac{1}{2})We prove that the ground-state eigenfunction for symmetric stable processes of order α∈(0,2) killed upon leaving the interval (−1,1) is concave on . We call this property “mid-concavity”. A similar statement holds for rectangles in ℝ^d, d>1. These result follow from similar results for finite-dimensional distributions of Brownian motion and subordination. Mathematics Subject Classification (2000) 30C45. Rodrigo Ba?uelos: R. Ba?uelos was supported in part by NSF grant # 9700585-DMS. Tadeusz Kulczycki: T. Kulczycki was supported by KBN grant 2 P03A 041 22 and RTN Harmonic Analysis and Related Problems, contract HPRN-CT-2001-00273-HARP.     

Multivariate central limit theorems for averages of fractional Volterra processes and applications to parameter estimation

The purpose of this paper is to establish the multivariate normal convergence for the average of certain Volterra processes constructed from a fractional Brownian motion with Hurst parameter \(H > \frac{1}{2}\). Some applications to parameter estimation are then discussed.     

Stratonovich’s signatures of Brownian motion determine Brownian sample paths

The signature of Brownian motion in $\mathbb R ^{d}$ over a running time interval $[0,T]$ is the collection of all iterated Stratonovich path integrals along the Brownian motion. We show that, in dimension $d\ge 2$ , almost all Brownian motion sample paths (running up to time $T$ ) are determined by their signature over $[0,T]$ .     

Delay Equations with Non-negativity Constraints Driven by a Hölder Continuous Function of Order \beta \in \left(\frac13,\frac12\right)

In this note we prove an existence and uniqueness result of solution for multidimensional delay differential equations with normal reflection and driven by a Hölder continuous function of order \(\beta \in (\frac13,\frac12)\) . We also obtain a bound for the supremum norm of this solution. As an application, we get these results for stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H \(\in (\frac13,\frac12)\) .     

On Double Stratonovich Fractional Integrals and Some Strong and Weak Approximations

Abstract

Double Stratonovich integrals with respect to the odd part and even part of the fractional Brownian motion are constructed. The first and the second moments of such integrals are explicitly identified. As application of double Stratonovich integrals a strong law of large numbers for efBm and ofBm is derived.

Riemann–Stieltjes integral approximations to double Stratonovich fractional integrals are also considered. The strong convergence (almost surely and mean square) is obtained for approximations based on explicit series expansions of the fractional Brownian processes. The weak convergence is derived for approximations by processes with absolutely continuous paths which converge weakly to the considered fractional Brownian processes. The above-mentioned convergences are obtained for deterministic integrands which are given by bimeasures.     

On a Stochastic Fourier Coefficient: Case of Noncausal Functions

Given a random function $f(t,\omega )$ and an orthonormal basis $\{\varphi _n \}$ in $L^2(0,1),$ we are concerned with the basic question whether the function can be reconstructed from the complete set of its stochastic Fourier coefficients $\{{\hat{f}}_n(\omega )\}$ which are defined by the following stochastic integral with respect to the Brownian motion $W.$ : ${\hat{f}}_n(\omega ):=\int _0^1 f(t,\omega ) \overline{\varphi _n(t)}{\text{ d}}_*W_t$ , where the symbol $\int {\text{ d}}_*W_t$ stands for the stochastic integral of noncausal type. In an earlier article (Stochastics, doi: 10.1080/17442508.2011.651621, 2012), Ogawa studied the question in the limited framework of homogeneous chaos and gave some affirmative answers when the random functions are causal and square integrable Wiener functionals for which the Itô integral is used for the definition of the stochastic Fourier coefficient. In this note, we aim to extend those results to the more general case where the functions are free from the causality restriction and the Skorokhod integral is employed instead of the Itô integral.     

Two families of approximations for the gamma function

In this paper, we establish two families of approximations for the gamma function: $$ \begin{array}{lll} {\varGamma}(x+1)&=\sqrt{2\pi x}{\left({\frac{x+a}{{\mathrm{e}}}}\right)}^x {\left({\frac{x+a}{x-a}}\right)}^{-\frac{x}{2}+\frac{1}{4}} {\left({\frac{x+b}{x-b}}\right)}^{\sum\limits_{k=0}^m\frac{{\beta}_k}{x^{2k}}+O{{\left(\frac{1}{x^{2m+2}}\right)}}},\\ {\varGamma}(x+1)&=\sqrt{2\pi x}\cdot(x+a)^{\frac{x}{2}+\frac{1}{4}}(x-a)^{\frac{x}{2}-\frac{1}{4}} {\left({\frac{x-1}{x+1}}\right)}^{\frac{x^2}{2}}\\ &\quad\times {\left({\frac{x-c}{x+c}}\right)}^{\sum\limits_{k=0}^m\frac{{\gamma}_k}{x^{2k}}+O{\left({\frac{1}{x^{2m+2}}}\right)}}, \end{array}$$ where the constants ${\beta }_k$ and ${\gamma }_k$ can be determined by recurrences, and $a$ , $b$ , $c$ are parameters. Numerical comparison shows that our results are more accurate than Stieltjes, Luschny and Nemes’ formulae, which, to our knowledge, are better than other approximations in the literature.     

Convergence rate of multiple fractional Stratonovich type integral for Hurst parameter less than 1/2

In this paper, we have investigated the problem of the convergence rate of the multiple integralwhere f ∈ Cn+1([0, T ]n) is a given function, π is a partition of the interval [0, T ] and {BtHi ,π} is a family of interpolation approximation of fractional Brownian motion BtH with Hurst parameter H < 1/2. The limit process is the multiple Stratonovich integral of the function f . In view of known results, the convergence rate is different for different multiplicity n. Under some mild conditions, we obtain that the uniform convergence rate is 2H in the mean square sense, where is the norm of the partition generating the approximations.     

Stochastic heat equation driven by fractional noise and local time

The aim of this paper is to study the d-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and has the covariance of a fractional Brownian motion with Hurst parameter H ∈ (0,1) in time. Two types of equations are considered. First we consider the equation in the Itô-Skorohod sense, and later in the Stratonovich sense. An explicit chaos expansion for the solution is obtained. On the other hand, the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the Brownian motion.     

On the infimum attained by the reflected fractional Brownian motion

Let {B _H (t):t≥0} be a fractional Brownian motion with Hurst parameter \(H\in (\frac {1}{2},1)\) . For the storage process \(Q_{B_{H}}(t)=\sup _{-\infty \le s\le t}\) \(\left (B_{H}(t)-B_{H}(s)-c(t-s)\right )\) we show that, for any T(u)>0 such that \(T(u)=o(u^{\frac {2H-1}{H}})\) , $$\mathbb P (\inf_{s\in[0,T(u)]} Q_{B_{H}}(s)>u)\sim\mathbb P(Q_{B_{H}}(0)>u),$$ as \(u\to \infty \) . This finding, known in the literature as the strong Piterbarg property, goes in line with previously observed properties of storage processes with self-similar and infinitely divisible input without Gaussian component.