A rigorous but gentle introduction for economists

This open access textbook is the first to provide Business and Economics Ph.D. students with a precise and intuitive introduction to the formal backgrounds of modern financial theory. It explains Brownian motion, random processes, measures, and Lebesgue integrals intuitively, but without sacrificing the necessary mathematical formalism, making them accessible for readers with little or no previous knowledge of the field. It also includes mathematical definitions and the hidden stories behind the terms discussing why the theories are presented in specific ways.

Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions

For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter $H>\frac{1}{2}$, it is known that the existing (naive) Euler scheme has the rate of convergence $n^{1-2H}$. Since the limit $H\rightarrow\frac{1}{2}$ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for It\^{o} SDEs for $H=\frac{1}{2}$, the convergence rate of the naive Euler scheme deteriorates for $H\rightarrow\frac{1}{2}$. In this paper we introduce a new (modified Euler) approximation scheme which is closer to the classical Euler scheme for Stratonovich SDEs for $H=\frac{1}{2}$, and it has the rate of convergence $\gamma_n^{-1}$, where $\gamma_n=n^{2H-{1}/2}$ when $H<\frac{3}{4}$, $\gamma_n=n/\sqrt{\log n}$ when $H=\frac{3}{4}$ and $\gamma_n=n$ if $H>\frac{3}{4}$. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if $\{X_t,0\le t\le T\}$ is the solution of a SDE driven by a fBm and if $\{X_t^n,0\le t\le T\}$ is its approximation obtained by the new modified Euler scheme, then we prove that $\gamma_n(X^n-X)$ converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when $H\in(\frac{1}{2},\frac{3}{4}]$. In the case $H>\frac{3}{4}$, we show the $L^p$ convergence of $n(X^n_t-X_t)$, and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme.Comment: Published at http://dx.doi.org/10.1214/15-AAP1114 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Partial Sums of the Least Squares Residuals of Spatial Observations Sampled According to a Probability Measure

A functional central limit theorem for a sequence of partial sums processes of the least squares residuals of a spatial linear regression model in which the observations are sampled according to a probability measure is established. Under mild assumptions to the model, the limit of the sequence of the least squares residual partial sums processes is explicitly derived. It is shown that the limit process which is a function of the Brownian sheet depends on the regression functions and the probability measure under which the design is constructed. Several examples ofthe limit processes when the model is true are presented. Lower and upper bounds for boundary crossing probabilities of signal plus noise models when the noises come from the residual partial sums processes are also investigated.DOI : http://dx.doi.org/10.22342/jims.19.1.127.23-4

The Brownian Motion

This open access textbook is the first to provide Business and Economics Ph.D. students with a precise and intuitive introduction to the formal backgrounds of modern financial theory. It explains Brownian motion, random processes, measures, and Lebesgue integrals intuitively, but without sacrificing the necessary mathematical formalism, making them accessible for readers with little or no previous knowledge of the field. It also includes mathematical definitions and the hidden stories behind the terms discussing why the theories are presented in specific ways

Rough path recursions and diffusion approximations

In this article, we consider diffusion approximations for a general class of stochastic recursions. Such recursions arise as models for population growth, genetics, financial securities, multiplicative time series, numerical schemes and MCMC algorithms. We make no particular probabilistic assumptions on the type of noise appearing in these recursions. Thus, our technique is well suited to recursions where the noise sequence is not a semi-martingale, even though the limiting noise may be. Our main theorem assumes a weak limit theorem on the noise process appearing in the random recursions and lifts it to diffusion approximation for the recursion itself. To achieve this, we approximate the recursion (pathwise) by the solution to a stochastic equation driven by piecewise smooth paths; this can be thought of as a pathwise version of backward error analysis for SDEs. We then identify the limit of this stochastic equation, and hence the original recursion, using tools from rough path theory. We provide several examples of diffusion approximations, both new and old, to illustrate this technique.Comment: Published at http://dx.doi.org/10.1214/15-AAP1096 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Pathwise stochastic control with applications to robust filtering

We study the problem of pathwise stochastic optimal control, where the optimization is performed for each fixed realisation of the driving noise, by phrasing the problem in terms of the optimal control of rough differential equations. We investigate the degeneracy phenomenon induced by directly controlling the coefficient of the noise term, and propose a simple procedure to resolve this degeneracy whilst retaining dynamic programming. As an application, we use pathwise stochastic control in the context of stochastic filtering to construct filters which are robust to parameter uncertainty, demonstrating an original application of rough path theory to statistics

Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion

We study the approximation of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H>1/2$. For the mean-square error at a single point we derive the optimal rate of convergence that can be achieved by any approximation method using an equidistant discretization of the driving fractional Brownian motion. We find that there are mainly two cases: either the solution can be approximated perfectly or the best possible rate of convergence is $n^{-H-1/2},$ where $n$ denotes the number of evaluations of the fractional Brownian motion. In addition, we present an implementable approximation scheme that obtains the optimal rate of convergence in the latter case.Comment: 49 page

Fractional Calculus - Theory and Applications

In recent years, fractional calculus has led to tremendous progress in various areas of science and mathematics. New definitions of fractional derivatives and integrals have been uncovered, extending their classical definitions in various ways. Moreover, rigorous analysis of the functional properties of these new definitions has been an active area of research in mathematical analysis. Systems considering differential equations with fractional-order operators have been investigated thoroughly from analytical and numerical points of view, and potential applications have been proposed for use in sciences and in technology. The purpose of this Special Issue is to serve as a specialized forum for the dissemination of recent progress in the theory of fractional calculus and its potential applications

Complex boundary value problems of nonlinear differential equations: Theory, computational methods, and applications

Editorial to the theme Complex Boundary Value Problems of Nonlinear Differential Equations: Theory, Computational Methods, and Application