Rough paths techniques give the ability to define solutions of stochastic differential equations driven by signals $X$ which are not semimartingales and whose $p$-variation is finite only for large values of $p$. In this context, rough integrals are
usually Riemann-Stieltjes integrals with correction terms that are sometimes seen as unnatural. As opposed to those somewhat artificial correction terms, our endeavor in this note is to produce a trapezoid rule for rough integrals driven by general $d$-dimensional Gaussian processes. Namely we shall approximate a generic rough integral $int y , dX$ by Riemann sums avoiding the usual higher order correction terms, making the expression easier to work with and more natural. Our approximations apply to all controlled processes $y$ and to a wide range of Gaussian processes $X$ including fractional Brownian motion with a Hurst parameter $H>1/4$. As a corollary of the trapezoid rule, we also consider the convergence of a midpoint rule for integrals of the form $int f(X) dX$.
In this paper, we consider an anticipative nonlinear filtering problem, in which the observation noise is correlated with the past of the signal. This new signal-observation model has its applications in both finance models with insider trading and i
n engineering. We derive a new equation for the filter in this context, analyzing both the nonlinear and the linear cases. We also handle the case of a finite filter with Volterra type observation. The performance of our algorithm is presented through numerical experiments.
In this note we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem
invoking two different methods, respectively based on variance computations and on path-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.
We study the Crank-Nicolson scheme for stochastic differential equations (SDEs) driven by multidimensional fractional Brownian motion $(B^{1}, dots, B^{m})$ with Hurst parameter $H in (frac 12,1)$. It is well-known that for ordinary differential equa
tions with proper conditions on the regularity of the coefficients, the Crank-Nicolson scheme achieves a convergence rate of $n^{-2}$, regardless of the dimension. In this paper we show that, due to the interactions between the driving processes $ B^{1}, dots, B^{m} $, the corresponding Crank-Nicolson scheme for $m$-dimensional SDEs has a slower rate than for the one-dimensional SDEs. Precisely, we shall prove that when $m=1$ and when the drift term is zero, the Crank-Nicolson scheme achieves the exact convergence rate $n^{-2H}$, while in the case $m=1$ and the drift term is non-zero, the exact rate turns out to be $n^{-frac12 -H}$. In the general case when $m>1$, the exact rate equals $n^{frac12 -2H}$. In all these cases the limiting distribution of the leading error is proved to satisfy some linear SDE driven by Brownian motions independent of the given fractional Brownian motions.
In this article, we consider limit theorems for some weighted type random sums (or discrete rough integrals). We introduce a general transfer principle from limit theorems for unweighted sums to limit theorems for weighted sums via rough path techniq
ues. As a by-product, we provide a natural explanation of the various new asymptotic behaviors in contrast with the classical unweighted random sum case. We apply our principle to derive some weighted type Breuer-Major theorems, which generalize previous results to random sums that do not have to be in a finite sum of chaos. In this context, a Breuer-Major type criterion in notion of Hermite rank is obtained. We also consider some applications to realized power variations and to Itos formulas in law. In the end, we study the asymptotic behavior of weighted quadratic variations for some multi-dimensional Gaussian processes.
In this article, we consider the so-called modified Euler scheme for stochastic differential equations (SDEs) driven by fractional Brownian motions (fBm) with Hurst parameter $frac13<H<frac12$. This is a first-order time-discrete numerical approximat
ion scheme, and has been recently introduced by Hu, Liu and Nualart in order to generalize the classical Euler scheme for It^o SDEs to the case $H>frac12$. The current contribution generalizes the modified Euler scheme to the rough case $frac13<H<frac12$. Namely, we show a convergence rate of order $n^{frac12-2H}$ for the scheme, and we argue that this rate is exact. We also derive a central limit theorem for the renormalized error of the scheme, thanks to some new techniques for asymptotics of weighted random sums. Our main idea is based on the following observation: the triple of processes obtained by considering the fBm, the scheme process and the normalized error process, can be lifted to a new rough path. In addition, the Holder norm of this new rough path has an estimate which is independent of the step-size of the scheme.
In this paper, we study two variations of the time discrete Taylor schemes for rough differential equations and for stochastic differential equations driven by fractional Brownian motions. One is the incomplete Taylor scheme which excludes some terms
of an Taylor scheme in its recursive computation so as to reduce the computation time. The other one is to add some deterministic terms to an incomplete Taylor scheme to improve the mean rate of convergence. Almost sure rate of convergence and $L_p$-rate of convergence are obtained for the incomplete Taylor schemes. Almost sure rate is expressed in terms of the Holder exponents of the driving signals and the $L_p$-rate is expressed by the Hurst parameters. Both rates involves with the incomplete Taylor scheme in a very explicit way and then provide us with the best incomplete schemes, depending on that one needs the almost sure convergence or one needs $L_p$-convergence. As in the smooth case, general Taylor schemes are always complicated to deal with. The incomplete Taylor scheme is even more sophisticated to analyze. A new feature of our approach is the explicit expression of the error functions which will be easier to study. Estimates for multiple integrals and formulas for the iterated vector fields are obtained to analyze the error functions and then to obtain the rates of convergence.
Neuron is the most important building block in our brain, and information processing in individual neuron involves the transformation of input synaptic spike trains into an appropriate output spike train. Hardware implementation of neuron by individu
al ionic/electronic hybrid device is of great significance for enhancing our understanding of the brain and solving sensory processing and complex recognition tasks. Here, we provide a proof-of-principle artificial neuron based on a proton conducting graphene oxide (GO) coupled oxide-based electric-double-layer (EDL) transistor with multiple driving inputs and one modulatory input terminal. Paired-pulse facilitation, dendritic integration and orientation tuning were successfully emulated. Additionally, neuronal gain control (arithmetic) in the scheme of rate coding is also experimentally demonstrated. Our results provide a new-concept approach for building brain-inspired cognitive systems.
We consider a stochastic differential equation with additive fractional noise with Hurst parameter $H>1/2$, and a non-linear drift depending on an unknown parameter. We show the Local Asymptotic Normality property (LAN) of this parametric model with
rate $sqrt{tau}$ as $taurightarrow infty$, when the solution is observed continuously on the time interval $[0,tau]$. The proof uses ergodic properties of the equation and a Girsanov-type transform. We analyse the particular case of the fractional Ornstein-Uhlenbeck process and show that the Maximum Likelihood Estimator is asymptotically efficient in the sense of the Minimax Theorem.
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 $Hrightarrowfrac{1
}{2}$ 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 $Hrightarrowfrac{1}{2}$. 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 ${X_t,0le tle T}$ is the solution of a SDE driven by a fBm and if ${X_t^n,0le tle T}$ 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 $Hin(frac{1}{2},frac{3}{4}]$. 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.
