Counterexamples to regularities for the derivative processes associated to stochastic evolution equations

In the recent years there has been an increased interest in studying regularity properties of the derivatives of stochastic evolution equations (SEEs) with respect to their initial values. In particular, in the scientific literature it has been shown for every natural number $n\in\mathbb{N}$ that if the nonlinear drift coefficient and the nonlinear diffusion coefficient of the considered SEE are $n$-times continuously Fr\'{e}chet differentiable, then the solution of the considered SEE is also $n$-times continuously Fr\'{e}chet differentiable with respect to its initial value and the corresponding derivative processes satisfy a suitable regularity property in the sense that the $n$-th derivative process can be extended continuously to $n$-linear operators on negative Sobolev-type spaces with regularity parameters $\delta_1,\delta_2,\ldots,\delta_n\in[0,1/2)$ provided that the condition $\sum^n_{i=1} \delta_i < 1/2$ is satisfied. The main contribution of this paper is to reveal that this condition can essentially not be relaxed

Random Bit Multilevel Algorithms for Stochastic Differential Equations

We study the approximation of expectations \E(f(X)) for solutions $X$ of SDEs and functionals $f \colon C([0,1],\R^r) \to \R$ by means of restricted Monte Carlo algorithms that may only use random bits instead of random numbers. We consider the worst case setting for functionals $f$ from the Lipschitz class w.r.t.\ the supremum norm. We construct a random bit multilevel Euler algorithm and establish upper bounds for its error and cost. Furthermore, we derive matching lower bounds, up to a logarithmic factor, that are valid for all random bit Monte Carlo algorithms, and we show that, for the given quadrature problem, random bit Monte Carlo algorithms are at least almost as powerful as general randomized algorithms

Random Bit Quadrature and Approximation of Distributions on Hilbert Spaces

We study the approximation of expectations \E(f(X)) for Gaussian random elements $X$ with values in a separable Hilbert space $H$ and Lipschitz continuous functionals $f \colon H \to \R$. We consider restricted Monte Carlo algorithms, which may only use random bits instead of random numbers. We determine the asymptotics (in some cases sharp up to multiplicative constants, in the other cases sharp up to logarithmic factors) of the corresponding $n$-th minimal error in terms of the decay of the eigenvalues of the covariance operator of $X$. It turns out that, within the margins from above, restricted Monte Carlo algorithms are not inferior to arbitrary Monte Carlo algorithms, and suitable random bit multilevel algorithms are optimal. The analysis of this problem leads to a variant of the quantization problem, namely, the optimal approximation of probability measures on $H$ by uniform distributions supported by a given, finite number of points. We determine the asymptotics (up to multiplicative constants) of the error of the best approximation for the one-dimensional standard normal distribution, for Gaussian measures as above, and for scalar autonomous SDEs

On arbitrarily slow convergence rates for strong numerical approximations of Cox–Ingersoll–Ross processes and squared Bessel processes

ISSN:0949-2984ISSN:1432-112

Evaluating balance, stability and gait symmetry of stroke patients using instrumented gait analysis techniques

Stroke is one of the most common brain disorders and has disastrous effects on the walking function of the patients. A post-stroke gait is characterized by asymmetry, instability and loss of balance. In this context, the evaluation of the gait of stroke patients can be performed though the quantification of balance, stability, and gait symmetry, which is the topic of the present study. The gait of 14 control subjects and 55 post stroke patients was measured in a gait laboratory. Gait features were extracted, and symmetry indexes obtained in the literature were computed. A statistical analysis was carried out to understand the relationship between gait features and medical scales used to evaluate the degree of impairment of the patients (traditional scores and a newly developed functional score termed the ReHabX score). The analysis took the form of a correlation matrix where every single item was correlated with the remaining ones. The results show that stroke patients tend to spare the affected limb and, the more impaired they are, the more the spatio-temporal features deviate from the wellestablished “normal values”. Also, the best indexes to calculate the degree of symmetry are the ratio I and the Limp Index. Also, the gait features of the non-affected side correlate better with the other gait features, the symmetry indexes, the conventional clinical scales and the ReHabX Score. Furthermore, it was concluded that symmetry, balance and stability are closely related, and that the ReHabX Score is the most suitable medical scale to evaluate the gait disorders as well as balance, stability and gait symmetry