We discuss the problem of the regularity in time of the map t 7→ Tt ∈ L(R,R;σ) where Tt is a transport map (optimal or not) from a reference measure σ to a measure μt which lies along an absolutely continuous curve t 7→ μt in the space (Pp(R),Wp). We prove that in most cases such a map is no more than 1 p -Hölder continuous.

In this paper, a simple and intuitive technique for analyzing clock jitter effect on bandpass Continuous-Time SigmaDelta (Σ∆) modulators is introduced. The power spectral density of the jitter noise for different feedback DAC shapes are derived and compared. It is shown that DAC output signal shapes used to reduce clock jitter sensitivity in lowpass Continuous-Time Σ∆ modulators may not be suit...

1. When the asset pays continuous dividend yield at the rate q, the expected rate of return of the asset is r − q under the risk neutral measure (see Chap 3 of Kwok's text for justification). Under the continuous Geometric Brownian process model, the logarithm of the asset price ratio over ∆t interval is normally distributed with mean r − q − σ 2 2 ∆t and variance σ 2 ∆t. Accordingly, the mean ...

We show that the displacement and strain formulations of the displacement-traction problem of three-dimensional linearized elasticity can be viewed as Legendre-Fenchel dual problems to the stress formulation of the same problem. We also show that each corresponding Lagrangian has a saddle-point, thus fully justifying this new approach to elasticity by means of Legendre-Fenchel duality. Résumé D...

Consider a continuous time, finite state Markov chain X = {Xt, t ∈ [0, T ]}. We identify the states of this process with the unit vectors ei in R N , where N is the number of states of the chain. We consider stochastic processes defined on the filtered probability space (Ω, F , {Ft}, P), where {Ft} is the completed natural filtration generated by the σ-fields Ft = σ({Xu, u ≤ t}, F ∈ FT : P(F ) ...

Let σ be a positive number and Cσ := C([−σ, 0], R) the Banach space of continuous functions from [−σ, 0] into R and let T (t) be the operator from C([−σ, T ], R) into Cσ, defined by (T (t)x)(s) := x(t + s), s ∈ [−σ, 0]. We prove the existence of solutions for functional differential inclusion(differential inclusions with memory) x′ ∈ F (T (t)x) + f(t, T (t)x) where F is upper semicontinuous, co...

This section is devoted to the discussion of two fundamental (partial) differential equations, that arise in the context of Markov diffusion processes. After giving a brief introduction of continuous-time continuous state Markov processes, we introduce the forward and backward equation, and provide a heuristic derivation of these equations for diffusion processes. We also discuss some examples ...

Let E be a set of inequalities between finite Σ-terms. Let Vω and Vr denote the varieties of all ω-continuous ordered Σ-algebras and regular ordered Σ-algebras satisfying E, respectively. We prove that the free Vr-algebra R(X) on generators X is the subalgebra of the corresponding free Vω-algebra Fω(X) determined by those elements of Fω(X) denoted by the regular Σ-coterms. We actually establish...

and Applied Analysis 3 provided this limit exists. A function f : T → R is said to be rd-continuous provided f is continuous at right-dense points and there exists a finite left limit at all left-dense points in T .The set of all such rd-continuous functions is denoted by Crd(T).The derivativef Δ off and the forward jump operator σ are related by the formula f σ (t) = f (σ (t)) = f (t) + μ (t) ...