1 Analytical analysis of the exploding and vanishing gradients problem 1.1 Linear model x t = W rec σ(x t−1) + W in u t + b (1) Let us consider the term g T k = ∂Et ∂xt ∂xt ∂x k ∂ + x k ∂θ for the linear version of the parametriza-tion in equation (1) (i.e. set σ to the identity function) and assume t goes to infinity and l = t − k. We have that: ∂x t ∂x k = W T rec l (2) By employing a generic...

In relation with Monte-Carlo methods to solve some integro-differential equations, we study the approximation problem of IEg(XT ) by IEg(X̄n T ), where (Xt, 0 ≤ t ≤ T ) is the solution of a stochastic differential equation governed by a Lévy process (Zt), (X̄n t ) is defined by the Euler discretization scheme with step Tn . With appropriate assumptions on g(·), we show that the error IEg(XT ) − I...

Consider the following distribution dependent SDE: dXt=?t(Xt,?Xt)dWt+bt(Xt,?Xt)dt, where ?Xt stands for of Xt. In this paper non-degenerate ?, we show strong well-posedness above SDE under some integrability assumptions in spatial variable and Lipschitz continuity ? about b ?. particular, extend results Krylov–Röckner (Probab. Theory Related Fields 131 (2005) 154–196) to case.

Let X = {(Xt ,Yt )}t∈Z be a stationary time series where Xt is binary valued and Yt , the noisy observation of Xt , is real valued. Letting P denote the probability measure governing the joint process {(Xt ,Yt )}, we characterize U(l,P), the optimal asymptotic average performance of a predictor allowed to base its prediction for Xt on Y1, . . . , Yt−1, where performance is evaluated using the l...

We solve the pricing problem for perpetual American puts and calls on dividend-paying assets. The dependence of a dividend process on the underlying stochastic factor is fairly general: any non-decreasing function is admissible. The stochastic factor follows a Lévy process. This specification allows us to consider assets that pay no dividends at all when the level of the underlying factor (say,...

Let f be a measurable, real function defined in a neighbourhood of infinity. The function f is said to be of generalised regular variation if there exist functions h 6≡ 0 and g > 0 such that f(xt)− f(t) = h(x)g(t) + o(g(t)) as t → ∞ for all x ∈ (0,∞). Zooming in on the remainder term o(g(t)) leads eventually to a relation of the form f(xt)− f(t) = h1(x)g1(t) + · · ·+ hn(x)gn(t) + o(gn(t)), each...

Low lactate dehydrogenase (LD; EC 1.1.1.27) activity and an abnormal LD pattern in electrophoretograms of LD isoenzymes in the sera of two patients were caused by inhibition of LD by immunoglobulin G. One of these showed inhibitor activity in the serum upon direct analysis, while the other showed activity only after the immunoglobulin was stripped from the LD. As judged from the LD isoenzyme pa...

Prospects for utilizing whole-genome association analysis in autogamous plant populations appear promising due to the reported high levels of linkage disequilibrium (LD). To determine the optimal strategies for implementing association analysis in soybean (Glycine max L. Merr.), we analyzed the structure of LD in three regions of the genome varying in length from 336 to 574 kb. This analysis wa...

Schaefer’s fixed point theorem is used to study the controllability in an infinite delay system x′(t)=G(t,xt) + (Bu)(t). A compact map or homotopy is constructed enabling us to show that if there is an a priori bound on all possible solutions of the companion control system x′(t) = λ[G(t,xt) + (Bu)(t)], 0 < λ < 1, then there exists a solution for λ = 1. The a priori bound is established by mean...