We address the following problem from the intersection of dynamical systems and stochastic analysis: Two SDE dxt = P m j=0 fj (xt) dW j t and dxt = P m j=0 gj(xt)dW j t in R d with smooth coeecients satisfying fj (0) = gj(0) = 0 are said to be smoothly equivalent if there is a smooth random diieomorphism (coordinate transformation) h(!) with h(!; 0) = 0 and Dh(!; 0) = id which conjugates the co...

In this paper, we study a class of stochastic differential equations with additive noise that contains a fractional Brownian motion (fBM) and a Poisson point process of class (QL). The differential equation of this kind is motivated by the reserve processes in a general insurance model, in which the long term dependence between the claim payment and the past history of liability becomes the mai...

Automated analysis of complex systems based on multiple readouts remains a challenge. Change point detection algorithms are aimed to locating abrupt changes in the time series behaviour process. In this paper, we present novel change algorithm Latent Neural Stochastic Differential Equations (SDE). Our method learns non-linear deep learning transformation process into latent space and estimates ...

Abstract A scheme for the analytical stochastization of ordinary differential equations (ODEs) is presented in this article. Using Itô calculus, an ODE transformed into a stochastic equation (SDE) such way that solutions obtained can be constructed. Furthermore, constructed trajectories remain bounded same interval as deterministic solutions. The proposed approach stark contrast to methods base...

In this paper we study coupled fast-slow ordinary differential equations (ODEs) with small time scale separation parameter $\epsilon$ such that, for every fixed value of the slow variable, fast dynamics are sufficiently chaotic ergodic invariant measure. Convergence process to solution a homogenized stochastic equation (SDE) in limit zero, explicit formulas drift and diffusion coefficients, has...

In this article, we consider the linear-quadratic time-inconsistent mean-field type leader-follower Stackelberg differential game with an adapted open-loop information structure. The objective functionals of leader and follower include conditional expectations state control (mean field) variables, cost parameters could be general nonexponential discounting depending on initial time. As stated i...

We present a nonlinear random-effects stochastic differential equation (SDE) model of combined basal and pulsatile hormone secretion with a series-specific hormone half-life and conditional pulse times. The construct uses a three-parameter pulse shape (generalized gamma function) to allow variably skewed secretory bursts superimposed on a finite basal hormone secretion rate. The analysis imbeds...

We present a method for simultaneous dimension reduction, model fitting and metastability analysis of high dimensional time series. The approach is based on the combination of hidden Markov models (HMMs) with localized principal component analysis (PCA) and fitting of multidimensional stochastic differential equations (SDE). We derive explicit estimators for PCA-SDE model parameters and employ ...

In this paper we consider multidimensional stochastic differential equations (SDEs) with discontinuous drift and possibly degenerate diffusion coefficient. We prove an existence and uniqueness result for this class of SDEs and we present a numerical method that converges with strong order 1/2. Our result is the first one that shows strong convergence for such a general class of SDEs. The proof ...