Stochastic approach to the Navier-Stokes equation

Take a stochastic approach to the Navier-Stokes equation. The pressure and flow velocity are used as probabilities, the xyz coordinates are replaced with molar concentrations, and the Navier-Stokes equation is transformed into the Fokker-Planck equation using Ito's lemma(formula). This made it possible to obtain the probability of turbulence from the Navier-Stokes equation. The molar concentration in the micro space can be obtained by separately solving the diffusion equation. Using these results, the probability of turbulence and the quantities such as fluid pressure and flow velocity can be analytically obtained.

Almost conservation laws for stochastic nonlinear Schrödinger equations

In this paper, we present a globalization argument for stochastic nonlinear dispersive PDEs with additive noises by adapting the I-method (= the method of almost conservation laws) to the stochastic setting. As a model example, we consider the defocusing stochastic cubic nonlinear Schrödinger equation (SNLS) on $${\mathbb {R}}^3$$ R 3 with additive stochastic forcing, white in time and correlated in space, such that the noise lies below the energy space. By combining the I-method with Ito’s lemma and a stopping time argument, we construct global-in-time dynamics for SNLS below the energy space.

EXTENSION OF VASICEK MODEL TO THE MODELLING OF INTEREST RATE

Interest rate modelling is an interesting aspect of stochastic processes. It has been observed that interest rates fluctuates at random times, hence the need for its modelling as a stochastic process. In this paper, we apply the existing Vasicek model, Itô’s lemma and least-square regression method in the modelling and providing dynamics for a given interest rate.

Brownian Motion and Stochastic Calculus

Brownian motion and concepts of the Itôs calculus are explained, including total variation, quadratic variation, Levy’s characterization of Brownian motion, the Itô integral, the difference between martingales and local martingales, the martingale (predictable) representation theorem , Itô’s formula (Itô’s lemma), geometric Brownian motion, covariation (joint variation) processes, the relationship between variance and expected quadratic variation, the relationship between covariance and expected covariation, and rotations of Brownian motions.