The Euler scheme is one of the standard schemes to obtain numerical approximations of stochastic differential equations (SDEs). Its convergence properties are well-known in the case of globally Lipschitz continuous coefficients. However, in many situations, relevant systems do not show a smooth behavior, which results in SDE models with discontinuous drift coefficient. In this work, we will analyze the long time properties of the Euler scheme applied to SDEs with a piecewise constant drift and a constant diffusion coefficient and carry out intensive numerical tests for its convergence properties. We will emphasize on numerical convergence rates and analyze how they depend on properties of the drift coefficient and the initial value. We will also give theoretical interpretations of some of the arising phenomena. For application purposes, we will study a rank-based stock market model describing the evolution of the capital distribution within the market and provide theoretical as well as numerical results on the long time ranking behavior.

The Euler scheme for stochastic differential equations with discontinuous drift coefficient: A numerical study of the convergence rate

Associated with a potential V and any Brownian path ω: (x, s)→ (x′,s′) there is the integral

The Brownian Bridge

This paper proposes a stochastic model with discrete delay coefficients for modeling population dynamics. This model is a memory-modulated version of the well known CIR process. Existence and uniqueness of the global solution and non-negativity of the solution are discussed. Significantly, results concerning persistence of the population and other longterm qualitative behaviors of these non-Markov processes are obtained using hitting times. Under particular parameter values, the model exhibits reflection at the zero boundary and so models populations that persist due to the influence of past population.

https://digitalcommons.lsu.edu/cgi/viewcontent.cgi?article=1011&context=josa

Memory-Modulated CIR Process with Discrete Delay Coefficients

We consider stochastic differential equations with additive noise and conditions on the coefficients in those equations that allow a time singularity in the drift coefficient. Given a maximum step size, , we specify variable (adaptive) step sizes relative to which decrease as the time node points approach the singularity. We use an Euler-type numerical scheme to produce an approximate solution and estimate the error in the approximation. When the solution is restricted to a fixed closed time interval excluding the singularity, we obtain a global pointwise error of order . An order of error for any is obtained when the approximation is run up to a time within of the singularity for an appropriate choice of exponent . We apply this scheme to Brownian bridge, which is defined as the nonanticipating solution of a stochastic differential equation of the type under consideration. In this special case, we show that the global pointwise error is of order , independent of how close to the singularity the approximation is considered.

/pdf/efficient-variable-step-size-approximations-for-strong-3pzo1tuvzk.pdf

Efficient Variable Step Size Approximations for Strong Solutions of Stochastic Differential Equations with Additive Noise and Time Singularity

We discuss a generalization of the well known squared Bessel process with real nonnegative parameter δ by introducing a predictable almost everywhere positive process γ(t, ω) into the drift and diffusion terms. The resulting generalized process is nonnegative with instantaneous reflection at zero when δ is positive. When δ is a positive integer, the process can be constructed from δ-dimensional Brownian motion. In particular, we consider γt = Xt−τ which makes the process a solution of a stochastic delay differential equation with a discrete delay. The solutions of these equations are constructed in successive steps on time intervals of length τ . We prove that if 0 < δ < 2, zero is an accessible boundary and the process is instantaneously reflecting at zero. If δ ≤ 2, lim inft→∞Xt = 0. Zero is inaccessible if δ ≥ 2.

/pdf/squared-bessel-process-with-delay-24d2aljoze.pdf

Pathiranage Lochana Siriwardena

Papers

Efficient Variable Step Size Approximations for Strong Solutions of Stochastic Differential Equations with Additive Noise and Time Singularity

Squared Bessel Process with Delay