Abstract
In this paper we study solutions to stochastic differential equations (SDEs) with discontinuous drift. In this paper we discussed The Euler-Maruyama method and this shows that a candidate density function based on the Euler-Maruyama method. The point of departure for this work is a particular SDE with discontinuous drift.
Highlights
A general one dimensional stochastic differential equations (SDEs) is given by dxt = b(t, xt)dt + σ(t, xt)dBt, x0 = c, (1)where x = xt is an R-valued stochastic process : [0, T ] → R, b, σ : [0, T ] × R → R are the drift and diffusion coefficient of x, B = Bt is an R-valued Wiener process, and c is a random variable independent of Bt − B0 for t ≥ 0
A general one dimensional SDE is given by dxt = b(t, xt)dt + σ(t, xt)dBt, x0 = c, (1)
This paper focuses on the special case for (1), where σ(t, xt) = 1 and b(t, xt) = −k sign(xt) with k > 0 a control gain and the sign-function defined by
Summary
The Euler-Maruyama method is applied to approximate solutions to (3) and to investigate a theoretical methods to obtain candidate density functions for solutions to (3). The Euler-Maruyama method is a simple time discrete approximation technique which is used to approximate solutions to SDEs of the type given in (1), by discretizing the time interval [0, T ] in steps 0 < t1 < · · · < tn < tn+1 · · · < tN with If the drift and diffusion coefficient in (1) are measurable, satisfy a Lipschitz condition and a growth bound, the Euler Maruyama method guarantee strong convergence to the solution of (1).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
