A Study on Stochastic Differential Equation

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.


Introduction
A general one dimensional SDE is given by where x = x t is an R-valued stochastic process : [0, T ] → R, b, σ : [0, T ] × R → R are the drift and diffusion coefficient of x, B = B t is an R-valued Wiener process, and c is a random variable independent of B t − B 0 for t ≥ 0. On [0, T ], existence and uniqueness of a solution x t , continuous with probability 1, to (1) is guaranteed whenever the drift b and diffusion σ are measurable functions satisfying a Lipschitz condition together with a growth bound, both uniformly in t. This paper focuses on the special case for (1), where σ(t, x t ) = 1 and b(t, x t ) = −k sign(x t ) with k > 0 a control gain and the sign-function defined by with c given.
In the next section, 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
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 < t 1 < · · · < t n < t n+1 · · · < t N with N = T h , where h = t n+1 − t n is the step-length. Each recursive step is determined via the following method, normal with mean zero and variance h, which we denote by W n ∼ N (0, h).
Given an initial condition x 0 = c, it is possible from (4) to approximate a solution to (1) by determination of x 1 , x 2 , . . . , x N . 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). Hence for SDEs with discontinuous drift we can, in general, not expect the Euler Maruyama method to produce meaningful results. Nevertheless, we will in the sequel apply this method to the special case (3) in order to obtain candidate solutions.

Analysis of the Deterministic Step
Application of the Euler-Maruyama method to the SDE in (3) gives the recursive step in most of the simulations. From this, we expect after a finite time 0 < t < ∞ that there exists N ∈ N such that x n+N ≤ 0. Similar result is obtained if x m < 0, then we expect that there exists M ∈ N such that x m+M ≥ 0. The influence from the control gain k determines how quick the evolution of the sequence {x n } n≥0 switches around zero. In other words, a big k minimizes the influence of the random variable W n . The Euler-Maruyama method is easy to implement in software, so following we have applied Matlab to simulate solutions to (3).

Numerical solutions to a SDE with Discontinuous Drift
We consider the recursive step in (5)     3 illustrates four different histograms of 500 simulations. Here h is 0.01, 0.001, 0.0001 and 0.00001 respectively and k = 1. It can be seen right away that the result narrows slightly around zero when h becomes smaller, but changing in the step-size does not immediately give big effect. In figure 4, the control gain is changing, k = 1, 2, 3, 4 and h = 0.001. Here it is clear that changing k as an influence on the result of x T . The variance of x T gets smaller when k increases. This is not surprising since the overall influence of the random variable W n is decreased when k increases as mentioned in previous section.   Consider the recursive determination of x n+1 in (5). Define an intermediate variable z n = x n + hksign(x n ) and let f zn and f n denote the density functions of z n and x n , respectively. Moreover, let N (0, h) denote the density function for W n . From probability theory we get Hence, we proceed by studying the density function f zn (z). Let the distribution function of z n be denoted by F zn such that F zn (z) = P (z n ≤ z) = P (x n − hksign(x n ) ≤ z), which can be expressed by P (z n ≤ z) = P (x n − hk ≤ z, x n > 0) + P (x n + hk ≤ z, x n < 0) + P (x n ≤ z, x n = 0) = P (x n ≤ z + hk, x n > 0) + P (x n ≤ z − hk, x n < 0) .
For different values of z, the probability P (z n ≤ z) can be expressed differently. If z < −hk and if −hk ≤ z ≤ hk.
By introducing the indicator function I, the expression of the distribution function of z n is By differentiating with respect to z, the density function of z n is By substituting the above into (6), the density function f n+1 is found from the density function f n . In the following section, the solution to (7) is investigated numerically.

Recursive Developing of the Density Function in Matlab
The recursive density function is given by Following, we apply Matlab to investigate the evolution of the function f n+1 (x) for n increasing. Assume that the density function for the initial condition x 0 = c is normal distributed with mean zero and variance h. The Euler-Maruyama method is expected to converge to a stochastic process (or a distribution of a stochastic process) when h → 0. (Under certain regularity conditions, so actually we cannot expect it here but only conjecture.) We hope that the developing of the recursive density functions in (8) will reach stationary condition for n → ∞. For this reason, the number n of simulations is chosen to depend on the step size, such that n = 1 h 1+α , where α > 0. This ensures that both convergence criteria are fulfilled. Equation (8) is simulated in Matlab for h = 0.01, α = 0.5, k = 1 such that n = 1000, the result is shown in figure 5. At the end of section, there is a comparison between the convergence of the recursive density function and the result obtained there is presented.  In the following, we continue the study of (7) under stationary assumptions.