[an error occurred while processing this directive]
This section is adapted from: “An Algorithmic Introduction to the Numerical Simulation of Stochastic Differential Equations”, by Desmond J. Higham, in SIAM Review, Vol. 43, No. 3, pp. 525546, 2001. and Financial Calculus: An introduction to derivative pricing by M Baxter, and A. Rennie, Cambridge University Press, 1996, pages 5262.
The straight line segment is the building block of differential calculus. Differentiable functions, no matter how difficult or how strange their global behavior, are on a local scale composed of segments which are nearly straight line segments. This is the basic idea behind all of differential calculus. In particular, this is the idea Euler’s method for approximating the differentiable functions which are defined by differential equations.
We know that rescaling (“zooming in” on) Brownian motion does not produce a straight line, it produces another image of Brownian motion. This selfsimilarity is ideal for an infinitesimal building block, for instance, we could build global Brownian motion out of lots of local “chunks” of Brownian motion. And then that suggests we could build other stochastic processes out of suitably scaled Brownian motion. In addition, if we include straight line segments we can overlay the behavior of differentiable functions onto the stochastic processes as well. Thus, straight line segments and “chunks” of Brownian motion are the building blocks of stochastic calculus.
With stochastic differential calculus, we can build a nice class of new stochastic processes by specifying how they are built up locally out of our base deterministic function, the straight line and our base stochastic process, Standard Brownian motion. We write the local change in value of the stochastic process over a time interval of (infinitesimal) length dt informally as
Note that we are not allowed to write
since Standard Brownian Motion is nowhere differentiable with probability 1. (Actually, the informal stochastic differential equation is really just a compact way of writing a rigorously defined, and equivalent, implicit Ito integral equation, but since we do not have the rigor required, we will approach the stochastic differential equation intuitively and informally.)
This says that the initial point (t_{0},X_{0}) is specified, perhaps with X_{0} a random variable with a given distribution. A deterministic component at each point has a slope determined through G at that point. In addition, there is some random perturbation that effects the evolution of the process. The variance of the random perturbation is determined at each point through the function H. This is a simple expression of a Stochastic Differential Equation (SDE) which determines a stochastic process, just as an Ordinary Differential Equation (ODE) determines a differentiable function. We infinitesimally extend the process with the incremental change information and repeat. This is an expression in words of the EulerMaruyama method for numerically simulating the stochastic differential expression.
Simple Example 1 The simplest stochastic differential equation is
where r is a constant. Take a deterministic initial condition to be X(0) = b. This process is the stochastic extension of the differential equation expression of a straight line. The new stochastic process X is drifting or trending at rate r with a random variation due to Brownian Motion perturbations around that trend. We will later show explicitly that the solution of this SDE is X(t) = b + rt + W(t) although it is seems intuitively clear that this should be the process. We will call this Brownian motion with drift.
Simple Example 2 The next simplest stochastic differential equation is
This stochastic differential equation says that the process is evolving as a multiple of Standard Brownian Motion. The solution may be easily guessed as X(t) = W(t) which has variance ^{2}t on increments of length t. Sometimes this is called Brownian motion (in contrast to Standard Brownian Motion which has variance t on increments of length t).
We can combine the previous two examples to consider
which would have solution X(t) = b + rt + W(t), a multiple of Brownian motion with drift r started at b. Sometimes this extension of Standard Brownian motion is called Brownian motion and some authors consider this process directly instead of the more special case we considered in the previous chapter.
Simple Example 3 The next simplest and first nontrivial differential equation is dX = X dW. Here the differential equation says that process is evolving like Brownian motion with a variance which is the same as the process value. Expressing the stochastic differential equation as dX/X = dW we may say that the relative change acts like Standard Brownian Motion. When the process is small, the variance is small, when the process is large, the variance is large. The resulting stochastic process is called geometric Brownian motion and it will figure extensively in what we consider later as models of security prices.
Simple Example 4 The next simplest differential equation is
Here the stochastic differential equation says that the growth of the process at a point is proportional to the process value, with a random perturbation proportional to the process value. Again looking ahead, we could write the differential equation as dX/X = rdt + dW and interpret it to say the relative rate of increase is proportional to the time observed together with a random perturbation like a Brownian segment proportional to the length of time.
The sample path that the EulerMaruyama method produces numerically is the analog of using the Euler method.
The formula for EulerMaruyama (EM) method is based on the definition of the Ito stochastic integral:
Note that we will not use Brownian motion directly to obtain the increments W(t_{j1} + dt)  W(t_{j1}), instead we will use coinflipping sequences of an appropriate length to create an approximation to W(t). Note that since the increments W(t_{j1} + dt)  W(t_{j1}) are independent and identically distributed, we will be able to use independent coinflip sequences to generate the approximation of the increments. For convenience, I generated the approximations using a random number generator on a computer, but I could as well have used actual coinflipping. I have not recorded the generation of the sequences, only the summed and scaled (independently sampled) outcomes for
For convenience, I will take dt = 1/10, N = 100, so I will be needing S(100 ^{.} (1/10))/ = S(10)/10. Also, I will take r = 2, b = 1, and = 1, so I am simulating the solution of








j  t_{j}  X_{j}  2X_{j} dt  dW  X_{j}dW  2X_{j} + X_{j}dW  X_{j} + 2X_{j} dt + X_{j}dW 








0  0  1  0.2  0  0  0.2  1.2 
1  0.1  1.2  0.24  0.2  0.24  0.48  1.68 
2  0.2  1.68  0.34  0.2  0.34  0.0  1.68 
3  0.3  1.68  0.34  0.4  0.67  1.01  2.69 
4  0.4  2.69  0.54  0.2  0.54  0.0  2.69 
5  0.5  2.69  0.54  0  0  0.54  3.23 
6  0.6  3.23  0.65  0.4  1.29  1.94  5.16 
7  0.7  5.16  1.03  0.4  2.06  3.1  8.26 
8  0.8  8.26  1.65  0.4  3.3  4.95  13.21 
9  0.9  13.21  2.64  0  0  2.64  15.85 
10  1.0  15.85  








Of course, it is easy to imagine that this can be programmed, and the step size made much smaller, presumably with better approximation properties. In fact, it is possible to consider kinds of convergence for the EM method comparable to the Strong Law of Large Numbers and the Weak Law of Large Numbers.
[an error occurred while processing this directive] [an error occurred while processing this directive]
Last modified: [an error occurred while processing this directive]