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Math 489/Math 889
Stochastic Processes and
Advanced Mathematical Finance
Dunbar, Fall 2010

Introduction to Stochastic Differential Equations

Key Concepts

Simulating stochastic differential equations numerically with an analog to Euler’s method, called the Euler-Maruyama (EM) method.

Vocabulary

A stochastic differential equation is a mathematical equation relating a stochastic process to its local deterministic and random components. The goal is to unravel the relation to find the stochastic process. Under mild conditions on the relationship, and with a specifying initial condition, solutions of stochastic differential equations exist and are unique.
The Euler-Maruyama (EM) method is a numerical method for simulating the solutions of a stochastic differential equation based on the definition of the Ito stochastic integral: Given $dX(t) = G(X(t))dt + H(X(t))dW (t), X(t0) = X0,$ and a step size dt, we approximate and simulate with $Xj = Xj -1 + G(Xj -1)dt + H(Xj - 1)(W (tj-1 + dt)- W (tj-1))$
Extensions and variants of Standard Brownian Motion defined through stochastic differential equations are Brownian Motion with drift, scaled Brownian Motion, and geometric Brownian Motion.

Mathematical Ideas

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. 525-546, 2001. and Financial Calculus: An introduction to derivative pricing by M Baxter, and A. Rennie, Cambridge University Press, 1996, pages 52-62.

Stochastic Differential Equations: Symbolically

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 self-similarity 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

dX = G(X(t)) dt + H(X(t)) dW (t),X(t0) = X0.

Note that we are not allowed to write

dX--= G(X(t)) + H(X(t)) dW--,X(t ) = X dt dt 0 0

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 (t₀,X₀) is specified, perhaps with X₀ 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 Euler-Maruyama method for numerically simulating the stochastic differential expression.

Simple Example 1 The simplest stochastic differential equation is

dX = r dt + dW

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

dX = sdW.

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 ²t 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

dX = rdt + sdW, X(0) = b

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 non-trivial 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

dX = rX dt + sX dW.

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.

Stochastic Differential Equations: Numerically

The sample path that the Euler-Maruyama method produces numerically is the analog of using the Euler method.

The formula for Euler-Maruyama (EM) method is based on the definition of the Ito stochastic integral:

Xj = Xj-1+G(Xj -1)dt+H(Xj -1)(W (tj-1+dt)) - W (tj- 1), tj = tj-1+dt

Note that we will not use Brownian motion directly to obtain the increments W(t_j-1 + dt) - W(t_j-1), instead we will use coin-flipping sequences of an appropriate length to create an approximation to W(t). Note that since the increments W(t_j-1 + dt) - W(t_j-1) are independent and identically distributed, we will be able to use independent coin-flip 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 coin-flipping. I have not recorded the generation of the sequences, only the summed and scaled (independently sampled) outcomes for

W (dt) ~~ W (dt) = S(N V~ --dt) = V~ dt-S( V~ N-dt). N N N dt

For convenience, I will take dt = 1/10, N = 100, so I will be needing S(100 ^. (1/10))/ V~ ---- 100 = S(10)/10. Also, I will take r = 2, b = 1, and = 1, so I am simulating the solution of

dX = 2X dt + XdW, X(0) = 1.


j	t_j	X_j	2X_j dt	dW	X_jdW	2X_j + X_jdW	X_j + 2X_j dt + X_jdW

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.

Problems to Work for Understanding

Simulate the solution of the stochastic differential equation $dY (t) = Y (t)dt + 2Y (t)dX$ on the interval [0, 1] with initial condition Y (0) = 1 and step size t = 1/10.
Solution
Simulate the solution of the stochastic differential equation $dY (t) = tY (t)dt + 2Y(t)dX$ Note the difference with the previous problem, now the multiplier of the dt term is a function of time.
Solution
Solution
Solution

Reading Suggestion:

“An Algorithmic Introduction to the Numerical Simulation of Stochastic Differential Equations”, by Desmond J. Higham, in SIAM Review, Vol. 43, No. 3, pp. 525-546, 2001.
Financial Calculus: An introduction to derivative pricing by M Baxter, and A. Rennie, Cambridge University Press, 1996, pages 52-62.

Outside Readings and Links:

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Math 489/Math 889 Stochastic Processes and Advanced Mathematical Finance Dunbar, Fall 2010