Brownian motion models
BM = bm(Mu, Sigma)
BM = bm(Mu, Sigma, 'Name1', Value1, 'Name2', Value2,
...)
This constructor creates and displays Brownian motion (sometimes
called arithmetic Brownian motion or generalized
Wiener process) objects that derive from the SDELD
(SDE
with drift rate expressed in linear form) class. Use BM
objects
to simulate sample paths of NVARS
state variables
driven by NBROWNS
sources of risk over NPERIODS
consecutive
observation periods, approximating continuoustime Brownian motion
stochastic processes. This enables you to transform a vector of NBROWNS
uncorrelated,
zerodrift, unitvariance rate Brownian components into a vector of NVARS
Brownian
components with arbitrary drift, variance rate, and correlation structure.
The bm
method allows you to simulate any
vectorvalued BM process of the form:
$$d{X}_{t}=\mu (t)dt+V(t)d{W}_{t}$$  (181) 
where:
X_{t} is an NVARS
by1
state vector of process variables.
μ is an NVARS
by1
driftrate vector.
V is an NVARS
byNBROWNS
instantaneous
volatility rate matrix.
dW_{t} is an NBROWNS
by1
vector of (possibly) correlated zerodrift/unitvariance rate Brownian
components.
Specify required input parameters as one of the following types:
A MATLAB^{®} array. Specifying an array indicates a static (nontimevarying) parametric specification. This array fully captures all implementation details, which are clearly associated with a parametric form.
A MATLAB function. Specifying a function provides indirect support for virtually any static, dynamic, linear, or nonlinear model. This parameter is supported via an interface, because all implementation details are hidden and fully encapsulated by the function.
Note: You can specify combinations of array and function input parameters as needed. Moreover, a parameter is identified as a deterministic function
of time if the function accepts a scalar time 
The required input parameters are:
Mu  Mu represents μ.
If you specify Mu as an array, it must be an NVARS by1 column
vector representing the drift rate (the expected instantaneous rate
of drift, or time trend). As a deterministic function of time, when Mu is
called with a realvalued scalar time t as its
only input, Mu must produce an NVARS byNVARS matrix.
If you specify Mu as a function of time and state,
it calculates the expected instantaneous rate of drift. This function
must generate an NVARS by1 column
vector when invoked with two inputs:

Sigma  Sigma represents the parameter V.
If you specify Sigma as an array, it must be an NVARS byNBROWNS matrix
of instantaneous volatility rates. In this case, each row of Sigma corresponds
to a particular state variable. Each column of Sigma corresponds
to a particular Brownian source of uncertainty, and associates the
magnitude of the exposure of state variables with sources of uncertainty.
As a deterministic function of time, when Sigma is
called with a realvalued scalar time t as its
only input, Sigma must produce an NVARS byNBROWNS matrix.
If you specify Sigma as a function of time and
state, it must generate an NVARS byNBROWNS matrix
of volatility rates when invoked with two inputs:
Although the constructor does not enforce restrictions
on the sign of this argument, 
Specify optional inputs as matching parameter name/value pairs as follows:
Specify the parameter name as a character string, followed by its corresponding value.
You can specify parameter name/value pairs in any order.
Parameter names are case insensitive.
You can specify unambiguous partial string matches.
Valid parameter names are:
StartTime  Scalar starting time of the first observation, applied to all
state variables. If you do not specify a value for StartTime ,
the default is 0 . 
StartState  Scalar, NVARS by1 column
vector, or NVARS byNTRIALS matrix
of initial values of the state variables. If If If If you do not specify
a value for 
Correlation  Correlation between Gaussian random variates drawn to generate
the Brownian motion vector (Wiener processes). Specify Correlation as
an NBROWNS byNBROWNS positive
semidefinite matrix, or as a deterministic function C(t) that
accepts the current time t and returns an NBROWNS byNBROWNS positive
semidefinite correlation matrix. A As a deterministic function
of time, If you do not specify a value for 
Simulation  A userdefined simulation function or SDE simulation method.
If you do not specify a value for Simulation , the
default method is simulation by Euler approximation (simByEuler ). 
BM  Object of class

When you specify the required input parameters as arrays, they are associated with a specific parametric form. By contrast, when you specify either required input parameter as a function, you can customize virtually any specification.
Accessing the output parameters with no inputs simply returns the original input specification. Thus, when you invoke these parameters with no inputs, they behave like simple properties and allow you to test the data type (double vs. function, or equivalently, static vs. dynamic) of the original input specification. This is useful for validating and designing methods.
When you invoke these parameters with inputs, they behave like
functions, giving the impression of dynamic behavior. The parameters
accept the observation time t and a state vector X_{t},
and return an array of appropriate dimension. Even if you originally
specified an input as an array, bm
treats it as
a static function of time and state, by that means guaranteeing that
all parameters are accessible by the same interface.
AitSahalia, Y., "Testing ContinuousTime Models of the Spot Interest Rate," The Review of Financial Studies, Spring 1996, Vol. 9, No. 2, pp. 385–426.
AitSahalia, Y., "Transition Densities for Interest Rate and Other Nonlinear Diffusions," The Journal of Finance, Vol. 54, No. 4, August 1999.
Glasserman, P., Monte Carlo Methods in Financial Engineering, New York: SpringerVerlag, 2004.
Hull, J. C., Options, Futures, and Other Derivatives, 5th ed. Englewood Cliffs, NJ: Prentice Hall, 2002.
Johnson, N. L., S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, Vol. 2, 2nd ed. New York: John Wiley & Sons, 1995.
Shreve, S. E., Stochastic Calculus for Finance II: ContinuousTime Models, New York: SpringerVerlag, 2004.