Documentation 
Create GBM model
GBM = gbm(Return, Sigma)
GBM = gbm(Return, Sigma, 'Name1', Value1, 'Name2', Value2, ...)
This function creates and displays geometric Brownian motion (GBM) models, which derive from the CEV (constant elasticity of variance) class. Use GBM models to simulate sample paths of NVARS state variables driven by NBROWNS Brownian motion sources of risk over NPERIODS consecutive observation periods, approximating continuoustime GBM stochastic processes.
This function allows simulation of vectorvalued GBM processes of the form:
$$d{X}_{t}=\mu (t){X}_{t}dt+D(t,{X}_{t})V(t)d{W}_{t}$$  (186) 
where:
X_{t} is an NVARSby1 state vector of process variables.
μ is an NVARSbyNVARS generalized expected instantaneous rate of return matrix.
D is an NVARSbyNVARS diagonal matrix, where each element along the main diagonal is the corresponding element of the state vector X_{t}.
V is an NVARSbyNBROWNS instantaneous volatility rate matrix.
dW_{t} is an NBROWNSby1 Brownian motion vector.
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 t as its only input argument. Otherwise, a parameter is assumed to be a function of time t and state X(t) and is invoked with both input arguments. 
The required input parameters are:
Return  Return represents the parameter μ.
If you specify Return as an array and it must be
an NVARSbyNVARS matrix representing
the expected (mean) instantaneous rate of return. As a deterministic
function of time, when Return is called with a
realvalued scalar time t as its only input, Return must
produce an NVARSbyNVARS matrix.
If you specify Return as a function of time and
state, it must return an NVARSbyNVARS matrix
when invoked with two inputs:

Sigma  Sigma represents the parameter V.
If you specify Sigma as an array, it must be an NVARSbyNBROWNS matrix
of instantaneous volatility rates or as a deterministic function of
time. In this case, each row of Sigma corresponds
to a particular state variable. Each column 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 NVARSbyNBROWNS matrix.
If you specify Sigma as a function of time and
state, it must return an NVARSbyNBROWNS matrix
of volatility rates when invoked with two inputs:
Although the gbm function enforces no restrictions on the sign of Sigma volatilities, they are usually specified as positive values. 
Specify optional input arguments as variablelength lists of matching parameter name/value pairs: 'Name1', Value1, 'Name2', Value2, ... and so on. The following rules apply when specifying parametername pairs:
Specify the parameter name as a character string, followed by its corresponding parameter 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, NVARSby1 column vector, or NVARSbyNTRIALS matrix
of initial values of the state variables. If StartState is a scalar, gbm applies the same initial value to all state variables on all trials. If StartState is a column vector, gbm applies a unique initial value to each state variable on all trials. If StartState is a matrix, gbm applies a unique initial value to each state variable on each trial. If you do not specify a value for StartState, all variables start at 1. 
Correlation  Correlation between Gaussian random variates drawn to generate
the Brownian motion vector (Wiener processes). Specify Correlation as
an NBROWNSbyNBROWNS positive
semidefinite matrix, or as a deterministic function C(t) that
accepts the current time t and returns an NBROWNSbyNBROWNS positive
semidefinite correlation matrix. A Correlation matrix represents a static condition. As a deterministic function of time, Correlation allows you to specify a dynamic correlation structure. If you do not specify a value for Correlation, the default is an NBROWNSbyNBROWNS identity matrix representing independent Gaussian processes. 
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). 
GBM  Geometric Brownian motion model with the following displayed
parameters:

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