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 continuous-time Brownian motion stochastic processes. This enables you to transform a vector of NBROWNS uncorrelated, zero-drift, unit-variance rate Brownian components into a vector of NVARS Brownian components with arbitrary drift, variance rate, and correlation structure.
Xt is an NVARS-by-1 state vector of process variables.
μ is an NVARS-by-1 drift-rate vector.
V is an NVARS-by-NBROWNS instantaneous volatility rate matrix.
dWt is an NBROWNS-by-1 vector of (possibly) correlated zero-drift/unit-variance rate Brownian components.
Specify required input parameters as one of the following types:
A MATLAB® array. Specifying an array indicates a static (non-time-varying) 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:
|Mu||Mu represents μ.
If you specify Mu as an array, it must be an NVARS-by-1 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 real-valued scalar time t as its
only input, Mu must produce an NVARS-by-NVARS 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-by-1 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-by-NBROWNS 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 real-valued scalar time t as its
only input, Sigma must produce an NVARS-by-NBROWNS matrix.
If you specify Sigma as a function of time and
state, it must generate an NVARS-by-NBROWNS matrix
of volatility rates when invoked with two inputs: |
Although the constructor does not enforce restrictions on the sign of this argument, Sigma is usually specified as a positive value.
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-by-1 column
vector, or NVARS-by-NTRIALS matrix
of initial values of the state variables. |
If StartState is a scalar, bm applies the same initial value to all state variables on all trials.
If StartState is a column vector, bm applies a unique initial value to each state variable on all trials.
If StartState is a matrix, bm 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 NBROWNS-by-NBROWNS positive
semidefinite matrix, or as a deterministic function C(t) that
accepts the current time t and returns an NBROWNS-by-NBROWNS 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 NBROWNS-by-NBROWNS identity matrix representing independent Gaussian processes.
|Simulation||A user-defined simulation function or SDE simulation method. If you do not specify a value for Simulation, the default method is simulation by Euler approximation (simByEuler).|
Object of class BM with the following displayed parameters:
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 Xt, 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, thereby guaranteeing that all parameters are accessible by the same interface.
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