Construct stochastic differential equation from mean-reverting drift-rate models
SDE = sdemrd(Speed, Level, Alpha, Sigma)
SDE = sdemrd(Speed, Level, Alpha, Sigma, 'Name1', Value1, 'Name2', Value2, ...)
This constructor creates and displays SDE objects whose drift rate is expressed in mean-reverting drift-rate form and which derive from the SDEDDO class (SDE from drift and diffusion objects). Use SDEMRD objects to simulate of sample paths of NVARS state variables expressed in mean-reverting drift-rate form, and provide a parametric alternative to the linear drift form (see sdeld). These state variables are driven by NBROWNS Brownian motion sources of risk over NPERIODS consecutive observation periods, approximating continuous-time stochastic processes with mean-reverting drift-rate functions.
Xt is an NVARS-by-1 state vector of process variables.
S is an NVARS-by-NVARS matrix of mean reversion speeds.
L is an NVARS-by-1 vector of mean reversion levels.
D is an NVARS-by-NVARS diagonal matrix, where each element along the main diagonal is the corresponding element of the state vector raised to the corresponding power of α.
V is an NVARS-by-NBROWNS instantaneous volatility rate matrix.
dWt is an NBROWNS-by-1 Brownian motion vector.
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.
The required input parameters are:
|Speed||Speed represents the parameter S.
If you specify Speed as an array, it represents
an NVARS-by-NVARS 2-dimensional
matrix of mean-reversion speeds (the rate or speed at which the state
vector reverts to its long-run average Level).
If you specify Speed as a function, Speed calculates
the speed of mean reversion. This function must generate an NVARS-by-NVARS matrix
of reversion rates when called with two inputs: |
|Level||Level represents the parameter L.
If you specify Level as an array, it must be an NVARS-by-1
column vector of reversion levels. If you specify Level as
a function, it must generate an NVARS-by-1 column
vector of reversion levels when invoked with two inputs: |
|Alpha||Alpha determines the format of the parameter D.
If you specify Alpha as an array, it must be an NVARS-by-1
column vector of exponents. If you specify it as a function, it must
return an NVARS-by-1 column vector of exponents
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 2-dimensional
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.
If you specify it as a function, it must generate an NVARS-by-NBROWNS matrix
of volatility rates when invoked with two inputs:|
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, sdemrd applies the same initial value to all state variables on all trials.
If StartState is a column vector, sdemrd applies a unique initial value to each state variable on all trials.
If StartState is a matrix, sdemrd 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 SDEMRD, with the following 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, sdemrd treats it as a static function of time and state, thereby guaranteeing that all parameters are accessible by the same interface.