Construct stochastic differential equation from linear drift-rate models
SDE = sdeld(A, B, Alpha, Sigma)
SDE = sdeld(A, B, Alpha, Sigma, 'Name1', Value1, 'Name2', Value2, ...)
This constructor creates and displays SDE objects whose drift rate is expressed in linear drift-rate form and that derive from the SDEDDO (SDE from drift and diffusion objects class).
Use SDELD objects to simulate sample paths of NVARS state variables expressed in linear drift-rate form. They provide a parametric alternative to the mean-reverting drift form (see sdemrd).
These state variables are driven by NBROWNS Brownian motion sources of risk over NPERIODS consecutive observation periods, approximating continuous-time stochastic processes with linear drift-rate functions.
Xt is an NVARS-by-1 state vector of process variables.
A is an NVARS-by-1 vector.
B is an NVARS-by-NVARS matrix.
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.
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:
|A||A represents the parameter A.
If you specify A as an array, it must be an NVARS-by-1 column
vector of intercepts. As a deterministic function of time, when A is
called with a real-valued scalar time t as its
only input, A must produce an NVARS-by-1 column
vector. If you specify A as a function of time
and state, it must generate an NVARS-by-1 column
vector of intercepts when invoked with two inputs:|
|B||B represents the parameter B.
If you specify B as an array, it must be an NVARS-by-NVARS matrix
of state vector coefficients. As a deterministic function of time,
when B is called with a real-valued scalar time t as
its only input, B must produce an NVARS-by-NVARS matrix.
If you specify B as a function of time and state,
it must generate an NVARS-by-NVARS matrix
of state vector coefficients when invoked with two inputs: |
|Alpha||Alpha determines the format of the parameter D.
If you specify Alpha as an array, it represents
an NVARS-by-1 column vector
of exponents. As a deterministic function of time, when Alpha is
called with a real-valued scalar time t as its
only input, Alpha must produce an NVARS-by-1 column
vector. If you specify it as a function of time and state, 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 represents
is 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. 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 it as a function of time and state, 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, sdeld applies the same initial value to all state variables on all trials.
If StartState is a column vector, sdeld applies a unique initial value to each state variable on all trials.
If StartState is a matrix, sdeld 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 sdeld 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, sdeld 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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