sde - Construct stochastic differential equation models

Synopsis

SDE = sde(DriftRate, DiffusionRate)

SDE = sde(DriftRate, DiffusionRate, 'Name1', Value1, 'Name2', Value2, ...)

Class

Description

This constructor creates and displays SDE objects. Use SDE objects to simulate sample paths of NVARS state variables driven by NBROWNS Brownian motion sources of risk over NPERIODS consecutive observation periods, approximating continuous-time stochastic processes.

This method enables you to simulate any vector-valued SDE of the form:

(11-10)

where:

X_t is an NVARS-by-1 state vector of process variables.
dW_t is an NBROWNS-by-1 Brownian motion vector.
F is an NVARS-by-1 vector-valued drift-rate function.
G is an NVARS-by-NBROWNS matrix-valued diffusion-rate function.

Input Arguments

DriftRate

User-defined drift-rate function, denoted by F. DriftRate is a function that returns an NVARS-by-1 drift-rate vector when called with two inputs:

A real-valued scalar observation time t.
An NVARS-by-1 state vector X_t.

Alternatively, DriftRate may also be an object of class Drift that encapsulates the drift-rate specification. In this case, however, sde uses only the Rate parameter of the object; it uses no other class information.

DiffusionRate

User-defined diffusion-rate function, denoted by G. DiffusionRate is a function that returns an NVARS-by-NBROWNS diffusion-rate matrix when called with two inputs:

A real-valued scalar observation time t.
An NVARS-by-1 state vector X_t.

Alternatively, DiffusionRate may also be an object of class Diffusion that encapsulates the diffusion-rate specification. In this case, however, sde uses only the Rate parameter of the object; it uses no other class information.

Optional Input Arguments

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, `sde` applies the same initial value to all state variables on all trials. If `StartState` is a column vector, `sde` applies a unique initial value to each state variable on all trials. If `StartState` is a matrix, `sde` 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`).

Output Arguments

SDE

Object of class sde with the following parameters:

StartTime: Initial observation time
StartState: Initial state at time StartTime
Correlation: Access function for the Correlation input argument, callable as a function of time
Drift: Composite drift-rate function, callable as a function of time and state
Diffusion: Composite diffusion-rate function, callable as a function of time and state
Simulation: A simulation function or method

Remarks

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, sde treats it as a static function of time and state, thereby guaranteeing that all parameters are accessible by the same interface.