The base SDE
class:
$$d{X}_{t}=F(t,{X}_{t})dt+G(t,{X}_{t})d{W}_{t}$$
represents the most general model.
Tip
The |
Constructing an SDE
object requires the following
inputs:
A drift-rate function F
.
This function returns an NVARS
-by-1 drift-rate
vector when run with the following inputs:
A real-valued scalar observation time t.
An NVARS
-by-1
state
vector X_{t}.
A diffusion-rate function G
. This
function returns an NVARS
-by-NBROWNS
diffusion-rate
matrix when run with the inputs t and X_{t}.
Evaluating object parameters by passing (t, X_{t}) to a common, published interface allows most parameters to be referenced by a common input argument list that reinforces common method programming. You can use this simple function evaluation approach to model or construct powerful analytics, as in the following example.
Construct an SDE
object obj
to
represent a univariate geometric Brownian Motion model of the form:
$$d{X}_{t}=0.1{X}_{t}dt+0.3{X}_{t}d{W}_{t}$$
Create drift and diffusion functions that are accessible by the common (t,X_{t}) interface:
F = @(t,X) 0.1 * X; G = @(t,X) 0.3 * X;
Pass the functions to the SDE
constructor
to create an object obj
of class SDE
:
obj = sde(F, G) % dX = F(t,X)dt + G(t,X)dW
obj = Class SDE: Stochastic Differential Equation ------------------------------------------- Dimensions: State = 1, Brownian = 1 ------------------------------------------- StartTime: 0 StartState: 1 Correlation: 1 Drift: drift rate function F(t,X(t)) Diffusion: diffusion rate function G(t,X(t)) Simulation: simulation method/function simByEuler
obj
displays like a MATLAB^{®} structure,
with the following information:
The object's class
A brief description of the object
A summary of the dimensionality of the model
The object's displayed parameters are as follows:
StartTime
: The initial observation
time (real-valued scalar)
StartState
: The initial state vector
(NVARS
-by-1 column vector)
Correlation
: The correlation structure
between Brownian process
Drift
: The drift-rate function F(t,X_{t})
Diffusion
: The diffusion-rate
function G(t,X_{t})
Simulation
: The simulation method
or function.
Of these displayed parameters, only Drift
and Diffusion
are
required inputs.
The only exception to the (t, X_{t})
evaluation interface is Correlation
. Specifically,
when you enter Correlation
as a function, the SDE
engine assumes that it is a deterministic function of time, C(t).
This restriction on Correlation
as a deterministic
function of time allows Cholesky factors to be computed and stored
before the formal simulation. This inconsistency dramatically improves
run-time performance for dynamic correlation structures. If Correlation
is
stochastic, you can also include it within the simulation architecture
as part of a more general random number generation function.
bm
| cev
| cir
| diffusion
| drift
| gbm
| heston
| hwv
| interpolate
| sde
| sdeddo
| sdeld
| sdemrd
| simByEuler
| simBySolution
| simulate
| ts2func