Construct stochastic differential equation from linear driftrate 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 driftrate 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 driftrate
form. They provide a parametric alternative to the meanreverting
drift form (see sdemrd
).
These state variables are driven by NBROWNS
Brownian
motion sources of risk over NPERIODS
consecutive
observation periods, approximating continuoustime stochastic processes
with linear driftrate functions.
This method allows you to simulate any vectorvalued SDE of the form:
$$d{X}_{t}=(A(t)+B(t){X}_{t})dt+D(t,{X}_{t}^{\alpha (t)})V(t)d{W}_{t}$$  (1810) 
where:
X_{t} is an NVARS
by1
state vector of process variables.
A is an NVARS
by1
vector.
B is an NVARS
byNVARS
matrix.
D is an NVARS
byNVARS
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
byNBROWNS
instantaneous
volatility rate matrix.
dW_{t} is an NBROWNS
by1
Brownian motion vector.
Specify required input parameters as one of the following types:
A MATLAB^{®} array. Specifying an array indicates a static (nontimevarying) 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 
The required input parameters are:
A  A represents the parameter A.
If you specify A as an array, it must be an NVARS by1 column
vector of intercepts. As a deterministic function of time, when A is
called with a realvalued scalar time t as its
only input, A must produce an NVARS by1 column
vector. If you specify A as a function of time
and state, it must generate an NVARS by1 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 byNVARS matrix
of state vector coefficients. As a deterministic function of time,
when B is called with a realvalued scalar time t as
its only input, B must produce an NVARS byNVARS matrix.
If you specify B as a function of time and state,
it must generate an NVARS byNVARS 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 by1 column vector
of exponents. As a deterministic function of time, when Alpha is
called with a realvalued scalar time t as its
only input, Alpha must produce an NVARS by1 column
vector. If you specify it as a function of time and state, it must
return an NVARS by1 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 byNBROWNS 2dimensional
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 realvalued
scalar time t as its only input, Sigma must
produce an NVARS byNBROWNS matrix.
If you specify it as a function of time and state, it must generate
an NVARS byNBROWNS matrix of
volatility rates when invoked with two inputs:

Note:
Although the constructor does not enforce restrictions on the
signs of 
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 by1 column vector, or NVARS byNTRIALS matrix
of initial values of the state variables. If If If If you do not specify
a value for 
Correlation  Correlation between Gaussian random variates drawn to generate
the Brownian motion vector (Wiener processes). Specify Correlation as
an NBROWNS byNBROWNS positive
semidefinite matrix, or as a deterministic function C(t) that
accepts the current time t and returns an NBROWNS byNBROWNS positive
semidefinite correlation matrix. A As a deterministic function
of time, If you do not specify a value for 
Simulation  A userdefined simulation function or SDE simulation method.
If you do not specify a value for Simulation , the
default method is simulation by Euler approximation (simByEuler ). 
SDE  Object of class

AitSahalia, Y., "Testing ContinuousTime Models of the Spot Interest Rate," The Review of Financial Studies, Spring 1996, Vol. 9, No. 2, pp. 385–426.
AitSahalia, Y., "Transition Densities for Interest Rate and Other Nonlinear Diffusions," The Journal of Finance, Vol. 54, No. 4, August 1999.
Glasserman, P., Monte Carlo Methods in Financial Engineering, New York: SpringerVerlag, 2004.
Hull, J. C., Options, Futures, and Other Derivatives, 5th ed. Englewood Cliffs, NJ: Prentice Hall, 2002.
Johnson, N. L., S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, Vol. 2, 2nd ed. New York: John Wiley & Sons, 1995.
Shreve, S. E., Stochastic Calculus for Finance II: ContinuousTime Models, New York: SpringerVerlag, 2004.