Diffusion-rate model component

The `diffusion`

object specifies the diffusion-rate component
of continuous-time stochastic differential equations (SDEs).

The diffusion-rate specification supports the simulation of sample paths of
`NVARS`

state variables driven by `NBROWNS`

Brownian motion sources of risk over `NPERIODS`

consecutive observation
periods, approximating continuous-time stochastic processes.

The diffusion-rate specification can be any
`NVARS`

-by-`NBROWNS`

matrix-valued function
*G* of the general form:

$$G(t,{X}_{t})=D(t,{X}_{t}^{\alpha (t)})V(t)$$ | (1) |

`D`

is an`NVARS`

-by-`NVARS`

diagonal matrix-valued function.Each diagonal element of

`D`

is the corresponding element of the state vector raised to the corresponding element of an exponent`Alpha`

, which is an`NVARS`

-by-`1`

vector-valued function.`V`

is an`NVARS`

-by-`NBROWNS`

matrix-valued volatility rate function`Sigma`

.`Alpha`

and`Sigma`

are also accessible using the (*t*,*X*) interface._{t}

And a diffusion-rate specification is associated with a vector-valued SDE of the form:

$$d{X}_{t}=F(t,{X}_{t})dt+G(t,{X}_{t})d{W}_{t}$$

where:

*X*is an_{t}`NVARS`

-by-`1`

state vector of process variables.*dW*is an_{t}`NBROWNS`

-by-`1`

Brownian motion vector.*D*is an`NVARS`

-by-`NVARS`

diagonal matrix, in which 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`

matrix-valued volatility rate function`Sigma`

.

The diffusion-rate specification is flexible, and provides direct parametric support for static volatilities and state vector exponents. It is also extensible, and provides indirect support for dynamic/nonlinear models via an interface. This enables you to specify virtually any diffusion-rate specification.

`DiffusionRate = diffusion(Alpha,Sigma)`

creates default `DiffusionRate`

= diffusion(`Alpha`

,`Sigma`

)`DiffusionRate`

model component.

Specify required input parameters `A`

and
`B`

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.

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 `diffusion`

object that you create encapsulates the
composite drift-rate specification and returns the following displayed parameters:

`Rate`

— The diffusion-rate function,*G*.`Rate`

is the diffusion-rate calculation engine. It accepts the current time*t*and an`NVARS`

-by-`1`

state vector*X*as inputs, and returns an_{t}`NVARS`

-by-`1`

diffusion-rate vector.`Alpha`

— Access function for the input argument`Alpha`

.`Sigma`

— Access function for the input argument`Sigma`

.

When you specify the input arguments `Alpha`

and
`Sigma`

as MATLAB arrays, they are associated with a specific parametric form. By contrast,
when you specify either `Alpha`

or `Sigma`

as a
function, you can customize virtually any diffusion-rate specification.

Accessing the output diffusion-rate parameters `Alpha`

and
`Sigma`

with no inputs simply returns the original input
specification. Thus, when you invoke diffusion-rate 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 diffusion-rate parameters with inputs, they behave like functions,
giving the impression of dynamic behavior. The parameters `Alpha`

and
`Sigma`

accept the observation time *t* and a
state vector *X _{t}*, and return an array of
appropriate dimension. Specifically, parameters

`Alpha`

and
`Sigma`

evaluate the corresponding diffusion-rate component. Even
if you originally specified an input as an array, `diffusion`

treats it
as a static function of time and state, by that means guaranteeing that all parameters
are accessible by the same interface.[1] Ait-Sahalia, Y. “Testing Continuous-Time Models of the Spot Interest
Rate.” *The Review of Financial Studies*, Spring 1996, Vol.
9, No. 2, pp. 385–426.

[2] Ait-Sahalia, Y. “Transition Densities for Interest Rate and Other
Nonlinear Diffusions.” *The Journal of Finance*, Vol. 54,
No. 4, August 1999.

[3] Glasserman, P. *Monte Carlo Methods in Financial
Engineering.* New York, Springer-Verlag, 2004.

[4] Hull, J. C. *Options, Futures, and Other Derivatives*, 5th
ed. Englewood Cliffs, NJ: Prentice Hall, 2002.

[5] Johnson, N. L., S. Kotz, and N. Balakrishnan. *Continuous Univariate
Distributions.* Vol. 2, 2nd ed. New York, John Wiley & Sons,
1995.

[6] Shreve, S. E. *Stochastic Calculus for Finance II: Continuous-Time
Models.* New York: Springer-Verlag, 2004.