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**Superclasses: **

Diffusion-rate model component

The `diffusion`

constructor
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:

$$G(t,{X}_{t})=D(t,{X}_{t}^{\alpha (t)})V(t)$$ | (18-1) |

`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:

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

*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)
```

constructs a default `diffusion`

object.

For more information on constructing a `diffusion`

object,
see `drift`

.

The following figure illustrates the inheritance relationships among SDE classes.

For more information, see SDE Class Hierarchy.

Value. To learn how value classes affect copy operations, see Copying Objects (MATLAB).

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.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.

Ait-Sahalia, 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, Springer-Verlag, 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:
Continuous-Time Models.* New York: Springer-Verlag, 2004.

- Simulating Equity Prices
- Simulating Interest Rates
- Stratified Sampling
- Pricing American Basket Options by Monte Carlo Simulation
- Base SDE Models
- Drift and Diffusion Models
- Linear Drift Models
- Parametric Models
- Class Attributes (MATLAB)
- Property Attributes (MATLAB)
- SDEs
- SDE Models
- SDE Class Hierarchy
- Performance Considerations

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