# Documentation

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# diffusion

Construct diffusion-rate model components

## Synopsis

`DiffusionRate = diffusion(Alpha, Sigma)`

## Class

`diffusion`

## Description

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:

 $G\left(t,{X}_{t}\right)=D\left(t,{X}_{t}^{\alpha \left(t\right)}\right)V\left(t\right)$ (18-2)
associated with a vector-valued SDE of the form:
`$d{X}_{t}=F\left(t,{X}_{t}\right)dt+G\left(t,{X}_{t}\right)d{W}_{t}$`
where:

• Xt is an `NVARS`-by-`1` state vector of process variables.

• dWt is an `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.

## Input Arguments

Specify required input parameters 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.

### 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 `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 required input parameters are:

 `Alpha` `Alpha` determines the format of the parameter D. If you specify `Alpha` as an array, it must be an `NVARS`-by-`1` column vector of exponents. As a deterministic function of time, when `Alpha` is called with a real-valued scalar time `t` as its only input, `Alpha` must produce an `NVARS`-by-`1` column vector. If you specify `Alpha` as a function of time and state, it must return an `NVARS`-by-`1` column vector of exponents when invoked with two inputs: A real-valued scalar observation time t.An `NVARS`-by-`1` state vector Xt. `Sigma` `Sigma` represents the parameter V. If you specify `Sigma` as an array, it must be an `NVARS`-by-`NBROWNS` 2-dimensional matrix of instantaneous volatility rates. In this case, each row of `Sigma` corresponds to a particular state variable. Each column 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 real-valued scalar time `t` as its only input, `Sigma` must produce an `NVARS`-by-`NBROWNS` matrix. If you specify `Sigma` as a function of time and state, it must return an `NVARS`-by-`NBROWNS` matrix of volatility rates when invoked with two inputs: A real-valued scalar observation time t.An `NVARS`-by-`1` state vector Xt.

### Note

Although the `diffusion` constructor enforces no restrictions on the signs of these volatility parameters, each parameter is usually specified as a positive value.

## Output Arguments

 `DiffusionRate` Object of class `diffusion` that encapsulates the composite diffusion-rate specification, with 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 Xt as inputs, and returns an `NVARS`-by-`1` diffusion-rate vector.`Alpha`: Access function for the input argument `Alpha`.`Sigma`: Access function for the input argument `Sigma`.

## Examples

collapse all

Create a diffusion-rate function `G`:

`G = diffusion(1, 0.3) % Diffusion rate function G(t,X)`
```G = Class DIFFUSION: Diffusion Rate Specification --------------------------------------------- Rate: diffusion rate function G(t,X(t)) Alpha: 1 Sigma: 0.3 ```

The `diffusion` object displays like a MATLAB® structure and contains supplemental information, namely, the object's class and a brief description. However, in contrast to the SDE representation, a summary of the dimensionality of the model does not appear, because the `diffusion` class creates a model component rather than a model. `G` does not contain enough information to characterize the dimensionality of a problem.

## Algorithms

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 Xt, 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.

## References

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.