Documentation

# diffusion

Diffusion-rate model component

## Description

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:

 $G\left(t,{X}_{t}\right)=D\left(t,{X}_{t}^{\alpha \left(t\right)}\right)V\left(t\right)$ (1)
where:

• 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, Xt) interface.

And a diffusion-rate specification is 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.

## Creation

### Description

example

DiffusionRate = diffusion(Alpha,Sigma) creates default 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.

### 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 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 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.

### Input Arguments

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Alpha represents the parameter D, specified as an array or deterministic function of time.

If you specify Alpha as an array, it represents 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 matrix.

If you specify it as a function of time and state, Alpha 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.

Data Types: double | function_handle

Sigma represents the parameter V, specified as an array or a deterministic function of time.

If you specify Sigma as an array, it must be an NVARS-by-NBROWNS two-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.

Data Types: double | function_handle

### Note

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

## Properties

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This property is read-only.

Composite diffusion-rate function, specified as: G(t,Xt)). The function stored in Rate fully encapsulates the combined effect of Alpha and Sigma where:

• Alpha is the state vector exponent, which determines the format of D(t,Xt) of G(t,Xt).

• Sigma is the volatility rate, V(t,Xt), of G(t,Xt).

Data Types: struct | double

## Examples

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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.

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## 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

[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.