# drift

Drift-rate model component

## Description

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

The drift-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 drift-rate specification can be any NVars-by-1 vector-valued function F of the general form:

$F\left(t,{X}_{t}\right)=A\left(t\right)+B\left(t\right){X}_{t}$

where:

• A is an NVars-by-1 vector-valued function accessible using the (t, Xt) interface.

• B is an NVars-by-NVars matrix-valued function accessible using the (t, Xt) interface.

And a drift-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.

• A and B are model parameters.

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

## Creation

### Description

example

DriftRate = drift(A,B) creates a default DriftRate 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 drift object that you create encapsulates the composite drift-rate specification and returns the following displayed parameters:

• Rate — The drift-rate function, F. Rate is the drift-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 drift-rate vector.

• A — Access function for the input argument A.

• B — Access function for the input argument B.

### Input Arguments

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

If you specify A as an array, it must be an NVars-by-1 column vector of intercepts.

As a deterministic function of time, when A is called with a real-valued scalar time t as its only input, A must produce an NVars-by-1 column vector. If you specify A as a function of time and state, it must generate an NVars-by-1 column vector of intercepts when invoked with two inputs:

• A real-valued scalar observation time t.

• An NVars-by-1 state vector Xt.

Data Types: double | function_handle

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

If you specify B as an array, it must be an NVars-by-NVars two-dimensional matrix of state vector coefficients.

As a deterministic function of time, when B is called with a real-valued scalar time t as its only input, B must produce an NVars-by-NVars matrix. If you specify B as a function of time and state, it must generate an NVars-by-NVars matrix of state vector coefficients when invoked with two inputs:

• A real-valued scalar observation time t.

• An NVars-by-1 state vector Xt.

Data Types: double | function_handle

## Properties

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Composite drift-rate function, specified as F(t,Xt). The function stored in Rate fully encapsulates the combined effect of A and B, where A and B are:

• A: The intercept term, A(t,Xt), of F(t,Xt)

• B: The first order term, B(t,Xt), of F(t,Xt)

Data Types: struct | double

## Examples

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Create a drift-rate function F:

F = drift(0, 0.1)   % Drift rate function F(t,X)
F =
Class DRIFT: Drift Rate Specification
-------------------------------------
Rate: drift rate function F(t,X(t))
A: 0
B: 0.1

The drift 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 drift class creates a model component rather than a model. F does not contain enough information to characterize the dimensionality of a problem.

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## Algorithms

When you specify the input arguments A and B as MATLAB arrays, they are associated with a linear drift parametric form. By contrast, when you specify either A or B as a function, you can customize virtually any drift-rate specification.

Accessing the output drift-rate parameters A and B with no inputs simply returns the original input specification. Thus, when you invoke drift-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 drift-rate parameters with inputs, they behave like functions, giving the impression of dynamic behavior. The parameters A and B accept the observation time t and a state vector Xt, and return an array of appropriate dimension. Specifically, parameters A and B evaluate the corresponding drift-rate component. Even if you originally specified an input as an array, drift 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] Aït-Sahalia, Yacine. “Testing Continuous-Time Models of the Spot Interest Rate.” Review of Financial Studies, vol. 9, no. 2, Apr. 1996, pp. 385–426.

[2] Aït-Sahalia, Yacine. “Transition Densities for Interest Rate and Other Nonlinear Diffusions.” The Journal of Finance, vol. 54, no. 4, Aug. 1999, pp. 1361–95.

[3] Glasserman, Paul. Monte Carlo Methods in Financial Engineering. Springer, 2004.

[4] Hull, John. Options, Futures and Other Derivatives. 7th ed, Prentice Hall, 2009.

[5] Johnson, Norman Lloyd, et al. Continuous Univariate Distributions. 2nd ed, Wiley, 1994.

[6] Shreve, Steven E. Stochastic Calculus for Finance. Springer, 2004.