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# 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

### Syntax

``DriftRate = drift(A,B)``

### 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. NoteYou 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] 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.