# Documentation

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# sdeddo class

Superclasses:

Stochastic Differential Equation (SDE) model from Drift and Diffusion components

## Description

The sdeddo constructor creates and displays sdeddo objects, instantiated with objects of class drift and diffusion. These restricted sdeddo objects contain the input drift and diffusion objects; therefore, you can directly access their displayed parameters.

This abstraction also generalizes the notion of drift and diffusion-rate objects as functions that sdeddo evaluates for specific values of time t and state Xt. Likesde objects, sdeddo objects allow you to simulate 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 sdeddo object enables you to simulate any 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.

• F is an NVARS-by-1 vector-valued drift-rate function.

• G is an NVARS-by-NBROWNS matrix-valued diffusion-rate function.

## Construction

SDE = sdeddo(DriftRate,DiffusionRate) constructs a default sdeddo object.

SDE = sdeddo(DriftRate,DiffusionRate,Name,Value) constructs a sdeddo object with additional options specified by one or more Name,Value pair arguments.

Name is a property name and Value is its corresponding value. Name must appear inside single quotes (''). You can specify several name-value pair arguments in any order as Name1,Value1,…,NameN,ValueN.

### Input Arguments

expand all

DriftRate is a user-defined drift-rate function and represents the parameter F, specified as a vector or object of class drift.

DriftRate is a function that returns an NVARS-by-1 drift-rate vector when called with two inputs:

• A real-valued scalar observation time t.

• An NVARS-by-1 state vector Xt.

Alternatively, DriftRate can also be an object of class drift that encapsulates the drift-rate specification. In this case, however, sde uses only the Rate parameter of the object. For more information on the drift object, see drift.

Data Types: double

DiffusionRate is a user-defined drift-rate function and represents the parameter G, specified as a matrix or object of class diffusion.

DiffusionRate is a function that returns an NVARS-by-NBROWNS diffusion-rate matrix when called with two inputs:

• A real-valued scalar observation time t.

• An NVARS-by-1 state vector Xt.

Alternatively, DiffusionRate can also be an object of class diffusion that encapsulates the diffusion-rate specification. In this case, however, sde uses only the Rate parameter of the object. For more information on the diffusion object, see diffusion.

Data Types: double

#### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

## Properties

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Drift rate component of continuous-time stochastic differential equations (SDEs), specified as a drift object or function accessible by (t, Xt.

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 class allows you to create drift-rate objects (using thedrift constructor) of the 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.

The drift object's displayed parameters are:

• Rate: The drift-rate function, F(t,Xt)

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

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

A and B enable you to query the original inputs. The function stored in Rate fully encapsulates the combined effect of A and B.

When specified as MATLAB® double arrays, the inputs A and B are clearly associated with a linear drift rate parametric form. However, specifying either A or B as a function allows you to customize virtually any drift rate specification.

### Note

You can express drift and diffusion classes in the most general form to emphasize the functional (t, Xt) interface. However, you can specify the components A and B as functions that adhere to the common (t, Xt) interface, or as MATLAB arrays of appropriate dimension.

Example: F = drift(0, 0.1) % Drift rate function F(t,X)

Attributes:

 SetAccess private GetAccess public

Data Types: struct | double

Diffusion rate component of continuous-time stochastic differential equations (SDEs), specified as a drift object or function accessible by (t, Xt.

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 class allows you to create diffusion-rate objects (using the constructor diffusion constructor):

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

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.

The diffusion object's displayed parameters are:

• Rate: The diffusion-rate function, G(t,Xt).

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

• Sigma: The volatility rate, V(t,Xt), of G(t,Xt).

Alpha and Sigma enable you to query the original inputs. (The combined effect of the individual Alpha and Sigma parameters is fully encapsulated by the function stored in Rate.) The Rate functions are the calculation engines for the drift and diffusion objects, and are the only parameters required for simulation.

### Note

You can express drift and diffusion classes in the most general form to emphasize the functional (t, Xt) interface. However, you can specify the components A and B as functions that adhere to the common (t, Xt) interface, or as MATLAB arrays of appropriate dimension.

Example: G = diffusion(1, 0.3) % Diffusion rate function G(t,X)

Attributes:

 SetAccess private GetAccess public

Data Types: struct | double

Starting time of first observation, applied to all state variables, specified as a scalar

Attributes:

 SetAccess public GetAccess public

Data Types: double

Initial values of state variables, specified as a scalar, column vector, or matrix.

If StartState is a scalar, the gbm constructor applies the same initial value to all state variables on all trials.

If StartState is a column vector, thegbm constructor applies a unique initial value to each state variable on all trials.

If StartState is a matrix, the gbm constructor applies a unique initial value to each state variable on each trial.

Attributes:

 SetAccess public GetAccess public

Data Types: double

User-defined simulation function or SDE simulation method, specified as a function or SDE simulation method.

Attributes:

 SetAccess public GetAccess public

Data Types: function_handle

## Inherited Methods

The following methods are inherited from the sde class.

interpolate

simulate

simByEuler

## Instance Hierarchy

The following figure illustrates the inheritance relationships among SDE classes.

## Copy Semantics

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

## Examples

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The sdeddo class derives from the base sde class. To use this class, you must pass drift and diffusion-rate objects to the sdeddo constructor. Create drift and diffusion rate objects:

F = drift(0, 0.1);      % Drift rate function F(t,X)
G = diffusion(1, 0.3);  % Diffusion rate function G(t,X)

Pass the functions to the sdeddo constructor to create an object obj of class sdeddo:

obj = sdeddo(F, G)      % dX = F(t,X)dt + G(t,X)dW
obj =
Class SDEDDO: SDE from Drift and Diffusion Objects
--------------------------------------------------
Dimensions: State = 1, Brownian = 1
--------------------------------------------------
StartTime: 0
StartState: 1
Correlation: 1
Drift: drift rate function F(t,X(t))
Diffusion: diffusion rate function G(t,X(t))
Simulation: simulation method/function simByEuler
A: 0
B: 0.1
Alpha: 1
Sigma: 0.3

In this example, the object displays the additional parameters associated with input drift and diffusion objects.

## Algorithms

When you specify the required input parameters as arrays, they are associated with a specific parametric form. By contrast, when you specify either required input parameter as a function, you can customize virtually any specification.

Accessing the output parameters with no inputs simply returns the original input specification. Thus, when you invoke these 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 these parameters with inputs, they behave like functions, giving the impression of dynamic behavior. The parameters accept the observation time t and a state vector Xt, and return an array of appropriate dimension. Even if you originally specified an input as an array, sdeddo 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.