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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. Like`sde` 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`.

For more information on constructing a `sdeddo` object, see `sdeddo`.

### Input Arguments

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`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`.

For more information on using optional name-value arguments, see `sdeddo`.

## 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 the`drift` 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, the`gbm` 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.