# Documentation

### This is machine translation

Translated by
Mouseover text to see original. Click the button below to return to the English verison of the page.

Note: This page has been translated by MathWorks. Please click here
To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

# sde

Construct SDE model from user-specified functions

## Synopsis

`SDE = sde(DriftRate, DiffusionRate)`

```SDE = sde(DriftRate, DiffusionRate, 'Name1', Value1, 'Name2', Value2, ...)```

`sde`

## Description

This constructor creates and displays general stochastic differential equation (`SDE`) models from user-defined drift and diffusion rate functions. Use `sde` objects 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.

This constructor enables you to simulate any vector-valued SDE of the form:

 $d{X}_{t}=F\left(t,{X}_{t}\right)dt+G\left(t,{X}_{t}\right)d{W}_{t}$ (18-5)

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.

## Input Arguments

 `DriftRate` User-defined drift-rate function, denoted by F. `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` may 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. `DiffusionRate` User-defined diffusion-rate function, denoted by G. `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` may 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.

## Optional Input Arguments

Specify optional inputs as matching parameter name/value pairs as follows:

• Specify the parameter name as a character vector, followed by its corresponding value.

• You can specify parameter name/value pairs in any order.

• Parameter names are case insensitive.

• You can specify unambiguous partial character vector matches.

Valid parameter names are:

 `StartTime` Scalar starting time of the first observation, applied to all state variables. If you do not specify a value for `StartTime`, the default is `0`. `StartState` Scalar, `NVARS`-by-`1` column vector, or `NVARS`-by-`NTRIALS` matrix of initial values of the state variables. If `StartState` is a scalar, `sde` applies the same initial value to all state variables on all trials.If `StartState` is a column vector, `sde` applies a unique initial value to each state variable on all trials.If `StartState` is a matrix, `sde` applies a unique initial value to each state variable on each trial. If you do not specify a value for `StartState`, all variables start at `1`. `Correlation` Correlation between Gaussian random variates drawn to generate the Brownian motion vector (Wiener processes). Specify `Correlation` as an `NBROWNS`-by-`NBROWNS` positive semidefinite matrix, or as a deterministic function C(t) that accepts the current time t and returns an `NBROWNS`-by-`NBROWNS` positive semidefinite correlation matrix. A `Correlation` matrix represents a static condition.As a deterministic function of time, `Correlation` allows you to specify a dynamic correlation structure.If you do not specify a value for `Correlation`, the default is an `NBROWNS`-by-`NBROWNS` identity matrix representing independent Gaussian processes. `Simulation` A user-defined simulation function or SDE simulation method. If you do not specify a value for `Simulation`, the default method is simulation by Euler approximation (`simByEuler`).

## Output Arguments

 `SDE` Stochastic differential equation model (`SDE`) with the following parameters: `StartTime`: Initial observation time`StartState`: Initial state at time `StartTime``Correlation`: Access function for the `Correlation` input argument, callable as a function of time `Drift`: Composite drift-rate function, callable as a function of time and state `Diffusion`: Composite diffusion-rate function, callable as a function of time and state `Simulation`: A simulation function or method

## 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, `sde` 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.

#### Introduced in R2008a

Was this topic helpful?

#### Financial Risk Management: Improving Model Governance with MATLAB

Download the white paper