# Documentation

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# 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 timeStartState: Initial state at time StartTimeCorrelation: 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.