# interpolate

Brownian interpolation of stochastic differential equations (SDEs) for
`SDE`

, `BM`

, `GBM`

,
`CEV`

, `CIR`

, `HWV`

,
`Heston`

, `SDEDDO`

, `SDELD`

, or
`SDEMRD`

models

## Description

## Examples

## Input Arguments

## Output Arguments

## Algorithms

This function performs a Brownian interpolation into a user-specified time series array, based on a piecewise-constant Euler sampling approach.

Consider a vector-valued SDE of the form:

$$d{X}_{t}=F(t,{X}_{t})dt+G(t,{X}_{t})d{W}_{t}$$

where:

*X*is an*NVars*-by-`1`

state vector.*F*is an*NVars*-by-`1`

drift-rate vector-valued function.*G*is an*NVars*-by-*NBrowns*diffusion-rate matrix-valued function.*W*is an*NBrowns*-by-`1`

Brownian motion vector.

Given a user-specified time series array associated with this equation, this function
performs a Brownian (stochastic) interpolation by sampling from a conditional Gaussian
distribution. This sampling technique is sometimes called a *Brownian
bridge*.

**Note**

Unlike simulation methods, the `interpolation`

function does not
support user-specified noise processes.

The

`interpolate`

function assumes that all model parameters are piecewise-constant, and evaluates them from the most recent observation time in`Times`

that precedes a specified interpolation time in`T`

. This is consistent with the Euler approach of Monte Carlo simulation.When an interpolation time falls outside the interval specified by

`Times`

, a Euler simulation extrapolates the time series by using the nearest available observation.The user-defined time series

`Paths`

and corresponding observation`Times`

must be fully observed (no missing observations denoted by`NaN`

s).The

`interpolate`

function assumes that the user-specified time series array`Paths`

is associated with the`sde`

object. For example, the`Times`

and`Paths`

input pair are the result of an initial course-grained simulation. However, the interpolation ignores the initial conditions of the`sde`

object (`StartTime`

and`StartState`

), allowing the user-specified`Times`

and`Paths`

input series to take precedence.

## 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.

## Version History

**Introduced in R2008a**