# simByMilstein

Simulate diagonal diffusion for `BM`

, `GBM`

,
`CEV`

, `HWV`

, `SDEDDO`

,
`SDELD`

, or `SDEMRD`

sample paths by Milstein
approximation

*Since R2023a*

## Description

`[`

simulates `Paths`

,`Times`

,`Z`

] = simByMilstein(`MDL`

,`NPeriods`

)`NTrials`

sample paths of `NVARS`

state
variables driven by the `BM`

, `GBM`

,
`CEV`

, `HWV`

, `SDEDDO`

,
`SDELD`

, or `SDEMRD`

process sources of risk
over `NPeriods`

consecutive observation periods, approximating
continuous-time by the Milstein approximation.

`simByMilstein`

provides a discrete-time approximation of the
underlying generalized continuous-time process. The simulation is derived directly
from the stochastic differential equation of motion; the discrete-time process
approaches the true continuous-time process only in the limit as
`DeltaTime`

approaches zero.

`[`

specifies options using one or more name-value pair arguments in addition to the
input arguments in the previous syntax.`Paths`

,`Times`

,`Z`

] = simByMilstein(___,`Name=Value`

)

You can perform quasi-Monte Carlo simulations using the name-value arguments for
`MonteCarloMethod`

, `QuasiSequence`

, and
`BrownianMotionMethod`

. For more information, see Quasi-Monte Carlo Simulation.

## Examples

### Quasi-Monte Carlo Simulation with Milstein Scheme Using GBM Model

This example shows how to use `simByMilstein`

with a GBM model to perform a quasi-Monte Carlo simulation. Quasi-Monte Carlo simulation is a Monte Carlo simulation that uses quasi-random sequences instead of pseudo random numbers.

Create a univariate`gbm`

object to represent the model: $d{X}_{t}=0.25{X}_{t}dt+0.3{X}_{t}d{W}_{t}$.

`GBM_obj = gbm(0.25, 0.3) % (B = Return, Sigma) `

GBM_obj = Class GBM: Generalized Geometric Brownian Motion ------------------------------------------------ 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 Return: 0.25 Sigma: 0.3

`gbm`

objects display the parameter `B`

as the more familiar `Return`

.

Perform a quasi-Monte Carlo simulation by using `simByMilstein`

with the optional name-value arguments for `MonteCarloMethod`

, `QuasiSequence`

, and `BrownianMotionMethod`

.

[paths,time,z] = simByMilstein(GBM_obj,10,ntrials=4096,montecarlomethod="quasi",quasisequence="sobol",BrownianMotionMethod="brownian-bridge");

## Input Arguments

`MDL`

— Stochastic differential equation model

`BM`

object | `GBM`

object | `CEV`

object | `HWV`

object | `SDEDDO`

object | `SDELD`

object | `SDEMRD`

object

`NPeriods`

— Number of simulation periods

positive scalar integer

Number of simulation periods, specified as a positive scalar integer. The
value of `NPeriods`

determines the number of rows of the
simulated output series.

**Data Types: **`double`

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

**Example: **```
[Paths,Times,Z] =
simByMilstein(GBM_obj,NPeriods,DeltaTime=dt,Method="reflection")
```

`Method`

— Method to handle negative values

`'basic'`

(default) | character vector with values`'basic'`

,
`'absorption'`

, `'reflection'`

,
`'partial-truncation'`

,
`'full-truncation'`

, or
`'higham-mao'`

| string with values `"basic"`

,
`"absorption"`

, `"reflection"`

,
`"partial-truncation"`

,
`"full-truncation"`

, or
`"higham-mao"`

Method to handle negative values, specified as
`Method`

and a character vector or string with a
supported value.

**Data Types: **`char`

| `string`

`NTrials`

— Simulated trials (sample paths) of `NPeriods`

observations each

`1`

(single path of correlated state
variables) (default) | positive scalar integer

Simulated trials (sample paths) of `NPeriods`

observations each, specified as `NTrials`

and a
positive scalar integer.

**Data Types: **`double`

`DeltaTime`

— Positive time increments between observations

`1`

(default) | scalar | column vector

Positive time increments between observations, specified as the
comma-separated pair consisting of `'DeltaTime'`

and a
scalar or a `NPeriods`

-by-`1`

column
vector.

`DeltaTime`

represents the familiar
*dt* found in stochastic differential equations,
and determines the times at which the simulated paths of the output
state variables are reported.

**Data Types: **`double`

`NSteps`

— Number of intermediate time steps within each time increment *dt* (specified as
`DeltaTime`

)

`1`

(indicating no intermediate
evaluation) (default) | positive scalar integer

Number of intermediate time steps within each time increment
*dt* (specified as `DeltaTime`

),
specified as `NSteps`

and a positive scalar
integer.

The `simByMilestein`

function partitions each time
increment *dt* into `NSteps`

subintervals of length *dt*/`NSteps`

,
and refines the simulation by evaluating the simulated state vector at
`NSteps − 1`

intermediate points. Although
`simByMilstein`

does not report the output state
vector at these intermediate points, the refinement improves accuracy by
allowing the simulation to more closely approximate the underlying
continuous-time process.

**Data Types: **`double`

`Antithetic`

— Flag to indicate whether `simByMilstein`

uses antithetic sampling to generate Gaussian random variates

`False`

(no antithetic
sampling) (default) | logical with values `True`

or
`False`

Flag that indicates whether `simByMilstein`

uses
antithetic sampling to generate the Gaussian random variates that drive
the Brownian motion vector (Wiener processes). This argument is
specified as `Antithetic`

and a scalar logical flag
with a value of `True`

or
`False`

.

When you specify `True`

,
`simByMilstein`

performs sampling such that all
primary and antithetic paths are simulated and stored in successive
matching pairs:

Odd trials

`(1,3,5,...)`

correspond to the primary Gaussian paths.Even trials

`(2,4,6,...)`

are the matching antithetic paths of each pair derived by negating the Gaussian draws of the corresponding primary (odd) trial.

**Note**

If you specify an input noise process (see
`Z`

), `simByMilstein`

ignores
the value of `Antithetic`

.

**Data Types: **`logical`

`Z`

— Direct specification of the dependent random noise process used to generate Brownian motion vector

generates correlated Gaussian variates based on the
`Correlation`

member of the `SDE`

object (default) | function | three-dimensional array of dependent random variates

Direct specification of the dependent random noise process used to
generate the Brownian motion vector (Wiener process) that drives the
simulation. This argument is specified as `Z`

and a
function or as an ```
(NPeriods ⨉
NSteps)
```

-by-`NBrowns`

-by-`NTrials`

three-dimensional array of dependent random variates.

**Note**

If you specify `Z`

as a function, it must return
an `NBrowns`

-by-`1`

column vector,
and you must call it with two inputs:

A real-valued scalar observation time

*t*.An

`NVars`

-by-`1`

state vector*X*._{t}

**Data Types: **`double`

| `function`

`StorePaths`

— Flag that indicates how the output array `Paths`

is stored and returned

`True`

(default) | logical with values `True`

or
`False`

Flag that indicates how the output array `Paths`

is
stored and returned, specified as `StorePaths`

and a
scalar logical flag with a value of `True`

or
`False`

.

If

`StorePaths`

is`True`

(the default value) or is unspecified,`simByMilstein`

returns`Paths`

as a three-dimensional time series array.If

`StorePaths`

is`False`

(logical`0`

),`simByMilstein`

returns the`Paths`

output array as an empty matrix.

**Data Types: **`logical`

`MonteCarloMethod`

— Monte Carlo method to simulate stochastic processes

`"standard"`

(default) | string with values `"standard"`

,
`"quasi"`

, or
`"randomized-quasi"`

| character vector with values `'standard'`

,
`'quasi'`

, or
`'randomized-quasi'`

Monte Carlo method to simulate stochastic processes, specified as
`MonteCarloMethod`

and a string or character vector
with one of the following values:

`"standard"`

— Monte Carlo using pseudo random numbers`"quasi"`

— Quasi-Monte Carlo using low-discrepancy sequences`"randomized-quasi"`

— Randomized quasi-Monte Carlo

**Note**

If you specify an input noise process (see
`Z`

), `simByMilstein`

ignores
the value of `MonteCarloMethod`

.

**Data Types: **`string`

| `char`

`QuasiSequence`

— Low discrepancy sequence to drive stochastic processes

`"sobol"`

(default) | string with value `"sobol"`

| character vector with value `'sobol'`

Low discrepancy sequence to drive the stochastic processes, specified
as `QuasiSequence`

and a string or character vector
with the following value:

`"sobol"`

— Quasi-random low-discrepancy sequences that use a base of two to form successively finer uniform partitions of the unit interval and then reorder the coordinates in each dimension.

**Note**

If `MonteCarloMethod`

option is not specified
or specified as`"standard"`

,
`QuasiSequence`

is ignored.

If you specify an input noise process (see
`Z`

), `simByMilstein`

ignores
the value of `QuasiSequence`

.

**Data Types: **`string`

| `char`

`BrownianMotionMethod`

— Brownian motion construction method

`"standard"`

(default) | string with value `"brownian-bridge"`

or
`"principal-components"`

| character vector with value `'brownian-bridge'`

or
`'principal-components'`

Brownian motion construction method, specified as
`BrownianMotionMethod`

and a string or character
vector with one of the following values:

`"standard"`

— The Brownian motion path is found by taking the cumulative sum of the Gaussian variates.`"brownian-bridge"`

— The last step of the Brownian motion path is calculated first, followed by any order between steps until all steps have been determined.`"principal-components"`

— The Brownian motion path is calculated by minimizing the approximation error.

**Note**

If an input noise process is specified using the
`Z`

input argument,
`BrownianMotionMethod`

is ignored.

The starting point for a Monte Carlo simulation is the construction of a Brownian motion sample path (or Wiener path). Such paths are built from a set of independent Gaussian variates, using either standard discretization, Brownian-bridge construction, or principal components construction.

Both standard discretization and Brownian-bridge construction share
the same variance and, therefore, the same resulting convergence when
used with the `MonteCarloMethod`

using pseudo random
numbers. However, the performance differs between the two when the
`MonteCarloMethod`

option
`"quasi"`

is introduced, with faster convergence
seen for `"brownian-bridge"`

construction option and
the fastest convergence when using the
`"principal-components"`

construction
option.

**Data Types: **`string`

| `char`

`Processes`

— Sequence of end-of-period processes or state vector adjustments

`simByEuler`

makes no adjustments and
performs no processing (default) | function | cell array of functions

Sequence of end-of-period processes or state vector adjustments of the
form, specified as `Processes`

and a function or cell
array of functions of the form

$${X}_{t}=P(t,{X}_{t})$$

The `simByMilstein`

function runs processing
functions at each interpolation time. They must accept the current
interpolation time *t*, and the current state vector
*X _{t}*, and return a state
vector that may be an adjustment to the input state.

If you specify more than one processing function,
`simByMilstein`

invokes the functions in the order
in which they appear in the cell array. You can use this argument to
specify boundary conditions, prevent negative prices, accumulate
statistics, plot graphs, and more.

**Data Types: **`cell`

| `function`

## Output Arguments

`Paths`

— Simulated paths of correlated state variables

array

Simulated paths of correlated state variables, returned as an
```
(NPeriods +
1)
```

-by-`NVars`

-by-`NTrials`

three-dimensional time series array.

For a given trial, each row of `Paths`

is the transpose
of the state vector
*X*_{t} at time
*t*. When the input flag
`StorePaths`

= `False`

,
`simByEuler`

returns `Paths`

as an
empty matrix.

`Times`

— Observation times associated with simulated paths

column vector

Observation times associated with the simulated paths, returned as an
`(NPeriods + 1)`

-by-`1`

column vector.
Each element of `Times`

is associated with the
corresponding row of `Paths`

.

`Z`

— Dependent random variates used to generate Brownian motion vector

array

Dependent random variates used to generate the Brownian motion vector
(Wiener processes) that drive the simulation, returned as an
```
(NPeriods ⨉
NSteps)
```

-by-`NBrowns`

-by-`NTrials`

three-dimensional time series array.

## More About

### Milstein Method

The *Milstein method* is a numerical method
for approximating solutions to stochastic differential equations (SDEs).

The Milstein method is an extension of the Euler-Maruyama method, which is a first-order numerical method for SDEs. The Milstein method adds a correction term to the Euler-Maruyama method that takes into account the second-order derivative of the SDE. This correction term improves the accuracy of the approximation, especially for SDEs with non-linearities.

### Antithetic Sampling

Simulation methods allow you to specify a popular
*variance reduction* technique called *antithetic
sampling*.

This technique attempts to replace one sequence of random observations with
another of the same expected value, but smaller variance. In a typical Monte Carlo
simulation, each sample path is independent and represents an independent trial.
However, antithetic sampling generates sample paths in pairs. The first path of the
pair is referred to as the *primary path*, and the second as the
*antithetic path*. Any given pair is independent of any other
pair, but the two paths within each pair are highly correlated. Antithetic sampling
literature often recommends averaging the discounted payoffs of each pair,
effectively halving the number of Monte Carlo trials.

This technique attempts to reduce variance by inducing negative dependence between paired input samples, ideally resulting in negative dependence between paired output samples. The greater the extent of negative dependence, the more effective antithetic sampling is.

## Algorithms

This function simulates any 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 of process variables (for example, short rates or equity prices) to simulate.*W*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.

`simByEuler`

simulates `NTrials`

sample
paths of `NVars`

correlated state variables driven by
`NBrowns`

Brownian motion sources of risk over
`NPeriods`

consecutive observation periods, using the Euler
approach to approximate continuous-time stochastic processes.

Consider the process *X* satisfying a stochastic differential
equation of the form.

$$d{X}_{t}=\mu ({X}_{t})dt+\sigma ({X}_{t})d{W}_{t}$$

The attempt of including a term of *O*(*dt*) in the
drift refines the Euler scheme and results in the algorithm derived by Milstein [1].

$${X}_{t+1}={X}_{t}+\mu ({X}_{t})dt+\sigma ({X}_{t})d{W}_{t}+\frac{1}{2}\sigma ({X}_{t}){\sigma}^{/}({X}_{t})(d{W}_{t}^{2}-dt)$$

## References

[1] Milstein, G.N. "A Method of
Second-Order Accuracy Integration of Stochastic Differential
Equations."*Theory of Probability and Its Applications*, 23,
1978, pp. 396–401.

## Version History

**Introduced in R2023a**

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