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Simulate diagonal diffusion for BM, GBM, CEV, HWV, SDEDDO, SDELD, or SDEMRD sample paths by Milstein approximation

Since R2023a



[Paths,Times,Z] = simByMilstein(MDL,NPeriods) simulates 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.


[Paths,Times,Z] = simByMilstein(___,Name=Value) specifies options using one or more name-value pair arguments in addition to the input arguments in the previous syntax.

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.


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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 univariategbm object to represent the model: dXt=0.25Xtdt+0.3XtdWt.

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

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Stochastic differential equation model, specified as a BM, GBM, CEV, HWV, SDEDDO, SDELD, or SDEMRD object. You can use the following to create a MDL object:

Data Types: object

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 to handle negative values, specified as Method and a character vector or string with a supported value.

Data Types: char | string

Simulated trials (sample paths) of NPeriods observations each, specified as NTrials and a positive scalar integer.

Data Types: double

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

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

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.


If you specify an input noise process (see Z), simByMilstein ignores the value of Antithetic.

Data Types: logical

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.


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 Xt.

Data Types: double | function

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

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


If you specify an input noise process (see Z), simByMilstein ignores the value of MonteCarloMethod.

Data Types: string | char

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.


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

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.


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

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


The simByMilstein function runs processing functions at each interpolation time. They must accept the current interpolation time t, and the current state vector Xt, 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

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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 Xt at time t. When the input flag StorePaths = False, simByEuler returns Paths as an empty matrix.

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.

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

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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.


This function simulates any vector-valued SDE of the form



  • 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.


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].



[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