# simBySolution

Simulate approximate solution of diagonal-drift `Merton` jump diffusion process

## Syntax

``[Paths,Times,Z,N] = simBySolution(MDL,NPeriods)``
``[Paths,Times,Z,N] = simBySolution(___,Name,Value)``

## Description

example

````[Paths,Times,Z,N] = simBySolution(MDL,NPeriods)` simulates `NNTrials` sample paths of `NVars` correlated state variables driven by `NBrowns` Brownian motion sources of risk and `NJumps` compound Poisson processes representing the arrivals of important events over `NPeriods` consecutive observation periods. The simulation approximates continuous-time Merton jump diffusion process by an approximation of the closed-form solution.```

example

````[Paths,Times,Z,N] = simBySolution(___,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.```

## Examples

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Simulate the approximate solution of diagonal-drift Merton process.

Create a `merton` object.

```AssetPrice = 80; Return = 0.03; Sigma = 0.16; JumpMean = 0.02; JumpVol = 0.08; JumpFreq = 2; mertonObj = merton(Return,Sigma,JumpFreq,JumpMean,JumpVol,... 'startstat',AssetPrice)```
```mertonObj = Class MERTON: Merton Jump Diffusion ---------------------------------------- Dimensions: State = 1, Brownian = 1 ---------------------------------------- StartTime: 0 StartState: 80 Correlation: 1 Drift: drift rate function F(t,X(t)) Diffusion: diffusion rate function G(t,X(t)) Simulation: simulation method/function simByEuler Sigma: 0.16 Return: 0.03 JumpFreq: 2 JumpMean: 0.02 JumpVol: 0.08 ```

Use `simBySolution` to simulate `NTrials` sample paths of `NVARS` correlated state variables driven by `NBrowns` Brownian motion sources of risk and `NJumps` compound Poisson processes representing the arrivals of important events over `NPeriods` consecutive observation periods. The function approximates continuous-time Merton jump diffusion process by an approximation of the closed-form solution.

```nPeriods = 100; [Paths,Times,Z,N] = simBySolution(mertonObj, nPeriods,'nTrials', 3)```
```Paths = Paths(:,:,1) = 1.0e+03 * 0.0800 0.0600 0.0504 0.0799 0.1333 0.1461 0.2302 0.2505 0.3881 0.4933 0.4547 0.4433 0.5294 0.6443 0.7665 0.6489 0.7220 0.7110 0.5815 0.5026 0.6523 0.7005 0.7053 0.4902 0.5401 0.4730 0.4242 0.5334 0.5821 0.6498 0.5982 0.5504 0.5290 0.5371 0.4789 0.4914 0.5019 0.3557 0.2950 0.3697 0.2906 0.2988 0.3081 0.3469 0.3146 0.3171 0.3588 0.3250 0.3035 0.2386 0.2533 0.2420 0.2315 0.2396 0.2143 0.2668 0.2115 0.1671 0.1784 0.1542 0.2046 0.1930 0.2011 0.2542 0.3010 0.3247 0.3900 0.4107 0.3949 0.4610 0.5725 0.5605 0.4541 0.5796 0.8199 0.5732 0.5856 0.7895 0.6883 0.6848 0.9059 1.0089 0.8429 0.9955 0.9683 0.8769 0.7120 0.7906 0.7630 1.2460 1.1703 1.2012 1.1109 1.1893 1.4346 1.4040 1.2365 1.0834 1.3315 0.8100 0.5558 Paths(:,:,2) = 80.0000 81.2944 71.3663 108.8305 111.4851 105.4563 160.2721 125.3288 158.3238 138.8899 157.9613 125.6819 149.8234 126.0374 182.5153 195.0861 273.1622 306.2727 301.3401 312.2173 298.2344 327.6944 288.9799 394.8951 551.4020 418.2258 404.1687 469.3555 606.4289 615.7066 526.6862 625.9683 474.4597 316.5110 407.9626 341.6552 475.0593 478.4058 545.3414 365.3404 513.2186 370.5371 444.0345 314.6991 257.4782 253.0259 237.6185 206.6325 334.5253 300.2284 328.9936 307.4059 248.7966 234.6355 183.9132 159.6084 169.1145 123.3256 148.1922 159.7083 104.0447 96.3935 92.4897 93.0576 116.3163 135.6249 120.6611 100.0253 109.7998 85.8078 81.5769 73.7983 65.9000 62.5120 62.9952 57.6044 54.2716 44.5617 42.2402 21.9133 18.0586 20.5171 22.5532 24.1654 26.8830 22.7864 34.5131 27.8362 27.7258 21.7367 20.8781 19.7174 14.9880 14.8903 19.3632 23.4230 27.7062 17.8347 16.8652 15.5675 15.5256 Paths(:,:,3) = 80.0000 79.6263 93.2979 63.1451 60.2213 54.2113 78.6114 96.6261 123.5584 126.5875 102.9870 83.2387 77.8567 79.3565 71.3876 80.5413 90.8709 77.5246 107.4194 114.4328 118.3999 148.0710 108.6207 110.0402 124.1150 104.5409 94.7576 98.9002 108.0691 130.7592 129.9744 119.9150 86.0303 96.9892 86.8928 106.8895 119.3219 197.7045 208.1930 197.1636 244.4438 166.4752 125.3896 128.9036 170.9818 140.2719 125.8948 87.0324 66.7637 48.4280 50.5766 49.7841 67.5690 62.8776 85.3896 67.9608 72.9804 59.0174 50.1132 45.2220 59.5469 58.4673 98.4790 90.0250 80.3092 86.9245 88.1303 95.4237 104.4456 99.1969 168.3980 146.8791 150.0052 129.7521 127.1402 113.3413 145.2281 153.1315 125.7882 111.9988 112.7732 118.9120 150.9166 120.0673 128.2727 185.9171 204.3474 194.5443 163.2626 183.9897 233.4125 318.9068 356.0077 380.4513 446.9518 484.9218 377.4244 470.3577 454.5734 297.0580 339.0796 ```
```Times = 101×1 0 1 2 3 4 5 6 7 8 9 ⋮ ```
```Z = Z(:,:,1) = -2.2588 -1.3077 3.5784 3.0349 0.7147 1.4897 0.6715 1.6302 0.7269 -0.7873 -1.0689 1.4384 1.3703 -0.2414 -0.8649 0.6277 -0.8637 -1.1135 -0.7697 1.1174 0.5525 0.0859 -1.0616 0.7481 -0.7648 0.4882 1.4193 1.5877 0.8351 -1.1658 0.7223 0.1873 -0.4390 -0.8880 0.3035 0.7394 -2.1384 -1.0722 1.4367 -1.2078 1.3790 -0.2725 0.7015 -0.8236 0.2820 1.1275 0.0229 -0.2857 -1.1564 0.9642 -0.0348 -0.1332 -0.2248 -0.8479 1.6555 -0.8655 -1.3320 0.3335 -0.1303 0.8620 -0.8487 1.0391 0.6601 -0.2176 0.0513 0.4669 0.1832 0.3071 0.2614 -0.1461 -0.8757 -1.1742 1.5301 1.6035 -1.5062 0.2761 0.3919 -0.7411 0.0125 1.2424 0.3503 -1.5651 0.0983 -0.0308 -0.3728 -2.2584 1.0001 -0.2781 0.4716 0.6524 1.0061 -0.9444 0.0000 0.5946 0.9298 -0.6516 -0.0245 0.8617 -2.4863 -2.3193 Z(:,:,2) = 0.8622 -0.4336 2.7694 0.7254 -0.2050 1.4090 -1.2075 0.4889 -0.3034 0.8884 -0.8095 0.3252 -1.7115 0.3192 -0.0301 1.0933 0.0774 -0.0068 0.3714 -1.0891 1.1006 -1.4916 2.3505 -0.1924 -1.4023 -0.1774 0.2916 -0.8045 -0.2437 -1.1480 2.5855 -0.0825 -1.7947 0.1001 -0.6003 1.7119 -0.8396 0.9610 -1.9609 2.9080 -1.0582 1.0984 -2.0518 -1.5771 0.0335 0.3502 -0.2620 -0.8314 -0.5336 0.5201 -0.7982 -0.7145 -0.5890 -1.1201 0.3075 -0.1765 -2.3299 0.3914 0.1837 -1.3617 -0.3349 -1.1176 -0.0679 -0.3031 0.8261 -0.2097 -1.0298 0.1352 -0.9415 -0.5320 -0.4838 -0.1922 -0.2490 1.2347 -0.4446 -0.2612 -1.2507 -0.5078 -3.0292 -1.0667 -0.0290 -0.0845 0.0414 0.2323 -0.2365 2.2294 -1.6642 0.4227 -1.2128 0.3271 -0.6509 -1.3218 -0.0549 0.3502 0.2398 1.1921 -1.9488 0.0012 0.5812 0.0799 Z(:,:,3) = 0.3188 0.3426 -1.3499 -0.0631 -0.1241 1.4172 0.7172 1.0347 0.2939 -1.1471 -2.9443 -0.7549 -0.1022 0.3129 -0.1649 1.1093 -1.2141 1.5326 -0.2256 0.0326 1.5442 -0.7423 -0.6156 0.8886 -1.4224 -0.1961 0.1978 0.6966 0.2157 0.1049 -0.6669 -1.9330 0.8404 -0.5445 0.4900 -0.1941 1.3546 0.1240 -0.1977 0.8252 -0.4686 -0.2779 -0.3538 0.5080 -1.3337 -0.2991 -1.7502 -0.9792 -2.0026 -0.0200 1.0187 1.3514 -0.2938 2.5260 -1.2571 0.7914 -1.4491 0.4517 -0.4762 0.4550 0.5528 1.2607 -0.1952 0.0230 1.5270 0.6252 0.9492 0.5152 -0.1623 1.6821 -0.7120 -0.2741 -1.0642 -0.2296 -0.1559 0.4434 -0.9480 -0.3206 -0.4570 0.9337 0.1825 1.6039 -0.7342 0.4264 2.0237 0.3376 -0.5900 -1.6702 0.0662 1.0826 0.2571 0.9248 0.9111 1.2503 -0.6904 -1.6118 1.0205 -0.0708 -2.1924 -0.9485 ```
```N = N(:,:,1) = 3 1 2 1 0 2 0 1 3 4 2 1 0 1 1 1 1 0 0 3 2 2 1 0 1 1 3 3 4 2 4 1 1 2 0 2 2 3 2 1 3 2 2 1 1 1 3 0 2 2 1 0 1 1 1 1 0 2 2 1 1 5 7 3 2 2 1 3 3 5 3 0 1 6 2 0 5 2 2 1 2 1 3 0 2 4 2 2 4 2 3 1 2 5 1 0 3 3 1 1 N(:,:,2) = 4 2 2 2 0 4 1 2 3 1 2 1 4 2 6 2 2 2 2 1 4 3 1 3 3 1 3 6 1 4 2 2 1 2 1 1 5 0 2 2 3 2 2 1 0 1 5 4 0 1 1 2 1 2 3 2 2 1 2 2 0 3 1 6 3 3 0 2 1 2 0 6 1 3 1 2 2 2 1 0 2 2 2 2 1 1 3 1 2 2 1 4 1 3 3 0 1 1 1 2 N(:,:,3) = 1 3 2 2 1 4 2 3 0 0 4 3 2 3 1 1 1 1 3 4 1 2 3 1 1 1 1 0 3 0 1 0 4 0 2 4 3 1 0 1 5 3 3 2 1 2 3 1 5 4 1 1 2 2 1 1 1 2 1 5 1 2 1 3 2 2 1 3 1 6 0 1 4 1 1 3 5 3 1 2 2 1 2 1 1 1 1 1 2 3 6 2 1 3 2 1 1 0 1 3 ```

This example shows how to use `simBySolution` with a Merton model to perform a quasi-Monte Carlo simulation. Quasi-Monte Carlo simulation is a Monte Carlo simulation that uses quasi-random sequences instead pseudo random numbers.

Create a `merton` object.

```AssetPrice = 80; Return = 0.03; Sigma = 0.16; JumpMean = 0.02; JumpVol = 0.08; JumpFreq = 2; Merton = merton(Return,Sigma,JumpFreq,JumpMean,JumpVol,'startstat',AssetPrice)```
```Merton = Class MERTON: Merton Jump Diffusion ---------------------------------------- Dimensions: State = 1, Brownian = 1 ---------------------------------------- StartTime: 0 StartState: 80 Correlation: 1 Drift: drift rate function F(t,X(t)) Diffusion: diffusion rate function G(t,X(t)) Simulation: simulation method/function simByEuler Sigma: 0.16 Return: 0.03 JumpFreq: 2 JumpMean: 0.02 JumpVol: 0.08 ```

Perform a quasi-Monte Carlo simulation by using `simBySolution` with the optional name-value arguments for `'MonteCarloMethod'`,`'QuasiSequence'`, and `'BrownianMotionMethod'`.

`[paths,time,z,n] = simBySolution(Merton, 10,'ntrials',4096,'montecarlomethod','quasi','QuasiSequence','sobol','BrownianMotionMethod','brownian-bridge');`

## Input Arguments

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Merton model, specified as a `merton` object. You can create a `merton` object using `merton`.

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.

Before R2021a, use commas to separate each name and value, and enclose `Name` in quotes.

Example: ```[Paths,Times,Z,N] = simBySolution(merton,NPeriods,'DeltaTimes',dt,'NNTrials',10)```

Simulated NTrials (sample paths) of `NPeriods` observations each, specified as the comma-separated pair consisting of `'NNTrials'` and a positive scalar integer.

Data Types: `double`

Positive time increments between observations, specified as the comma-separated pair consisting of `'DeltaTimes'` and a scalar or an `NPeriods`-by-`1` column vector.

`DeltaTimes` 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 `DeltaTimes`), specified as the comma-separated pair consisting of `'NSteps'` and a positive scalar integer.

The `simBySolution` 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 `simBySolution` 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 to use antithetic sampling to generate the Gaussian random variates that drive the Brownian motion vector (Wiener processes), specified as the comma-separated pair consisting of `'Antithetic'` and a scalar numeric or logical `1` (`true`) or `0` (`false`).

When you specify `true`, `simBySolution` performs sampling such that all primary and antithetic paths are simulated and stored in successive matching pairs:

• Odd NTrials `(1,3,5,...)` correspond to the primary Gaussian paths.

• Even NTrials `(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`), `simBySolution` ignores the value of `Antithetic`.

Data Types: `logical`

Monte Carlo method to simulate stochastic processes, specified as the comma-separated pair consisting of `'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` and `N`), `simBySolution` ignores the value of `MonteCarloMethod`.

Data Types: `string` | `char`

Low discrepancy sequence to drive the stochastic processes, specified as the comma-separated pair consisting of `'QuasiSequence'` and a string or character vector with one of the following values:

• `"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`), `simBySolution` ignores the value of `QuasiSequence`.

Data Types: `string` | `char`

Brownian motion construction method, specified as the comma-separated pair consisting of `'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`

Direct specification of the dependent random noise process for generating the Brownian motion vector (Wiener process) that drives the simulation, specified as the comma-separated pair consisting of `'Z'` and a function or an ```(NPeriods * NSteps)```-by-`NBrowns`-by-`NNTrials` three-dimensional array of dependent random variates.

The input argument `Z` allows you to directly specify the noise generation process. This process takes precedence over the `Correlation` parameter of the input `merton` object and the value of the `Antithetic` input flag.

Specifically, when `Z` is specified, `Correlation` is not explicitly used to generate the Gaussian variates that drive the Brownian motion. However, `Correlation` is still used in the expression that appears in the exponential term of the log[Xt] Euler scheme. Thus, you must specify `Z` as a correlated Gaussian noise process whose correlation structure is consistently captured by `Correlation`.

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 Xt

Data Types: `double` | `function`

Dependent random counting process for generating the number of jumps, specified as the comma-separated pair consisting of `'N'` and a function or an (`NPeriods``NSteps`) -by-`NJumps`-by-`NNTrials` three-dimensional array of dependent random variates. If you specify a function, `N` must return an `NJumps`-by-`1` column vector, and you must call it with two inputs: a real-valued scalar observation time t followed by 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 the comma-separated pair consisting of `'StorePaths'` and a scalar numeric or logical `1` (`true`) or `0` (`false`).

If `StorePaths` is `true` (the default value) or is unspecified, `simBySolution` returns `Paths` as a three-dimensional time series array.

If `StorePaths` is `false` (logical `0`), `simBySolution` returns `Paths` as an empty matrix.

Data Types: `logical`

Sequence of end-of-period processes or state vector adjustments, specified as the comma-separated pair consisting of `'Processes'` and a function or cell array of functions of the form

`${X}_{t}=P\left(t,{X}_{t}\right)$`

`simBySolution` applies processing functions at the end of each observation period. These functions must accept the current observation time t and the current state vector Xt, and return a state vector that can be an adjustment to the input state.

The end-of-period `Processes` argument allows you to terminate a given trial early. At the end of each time step, `simBySolution` tests the state vector Xt for an all-`NaN` condition. Thus, to signal an early termination of a given trial, all elements of the state vector Xt must be `NaN`. This test enables a user-defined `Processes` function to signal early termination of a trial, and offers significant performance benefits in some situations (for example, pricing down-and-out barrier options).

If you specify more than one processing function, `simBySolution` 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-`NNTrials` 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 `StorePaths` is set to `false`, `simBySolution` 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 for generating the Brownian motion vector (Wiener processes) that drive the simulation, returned as a ```(NPeriods * NSteps)```-by-`NBrowns`-by-`NNTrials` three-dimensional time-series array.

Dependent random variates for generating the jump counting process vector, returned as an ```(NPeriods ⨉ NSteps)```-by-`NJumps`-by-`NNTrials` three-dimensional time-series array.

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### 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 that has the same expected value but a 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 other pairs, 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 NTrials.

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

The `simBySolution` function simulates the state vector Xt by an approximation of the closed-form solution of diagonal drift Merton jump diffusion models. Specifically, it applies a Euler approach to the transformed `log`[Xt] process (using Ito's formula). In general, this is not the exact solution to the Merton jump diffusion model because the probability distributions of the simulated and true state vectors are identical only for piecewise constant parameters.

This function simulates any vector-valued `merton` process of the form

`$d{X}_{t}=B\left(t,{X}_{t}\right){X}_{t}dt+D\left(t,{X}_{t}\right)V\left(t,{x}_{t}\right)d{W}_{t}+Y\left(t,{X}_{t},{N}_{t}\right){X}_{t}d{N}_{t}$`

Here:

• Xt is an `NVars`-by-`1` state vector of process variables.

• B(t,Xt) is an `NVars`-by-`NVars` matrix of generalized expected instantaneous rates of return.

• `D(t,Xt)` is an `NVars`-by-`NVars` diagonal matrix in which each element along the main diagonal is the corresponding element of the state vector.

• `V(t,Xt)` is an `NVars`-by-`NVars` matrix of instantaneous volatility rates.

• dWt is an `NBrowns`-by-`1` Brownian motion vector.

• `Y(t,Xt,Nt)` is an `NVars`-by-`NJumps` matrix-valued jump size function.

• dNt is an `NJumps`-by-`1` counting process vector.

## References

[1] Aït-Sahalia, Yacine. “Testing Continuous-Time Models of the Spot Interest Rate.” Review of Financial Studies 9, no. 2 ( Apr. 1996): 385–426.

[2] Aït-Sahalia, Yacine. “Transition Densities for Interest Rate and Other Nonlinear Diffusions.” The Journal of Finance 54, no. 4 (Aug. 1999): 1361–95.

[3] Glasserman, Paul. Monte Carlo Methods in Financial Engineering. New York: Springer-Verlag, 2004.

[4] Hull, John C. Options, Futures and Other Derivatives. 7th ed, Prentice Hall, 2009.

[5] Johnson, Norman Lloyd, Samuel Kotz, and Narayanaswamy Balakrishnan. Continuous Univariate Distributions. 2nd ed. Wiley Series in Probability and Mathematical Statistics. New York: Wiley, 1995.

[6] Shreve, Steven E. Stochastic Calculus for Finance. New York: Springer-Verlag, 2004.

## Version History

Introduced in R2020a

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