SDEs

• Git Homepage

• Pip Repository

• Simulating SDEs using Euler discretization.

• Euler scheme does not require derivatives, but Milstein and Runge-Kutta discretization schemes do...

• List of processes implemented:

  • 'BM', 'GBM', 'OU', 'ExponentialOU', '(Heston) GBMSA', 'srGBM', 'SBM', 'BrownianBridge', 'BrownianMeander', 'BrownianExcursion', 'DysonBM', 'StickyBM', 'ReflectingBM', 'CorrelatedBM', 'CorrelatedGBM', 'CircleBM', 'dDimensionalBM', '3dGBM', '3dOU', 'StochasticLogisticGrowth', 'StochasticExponentialDecay', 'SinCosVectorNoiseIto', 'PolynomialItoProcess', 'WrightFisherDiffusion', 'WeibullDiffusion', 'WeibullDiffusion2', 'ExpTC', 'MRSqrtDiff',

    'VG', 'VGSA', 'Merton', 'ATSM', 'ATSM_SV', 'CEV', 'CIR', 'Vasicek', 'ExponentialVasicek', 'SABR', 'ShiftedSABR',
    'DTDG', 'CKLS', 'HullWhite', 'LotkaVolterra', 'TwoFactorHullWhite', 'BlackKarasinski', 'Chen', 'LongstaffSchwartz', 'BDT', 'HoLee', 'CIR++', 'CIR2++', 'KWF',

    'Bates','GeneralBergomi', 'OneFactorBergomi', 'RoughBergomi', 'RoughVolatility', 'RfSV', 'SinRFSV', 'TanhRFSV', 'ARIMA', 'GARCH', 'GARCHJump', 'VIX', 'GaussTanhPolyRFSV', 'LaplaceTanhPolyRFSV', 't_TanhPolyRFSV', 'CauchyTanhPolyRFSV', 'triangularTanhPolyRFSV', 'GumbelTanhPolyRFSV', 'LogisticTanhPolyRFSV',

    'SCP_mean_reverting', 'SCP_modified_OU', 'SCP_tanh_Ito', 'SCP_arctan_Ito', 'SCQuanto', 'WrightFisherSC', 'Jacobi',

    'fBM', 'GFBM', 'fOU', 'fIM', 'tanh_fOU', 'Poly_fOU', 'fStochasticLogisticGrowth', 'fStochasticExponentialDecay', 'fBM_WrightFisherDiffusion', 'fSV', 'Bessel', 'SquaredBessel', 'ConicDiffusionMartingale', 'ConicUnifDiffusionMartingale', 'ConicHalfUnifDiffusionMartingale', 'SinFourierDecompBB', 'MixedFourierDecompBB', 'kCorrelatedGBMs', 'kCorrelatedBMs'

• StochasticProcessSimulator.py script can be used to obtain paths from the desired SDE/process.

• The main function of interest is StochasticProcessSimulator() which produces the paths and includes the options of plotting and recording execution time(s).

• READ_ME_SPS.txt describes all of the function arguments and their default settings.

• Simulating_SDEs_Euler_all_plots.ipynb shows an example of how this function can be used to produce plots of all the types of SDEs.

• Simulating_SDEs_Euler_simple_examples.ipynb (versions 1-4 of notebook) shows some simple examples of how this function can be used to produce stylized plots...

• NB: In StochasticProcessSimulator(), 'kCorrelatedGBMs' and 'kCorrelatedBMs' are implemented without the option of producing plots...

• 3dBMCube.ipynb and 3dBMSphere.ipynb are two examples demonstrating how StochasticProcessSimulator() can be integrated with more specific plotting requirements to produce plots of 3d BM paths constrained to a cube and sphere, respectively.

Installation

StochasticProcessSimulator is available on pypi and can be installed as follows

pip install StochasticProcessSimulator

Dependencies

StochasticProcessSimulator relies on

• numpy for random number generation and math functions,
• scipy and statsmodels for support for some specific distributions,
• matplotlib for creating visualisations.

Quick-Start

StochasticProcessSimulator allows you to simulate and plot paths from different stochastic processes in a simple way.

For instance, the following code

from StochasticProcessSimulator import StochasticProcessSimulator as SPS
simulator = SPS(
            process_type='GBM', do_plot=True)
paths = simulator.simulate()

produces the following output:

Future work:

• add handling of parameter inputs that break validity conditions...
• add Milstein and Runge-Kutta discretization schemes...
• add more SDEs/processes...
• This is a very rough implementation, and there are lots of improvements to be made. Any suggestions and improvements are welcome and appreciated.