RdFokkerPlanck.java
package org.drip.dynamics.kolmogorov;
/*
* -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
*/
/*!
* Copyright (C) 2020 Lakshmi Krishnamurthy
* Copyright (C) 2019 Lakshmi Krishnamurthy
*
* This file is part of DROP, an open-source library targeting analytics/risk, transaction cost analytics,
* asset liability management analytics, capital, exposure, and margin analytics, valuation adjustment
* analytics, and portfolio construction analytics within and across fixed income, credit, commodity,
* equity, FX, and structured products. It also includes auxiliary libraries for algorithm support,
* numerical analysis, numerical optimization, spline builder, model validation, statistical learning,
* and computational support.
*
* https://lakshmidrip.github.io/DROP/
*
* DROP is composed of three modules:
*
* - DROP Product Core - https://lakshmidrip.github.io/DROP-Product-Core/
* - DROP Portfolio Core - https://lakshmidrip.github.io/DROP-Portfolio-Core/
* - DROP Computational Core - https://lakshmidrip.github.io/DROP-Computational-Core/
*
* DROP Product Core implements libraries for the following:
* - Fixed Income Analytics
* - Loan Analytics
* - Transaction Cost Analytics
*
* DROP Portfolio Core implements libraries for the following:
* - Asset Allocation Analytics
* - Asset Liability Management Analytics
* - Capital Estimation Analytics
* - Exposure Analytics
* - Margin Analytics
* - XVA Analytics
*
* DROP Computational Core implements libraries for the following:
* - Algorithm Support
* - Computation Support
* - Function Analysis
* - Model Validation
* - Numerical Analysis
* - Numerical Optimizer
* - Spline Builder
* - Statistical Learning
*
* Documentation for DROP is Spread Over:
*
* - Main => https://lakshmidrip.github.io/DROP/
* - Wiki => https://github.com/lakshmiDRIP/DROP/wiki
* - GitHub => https://github.com/lakshmiDRIP/DROP
* - Repo Layout Taxonomy => https://github.com/lakshmiDRIP/DROP/blob/master/Taxonomy.md
* - Javadoc => https://lakshmidrip.github.io/DROP/Javadoc/index.html
* - Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
* - Release Versions => https://lakshmidrip.github.io/DROP/version.html
* - Community Credits => https://lakshmidrip.github.io/DROP/credits.html
* - Issues Catalog => https://github.com/lakshmiDRIP/DROP/issues
* - JUnit => https://lakshmidrip.github.io/DROP/junit/index.html
* - Jacoco => https://lakshmidrip.github.io/DROP/jacoco/index.html
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
*
* You may obtain a copy of the License at
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
*
* See the License for the specific language governing permissions and
* limitations under the License.
*/
/**
* <i>RdFokkerPlanck</i> exposes the R<sup>d</sup> Fokker-Planck Probability Density Function Evolution
* Equation. The References are:
*
* <br><br>
* <ul>
* <li>
* Bogoliubov, N. N., and D. P. Sankevich (1994): N. N. Bogoliubov and Statistical Mechanics
* <i>Russian Mathematical Surveys</i> <b>49 (5)</b> 19-49
* </li>
* <li>
* Holubec, V., K. Kroy, and S. Steffenoni (2019): Physically Consistent Numerical Solver for
* Time-dependent Fokker-Planck Equations <i>Physical Review E</i> <b>99 (4)</b> 032117
* </li>
* <li>
* Kadanoff, L. P. (2000): <i>Statistical Physics: Statics, Dynamics, and Re-normalization</i>
* <b>World Scientific</b>
* </li>
* <li>
* Ottinger, H. C. (1996): <i>Stochastic Processes in Polymeric Fluids</i> <b>Springer-Verlag</b>
* Berlin-Heidelberg
* </li>
* <li>
* Wikipedia (2019): Fokker-Planck Equation
* https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation
* </li>
* </ul>
*
* <br><br>
* <ul>
* <li><b>Module </b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/ProductCore.md">Product Core Module</a></li>
* <li><b>Library</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/FixedIncomeAnalyticsLibrary.md">Fixed Income Analytics</a></li>
* <li><b>Project</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/dynamics/README.md">HJM, Hull White, LMM, and SABR Dynamic Evolution Models</a></li>
* <li><b>Package</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/dynamics/kolmogorov/README.md">Fokker Planck Kolmogorov Forward/Backward</a></li>
* </ul>
*
* @author Lakshmi Krishnamurthy
*/
public class RdFokkerPlanck
{
private org.drip.dynamics.ito.DiffusionTensor _diffusionTensor = null;
private org.drip.dynamics.ito.RdToR1Drift[] _driftFunctionArray = null;
private org.drip.dynamics.kolmogorov.RiskenOmegaEstimator _riskenOmegaEstimator = null;
/**
* RdFokkerPlanck Constructor
*
* @param driftFunctionArray Drift Function Array
* @param diffusionTensor Diffusion Tensor
* @param riskenOmegaEstimator Risken Omega Estimator
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public RdFokkerPlanck (
final org.drip.dynamics.ito.RdToR1Drift[] driftFunctionArray,
final org.drip.dynamics.ito.DiffusionTensor diffusionTensor,
final org.drip.dynamics.kolmogorov.RiskenOmegaEstimator riskenOmegaEstimator)
throws java.lang.Exception
{
if (null == (_driftFunctionArray = driftFunctionArray) ||
null == (_diffusionTensor = diffusionTensor) ||
null == (_riskenOmegaEstimator = riskenOmegaEstimator)
)
{
throw new java.lang.Exception (
"RdFokkerPlanck Constructor => Invalid Inputs"
);
}
int dimension = _driftFunctionArray.length;
if (dimension != _diffusionTensor.dimension())
{
throw new java.lang.Exception (
"RdFokkerPlanck Constructor => Invalid Inputs"
);
}
for (int dimensionIndex = 0;
dimensionIndex < dimension;
++dimensionIndex)
{
if (null == _driftFunctionArray[dimensionIndex])
{
throw new java.lang.Exception (
"RdFokkerPlanck Constructor => Invalid Inputs"
);
}
}
}
/**
* Retrieve the Drift Function Array
*
* @return The Drift Function Array
*/
public org.drip.dynamics.ito.RdToR1Drift[] driftFunctionArray()
{
return _driftFunctionArray;
}
/**
* Retrieve the Diffusion Tensor
*
* @return The Diffusion Tensor
*/
public org.drip.dynamics.ito.DiffusionTensor diffusionTensor()
{
return _diffusionTensor;
}
/**
* Retrieve the Risken Omega Estimator
*
* @return The Risken Omega Estimator
*/
public org.drip.dynamics.kolmogorov.RiskenOmegaEstimator riskenOmegaEstimator()
{
return _riskenOmegaEstimator;
}
/**
* Compute the Next Incremental Time Derivative of the PDF
*
* @param probabilityDensityFunction The PDF
* @param timeRdVertex The R<sup>d</sup> Time Vertex
*
* @return Next Incremental Time Derivative of the PDF
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double pdfDot (
final org.drip.dynamics.process.RdProbabilityDensityFunction probabilityDensityFunction,
final org.drip.dynamics.ito.TimeRdVertex timeRdVertex)
throws java.lang.Exception
{
if (null == probabilityDensityFunction ||
null == timeRdVertex)
{
throw new java.lang.Exception (
"RdFokkerPlanck::pdfDot => Invalid Inputs"
);
}
final int dimension = _diffusionTensor.dimension();
final double time = timeRdVertex.t();
double pdfDot = 0.;
for (int dimensionIndex = 0;
dimensionIndex < dimension;
++dimensionIndex)
{
final int index = dimensionIndex;
pdfDot = pdfDot - new org.drip.function.definition.RdToR1 (
null
)
{
@Override public int dimension()
{
return dimension;
}
@Override public double evaluate (
final double[] xArray)
throws java.lang.Exception
{
org.drip.dynamics.ito.TimeRdVertex localTimeRdVertex =
new org.drip.dynamics.ito.TimeRdVertex (
time,
xArray
);
return _driftFunctionArray[index].drift (
localTimeRdVertex
) * probabilityDensityFunction.density (
localTimeRdVertex
);
}
}.derivative (
timeRdVertex.xArray(),
dimensionIndex,
1
);
}
for (int dimensionIndexI = 0;
dimensionIndexI < dimension;
++dimensionIndexI)
{
final int indexI = dimensionIndexI;
for (int dimensionIndexJ = 0;
dimensionIndexJ < dimension;
++dimensionIndexJ)
{
final int indexJ = dimensionIndexJ;
pdfDot = pdfDot + new org.drip.function.definition.RdToR1 (
null
)
{
@Override public int dimension()
{
return dimension;
}
@Override public double evaluate (
final double[] xArray)
throws java.lang.Exception
{
org.drip.dynamics.ito.TimeRdVertex localTimeRdVertex =
new org.drip.dynamics.ito.TimeRdVertex (
time,
xArray
);
return _diffusionTensor.diffusionCoefficient (
localTimeRdVertex,
indexI,
indexJ
) * probabilityDensityFunction.density (
localTimeRdVertex
);
}
}.hessian (
timeRdVertex.xArray()
)[indexI][indexJ];
};
}
return pdfDot;
}
/**
* Compute the Temporal Probability Distribution Function, if any
*
* @param intialProbabilityDensityFunction The Initial Probability Density Function
*
* @return The Temporal Probability Distribution Function
*/
public org.drip.dynamics.process.RdProbabilityDensityFunction temporalPDF (
final org.drip.function.definition.RdToR1 intialProbabilityDensityFunction)
{
return null;
}
/**
* Compute the Steady-State Probability Distribution Function, if any
*
* @return The Steady-State Probability Distribution Function
*/
public org.drip.function.definition.RdToR1 steadyStatePDF()
{
double[][] omega = _riskenOmegaEstimator.estimateOmega (
_diffusionTensor,
_driftFunctionArray
);
final double[][] omegaInverse =
org.drip.numerical.linearalgebra.Matrix.InvertUsingGaussianElimination (
omega
);
if (null == omegaInverse)
{
return null;
}
final int dimension = _diffusionTensor.dimension();
double rdNormalizer = java.lang.Double.NaN;
try
{
rdNormalizer = java.lang.Math.sqrt (
java.lang.Math.pow (
2. * java.lang.Math.PI,
-1. * dimension
) / new org.drip.function.matrix.Square (
omega
).determinant()
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
final double rdNormalizerFinal = rdNormalizer;
final org.drip.function.definition.R1ToR1 r1ToR1Exponential =
new org.drip.function.definition.R1ToR1 (
null
)
{
@Override public double evaluate (
final double x)
throws java.lang.Exception
{
return java.lang.Math.exp (
-0.5 * x
);
}
};
return new org.drip.function.definition.RdToR1 (
null
)
{
@Override public int dimension()
{
return dimension;
}
@Override public double evaluate (
final double[] xArray)
throws java.lang.Exception
{
return rdNormalizerFinal * r1ToR1Exponential.evaluate (
org.drip.numerical.linearalgebra.Matrix.DotProduct (
xArray,
org.drip.numerical.linearalgebra.Matrix.Product (
omegaInverse,
xArray
)
)
);
}
};
}
}