R1FokkerPlanck.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>R1FokkerPlanck</i> exposes the R<sup>1</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 R1FokkerPlanck
{
private org.drip.dynamics.ito.R1ToR1Drift _driftFunction = null;
private org.drip.dynamics.ito.R1ToR1Volatility _volatilityFunction = null;
/**
* R1FokkerPlanck Constructor
*
* @param driftFunction The Drift Function
* @param volatilityFunction The Volatility Function
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public R1FokkerPlanck (
final org.drip.dynamics.ito.R1ToR1Drift driftFunction,
final org.drip.dynamics.ito.R1ToR1Volatility volatilityFunction)
throws java.lang.Exception
{
if (null == (_driftFunction = driftFunction) ||
null == (_volatilityFunction = volatilityFunction))
{
throw new java.lang.Exception (
"R1FokkerPlanck Constructor => Invalid Inputs"
);
}
}
/**
* Retrieve the Drift Function
*
* @return The Drift Function
*/
public org.drip.dynamics.ito.R1ToR1Drift driftFunction()
{
return _driftFunction;
}
/**
* Retrieve the Volatility Function
*
* @return The Volatility Function
*/
public org.drip.dynamics.ito.R1ToR1Volatility volatilityFunction()
{
return _volatilityFunction;
}
/**
* Compute the Next Incremental Time Derivative of the PDF
*
* @param probabilityDensityFunction The PDF
* @param timeR1Vertex The R<sup>1</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.R1ProbabilityDensityFunction probabilityDensityFunction,
final org.drip.dynamics.ito.TimeR1Vertex timeR1Vertex)
throws java.lang.Exception
{
if (null == probabilityDensityFunction ||
null == timeR1Vertex)
{
throw new java.lang.Exception (
"R1FokkerPlanck::pdfDot => Invalid Inputs"
);
}
final double time = timeR1Vertex.t();
return new org.drip.function.definition.R1ToR1 (
null
)
{
@Override public double evaluate (
final double x)
throws java.lang.Exception
{
org.drip.dynamics.ito.TimeR1Vertex localTimeR1Vertex =
new org.drip.dynamics.ito.TimeR1Vertex (
time,
x
);
return _driftFunction.drift (
localTimeR1Vertex
) * probabilityDensityFunction.density (
localTimeR1Vertex
);
}
}.derivative (
timeR1Vertex.x(),
1
) + new org.drip.function.definition.R1ToR1 (
null
)
{
@Override public double evaluate (
final double x)
throws java.lang.Exception
{
org.drip.dynamics.ito.TimeR1Vertex localTimeR1Vertex =
new org.drip.dynamics.ito.TimeR1Vertex (
time,
x
);
double volatility = _volatilityFunction.volatility (
localTimeR1Vertex
);
return 0.5 * volatility * volatility * probabilityDensityFunction.density (
localTimeR1Vertex
);
}
}.derivative (
timeR1Vertex.x(),
2
);
}
/**
* 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.R1ProbabilityDensityFunction temporalPDF (
final org.drip.function.definition.R1ToR1 intialProbabilityDensityFunction)
{
return null;
}
/**
* Compute the Steady-State Probability Distribution Function, if any
*
* @return The Steady-State Probability Distribution Function
*/
public org.drip.function.definition.R1ToR1 steadyStatePDF()
{
return null;
}
/**
* Compute the Temporal Probability Distribution Function given the Delta 0 Starting PDF
*
* @param x0 The X Anchor for the Delta Function
*
* @return The Temporal Probability Distribution Function given the Delta 0 Starting PDF
*/
public org.drip.dynamics.process.R1ProbabilityDensityFunction deltaStartTemporalPDF (
final double x0)
{
return null;
}
}