R1FokkerPlanckBrownian.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>R1FokkerPlanckBrownian</i> exposes the R<sup>1</sup> Brownian 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 R1FokkerPlanckBrownian
    extends org.drip.dynamics.kolmogorov.R1FokkerPlanck
{

    /**
     * R1FokkerPlanckBrownian Constructor
     * 
     * @throws java.lang.Exception Thrown if R1FokkerPlanckBrownian cannot be constructed
     */

    public R1FokkerPlanckBrownian()
        throws java.lang.Exception
    {
        super (
            new org.drip.dynamics.ito.R1ToR1Drift()
            {
                @Override public double drift (
                    final org.drip.dynamics.ito.TimeR1Vertex r1TimeVertex)
                    throws java.lang.Exception
                {
                    return 0.;
                }
            },
            new org.drip.dynamics.ito.R1ToR1Volatility()
            {
                @Override public double volatility (
                    final org.drip.dynamics.ito.TimeR1Vertex r1TimeVertex)
                    throws java.lang.Exception
                {
                    return 1.;
                }
            }
        );
    }

    @Override public org.drip.dynamics.process.R1ProbabilityDensityFunction deltaStartTemporalPDF (
        final double x0)
    {
        return org.drip.numerical.common.NumberUtil.IsValid (
            x0
        ) ? new org.drip.dynamics.process.R1ProbabilityDensityFunction()
        {
            @Override public double density (
                final org.drip.dynamics.ito.TimeR1Vertex r1TimeVertex)
                throws java.lang.Exception
            {
                if (null == r1TimeVertex)
                {
                    throw new java.lang.Exception (
                        "R1FokkerPlanckBrownian::deltaStartTemporalPDF::density"
                    );
                }

                double t = r1TimeVertex.t();

                double xGap = r1TimeVertex.x() - x0;

                return java.lang.Math.exp (
                    -0.5 * xGap * xGap / t
                ) / java.lang.Math.sqrt (
                    2. * java.lang.Math.PI * t
                );
            }
        } : null;
    }
}