R1FokkerPlanck.java

  1. package org.drip.dynamics.kolmogorov;

  3. /*
  4.  * -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
  5.  */

  7. /*!
  8.  * Copyright (C) 2020 Lakshmi Krishnamurthy
  9.  * Copyright (C) 2019 Lakshmi Krishnamurthy
  10.  * 
  11.  *  This file is part of DROP, an open-source library targeting analytics/risk, transaction cost analytics,
  12.  *      asset liability management analytics, capital, exposure, and margin analytics, valuation adjustment
  13.  *      analytics, and portfolio construction analytics within and across fixed income, credit, commodity,
  14.  *      equity, FX, and structured products. It also includes auxiliary libraries for algorithm support,
  15.  *      numerical analysis, numerical optimization, spline builder, model validation, statistical learning,
  16.  *      and computational support.
  17.  *  
  18.  *      https://lakshmidrip.github.io/DROP/
  19.  *  
  20.  *  DROP is composed of three modules:
  21.  *  
  22.  *  - DROP Product Core - https://lakshmidrip.github.io/DROP-Product-Core/
  23.  *  - DROP Portfolio Core - https://lakshmidrip.github.io/DROP-Portfolio-Core/
  24.  *  - DROP Computational Core - https://lakshmidrip.github.io/DROP-Computational-Core/
  25.  * 
  26.  *  DROP Product Core implements libraries for the following:
  27.  *  - Fixed Income Analytics
  28.  *  - Loan Analytics
  29.  *  - Transaction Cost Analytics
  30.  * 
  31.  *  DROP Portfolio Core implements libraries for the following:
  32.  *  - Asset Allocation Analytics
  33.  *  - Asset Liability Management Analytics
  34.  *  - Capital Estimation Analytics
  35.  *  - Exposure Analytics
  36.  *  - Margin Analytics
  37.  *  - XVA Analytics
  38.  * 
  39.  *  DROP Computational Core implements libraries for the following:
  40.  *  - Algorithm Support
  41.  *  - Computation Support
  42.  *  - Function Analysis
  43.  *  - Model Validation
  44.  *  - Numerical Analysis
  45.  *  - Numerical Optimizer
  46.  *  - Spline Builder
  47.  *  - Statistical Learning
  48.  * 
  49.  *  Documentation for DROP is Spread Over:
  50.  * 
  51.  *  - Main                     => https://lakshmidrip.github.io/DROP/
  52.  *  - Wiki                     => https://github.com/lakshmiDRIP/DROP/wiki
  53.  *  - GitHub                   => https://github.com/lakshmiDRIP/DROP
  54.  *  - Repo Layout Taxonomy     => https://github.com/lakshmiDRIP/DROP/blob/master/Taxonomy.md
  55.  *  - Javadoc                  => https://lakshmidrip.github.io/DROP/Javadoc/index.html
  56.  *  - Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
  57.  *  - Release Versions         => https://lakshmidrip.github.io/DROP/version.html
  58.  *  - Community Credits        => https://lakshmidrip.github.io/DROP/credits.html
  59.  *  - Issues Catalog           => https://github.com/lakshmiDRIP/DROP/issues
  60.  *  - JUnit                    => https://lakshmidrip.github.io/DROP/junit/index.html
  61.  *  - Jacoco                   => https://lakshmidrip.github.io/DROP/jacoco/index.html
  62.  * 
  63.  *  Licensed under the Apache License, Version 2.0 (the "License");
  64.  *      you may not use this file except in compliance with the License.
  65.  *   
  66.  *  You may obtain a copy of the License at
  67.  *      http://www.apache.org/licenses/LICENSE-2.0
  68.  *  
  69.  *  Unless required by applicable law or agreed to in writing, software
  70.  *      distributed under the License is distributed on an "AS IS" BASIS,
  71.  *      WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  72.  *  
  73.  *  See the License for the specific language governing permissions and
  74.  *      limitations under the License.
  75.  */

  77. /**
  78.  * <i>R1FokkerPlanck</i> exposes the R<sup>1</sup> Fokker-Planck Probability Density Function Evolution
  79.  *  Equation. The References are:
  80.  *  
  81.  *  <br><br>
  82.  *  <ul>
  83.  *      <li>
  84.  *          Bogoliubov, N. N., and D. P. Sankevich (1994): N. N. Bogoliubov and Statistical Mechanics
  85.  *              <i>Russian Mathematical Surveys</i> <b>49 (5)</b> 19-49
  86.  *      </li>
  87.  *      <li>
  88.  *          Holubec, V., K. Kroy, and S. Steffenoni (2019): Physically Consistent Numerical Solver for
  89.  *              Time-dependent Fokker-Planck Equations <i>Physical Review E</i> <b>99 (4)</b> 032117
  90.  *      </li>
  91.  *      <li>
  92.  *          Kadanoff, L. P. (2000): <i>Statistical Physics: Statics, Dynamics, and Re-normalization</i>
  93.  *              <b>World Scientific</b>
  94.  *      </li>
  95.  *      <li>
  96.  *          Ottinger, H. C. (1996): <i>Stochastic Processes in Polymeric Fluids</i> <b>Springer-Verlag</b>
  97.  *              Berlin-Heidelberg
  98.  *      </li>
  99.  *      <li>
  100.  *          Wikipedia (2019): Fokker-Planck Equation
  101.  *              https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation
  102.  *      </li>
  103.  *  </ul>
  104.  *
  105.  *  <br><br>
  106.  *  <ul>
  107.  *      <li><b>Module </b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/ProductCore.md">Product Core Module</a></li>
  108.  *      <li><b>Library</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/FixedIncomeAnalyticsLibrary.md">Fixed Income Analytics</a></li>
  109.  *      <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>
  110.  *      <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>
  111.  *  </ul>
  112.  *
  113.  * @author Lakshmi Krishnamurthy
  114.  */

  116. public class R1FokkerPlanck
  117. {
  118.     private org.drip.dynamics.ito.R1ToR1Drift _driftFunction = null;
  119.     private org.drip.dynamics.ito.R1ToR1Volatility _volatilityFunction = null;

  121.     /**
  122.      * R1FokkerPlanck Constructor
  123.      * 
  124.      * @param driftFunction The Drift Function
  125.      * @param volatilityFunction The Volatility Function
  126.      * 
  127.      * @throws java.lang.Exception Thrown if the Inputs are Invalid
  128.      */

  130.     public R1FokkerPlanck (
  131.         final org.drip.dynamics.ito.R1ToR1Drift driftFunction,
  132.         final org.drip.dynamics.ito.R1ToR1Volatility volatilityFunction)
  133.         throws java.lang.Exception
  134.     {
  135.         if (null == (_driftFunction = driftFunction) ||
  136.             null == (_volatilityFunction = volatilityFunction))
  137.         {
  138.             throw new java.lang.Exception (
  139.                 "R1FokkerPlanck Constructor => Invalid Inputs"
  140.             );
  141.         }
  142.     }

  144.     /**
  145.      * Retrieve the Drift Function
  146.      * 
  147.      * @return The Drift Function
  148.      */

  150.     public org.drip.dynamics.ito.R1ToR1Drift driftFunction()
  151.     {
  152.         return _driftFunction;
  153.     }

  155.     /**
  156.      * Retrieve the Volatility Function
  157.      * 
  158.      * @return The Volatility Function
  159.      */

  161.     public org.drip.dynamics.ito.R1ToR1Volatility volatilityFunction()
  162.     {
  163.         return _volatilityFunction;
  164.     }

  166.     /**
  167.      * Compute the Next Incremental Time Derivative of the PDF
  168.      * 
  169.      * @param probabilityDensityFunction The PDF
  170.      * @param timeR1Vertex The R<sup>1</sup> Time Vertex
  171.      * 
  172.      * @return Next Incremental Time Derivative of the PDF
  173.      * 
  174.      * @throws java.lang.Exception Thrown if the Inputs are Invalid
  175.      */

  177.     public double pdfDot (
  178.         final org.drip.dynamics.process.R1ProbabilityDensityFunction probabilityDensityFunction,
  179.         final org.drip.dynamics.ito.TimeR1Vertex timeR1Vertex)
  180.         throws java.lang.Exception
  181.     {
  182.         if (null == probabilityDensityFunction ||
  183.             null == timeR1Vertex)
  184.         {
  185.             throw new java.lang.Exception (
  186.                 "R1FokkerPlanck::pdfDot => Invalid Inputs"
  187.             );
  188.         }

  190.         final double time = timeR1Vertex.t();

  192.         return new org.drip.function.definition.R1ToR1 (
  193.             null
  194.         )
  195.         {
  196.             @Override public double evaluate (
  197.                 final double x)
  198.                 throws java.lang.Exception
  199.             {
  200.                 org.drip.dynamics.ito.TimeR1Vertex localTimeR1Vertex =
  201.                     new org.drip.dynamics.ito.TimeR1Vertex (
  202.                         time,
  203.                         x
  204.                     );

  206.                 return _driftFunction.drift (
  207.                     localTimeR1Vertex
  208.                 ) * probabilityDensityFunction.density (
  209.                     localTimeR1Vertex
  210.                 );
  211.             }
  212.         }.derivative (
  213.             timeR1Vertex.x(),
  214.             1
  215.         ) +  new org.drip.function.definition.R1ToR1 (
  216.             null
  217.         )
  218.         {
  219.             @Override public double evaluate (
  220.                 final double x)
  221.                 throws java.lang.Exception
  222.             {
  223.                 org.drip.dynamics.ito.TimeR1Vertex localTimeR1Vertex =
  224.                     new org.drip.dynamics.ito.TimeR1Vertex (
  225.                         time,
  226.                         x
  227.                     );

  229.                 double volatility = _volatilityFunction.volatility (
  230.                     localTimeR1Vertex
  231.                 );

  233.                 return 0.5 * volatility * volatility * probabilityDensityFunction.density (
  234.                     localTimeR1Vertex
  235.                 );
  236.             }
  237.         }.derivative (
  238.             timeR1Vertex.x(),
  239.             2
  240.         );
  241.     }

  243.     /**
  244.      * Compute the Temporal Probability Distribution Function, if any
  245.      * 
  246.      * @param intialProbabilityDensityFunction The Initial Probability Density Function
  247.      * 
  248.      * @return The Temporal Probability Distribution Function
  249.      */

  251.     public org.drip.dynamics.process.R1ProbabilityDensityFunction temporalPDF (
  252.         final org.drip.function.definition.R1ToR1 intialProbabilityDensityFunction)
  253.     {
  254.         return null;
  255.     }

  257.     /**
  258.      * Compute the Steady-State Probability Distribution Function, if any
  259.      * 
  260.      * @return The Steady-State Probability Distribution Function
  261.      */

  263.     public org.drip.function.definition.R1ToR1 steadyStatePDF()
  264.     {
  265.         return null;
  266.     }

  268.     /**
  269.      * Compute the Temporal Probability Distribution Function given the Delta 0 Starting PDF
  270.      * 
  271.      * @param x0 The X Anchor for the Delta Function
  272.      * 
  273.      * @return The Temporal Probability Distribution Function given the Delta 0 Starting PDF
  274.      */

  276.     public org.drip.dynamics.process.R1ProbabilityDensityFunction deltaStartTemporalPDF (
  277.         final double x0)
  278.     {
  279.         return null;
  280.     }
  281. }
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