RdFokkerPlanck.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>RdFokkerPlanck</i> exposes the R<sup>d</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 RdFokkerPlanck
  117. {
  118.     private org.drip.dynamics.ito.DiffusionTensor _diffusionTensor = null;
  119.     private org.drip.dynamics.ito.RdToR1Drift[] _driftFunctionArray = null;
  120.     private org.drip.dynamics.kolmogorov.RiskenOmegaEstimator _riskenOmegaEstimator = null;

  122.     /**
  123.      * RdFokkerPlanck Constructor
  124.      * 
  125.      * @param driftFunctionArray Drift Function Array
  126.      * @param diffusionTensor Diffusion Tensor
  127.      * @param riskenOmegaEstimator Risken Omega Estimator
  128.      * 
  129.      * @throws java.lang.Exception Thrown if the Inputs are Invalid
  130.      */

  132.     public RdFokkerPlanck (
  133.         final org.drip.dynamics.ito.RdToR1Drift[] driftFunctionArray,
  134.         final org.drip.dynamics.ito.DiffusionTensor diffusionTensor,
  135.         final org.drip.dynamics.kolmogorov.RiskenOmegaEstimator riskenOmegaEstimator)
  136.         throws java.lang.Exception
  137.     {
  138.         if (null == (_driftFunctionArray = driftFunctionArray) ||
  139.             null == (_diffusionTensor = diffusionTensor) ||
  140.             null == (_riskenOmegaEstimator = riskenOmegaEstimator)
  141.         )
  142.         {
  143.             throw new java.lang.Exception (
  144.                 "RdFokkerPlanck Constructor => Invalid Inputs"
  145.             );
  146.         }

  148.         int dimension = _driftFunctionArray.length;

  150.         if (dimension != _diffusionTensor.dimension())
  151.         {
  152.             throw new java.lang.Exception (
  153.                 "RdFokkerPlanck Constructor => Invalid Inputs"
  154.             );
  155.         }

  157.         for (int dimensionIndex = 0;
  158.             dimensionIndex < dimension;
  159.             ++dimensionIndex)
  160.         {
  161.             if (null == _driftFunctionArray[dimensionIndex])
  162.             {
  163.                 throw new java.lang.Exception (
  164.                     "RdFokkerPlanck Constructor => Invalid Inputs"
  165.                 );
  166.             }
  167.         }
  168.     }

  170.     /**
  171.      * Retrieve the Drift Function Array
  172.      * 
  173.      * @return The Drift Function Array
  174.      */

  176.     public org.drip.dynamics.ito.RdToR1Drift[] driftFunctionArray()
  177.     {
  178.         return _driftFunctionArray;
  179.     }

  181.     /**
  182.      * Retrieve the Diffusion Tensor
  183.      * 
  184.      * @return The Diffusion Tensor
  185.      */

  187.     public org.drip.dynamics.ito.DiffusionTensor diffusionTensor()
  188.     {
  189.         return _diffusionTensor;
  190.     }

  192.     /**
  193.      * Retrieve the Risken Omega Estimator
  194.      * 
  195.      * @return The Risken Omega Estimator
  196.      */

  198.     public org.drip.dynamics.kolmogorov.RiskenOmegaEstimator riskenOmegaEstimator()
  199.     {
  200.         return _riskenOmegaEstimator;
  201.     }

  203.     /**
  204.      * Compute the Next Incremental Time Derivative of the PDF
  205.      * 
  206.      * @param probabilityDensityFunction The PDF
  207.      * @param timeRdVertex The R<sup>d</sup> Time Vertex
  208.      * 
  209.      * @return Next Incremental Time Derivative of the PDF
  210.      * 
  211.      * @throws java.lang.Exception Thrown if the Inputs are Invalid
  212.      */

  214.     public double pdfDot (
  215.         final org.drip.dynamics.process.RdProbabilityDensityFunction probabilityDensityFunction,
  216.         final org.drip.dynamics.ito.TimeRdVertex timeRdVertex)
  217.         throws java.lang.Exception
  218.     {
  219.         if (null == probabilityDensityFunction ||
  220.             null == timeRdVertex)
  221.         {
  222.             throw new java.lang.Exception (
  223.                 "RdFokkerPlanck::pdfDot => Invalid Inputs"
  224.             );
  225.         }

  227.         final int dimension = _diffusionTensor.dimension();

  229.         final double time = timeRdVertex.t();

  231.         double pdfDot = 0.;

  233.         for (int dimensionIndex = 0;
  234.             dimensionIndex < dimension;
  235.             ++dimensionIndex)
  236.         {
  237.             final int index = dimensionIndex;

  239.             pdfDot = pdfDot - new org.drip.function.definition.RdToR1 (
  240.                 null
  241.             )
  242.             {
  243.                 @Override public int dimension()
  244.                 {
  245.                     return dimension;
  246.                 }

  248.                 @Override public double evaluate (
  249.                     final double[] xArray)
  250.                     throws java.lang.Exception
  251.                 {
  252.                     org.drip.dynamics.ito.TimeRdVertex localTimeRdVertex =
  253.                         new org.drip.dynamics.ito.TimeRdVertex (
  254.                             time,
  255.                             xArray
  256.                         );

  258.                     return _driftFunctionArray[index].drift (
  259.                         localTimeRdVertex
  260.                     ) * probabilityDensityFunction.density (
  261.                         localTimeRdVertex
  262.                     );
  263.                 }
  264.             }.derivative (
  265.                 timeRdVertex.xArray(),
  266.                 dimensionIndex,
  267.                 1
  268.             );
  269.         }

  271.         for (int dimensionIndexI = 0;
  272.             dimensionIndexI < dimension;
  273.             ++dimensionIndexI)
  274.         {
  275.             final int indexI = dimensionIndexI;

  277.             for (int dimensionIndexJ = 0;
  278.                 dimensionIndexJ < dimension;
  279.                 ++dimensionIndexJ)
  280.             {
  281.                 final int indexJ = dimensionIndexJ;

  283.                 pdfDot = pdfDot + new org.drip.function.definition.RdToR1 (
  284.                     null
  285.                 )
  286.                 {
  287.                     @Override public int dimension()
  288.                     {
  289.                         return dimension;
  290.                     }

  292.                     @Override public double evaluate (
  293.                         final double[] xArray)
  294.                         throws java.lang.Exception
  295.                     {
  296.                         org.drip.dynamics.ito.TimeRdVertex localTimeRdVertex =
  297.                             new org.drip.dynamics.ito.TimeRdVertex (
  298.                                 time,
  299.                                 xArray
  300.                             );

  302.                         return _diffusionTensor.diffusionCoefficient (
  303.                             localTimeRdVertex,
  304.                             indexI,
  305.                             indexJ
  306.                         ) * probabilityDensityFunction.density (
  307.                             localTimeRdVertex
  308.                         );
  309.                     }
  310.                 }.hessian (
  311.                     timeRdVertex.xArray()
  312.                 )[indexI][indexJ];
  313.             };
  314.         }

  316.         return pdfDot;
  317.     }

  319.     /**
  320.      * Compute the Temporal Probability Distribution Function, if any
  321.      * 
  322.      * @param intialProbabilityDensityFunction The Initial Probability Density Function
  323.      * 
  324.      * @return The Temporal Probability Distribution Function
  325.      */

  327.     public org.drip.dynamics.process.RdProbabilityDensityFunction temporalPDF (
  328.         final org.drip.function.definition.RdToR1 intialProbabilityDensityFunction)
  329.     {
  330.         return null;
  331.     }

  333.     /**
  334.      * Compute the Steady-State Probability Distribution Function, if any
  335.      * 
  336.      * @return The Steady-State Probability Distribution Function
  337.      */

  339.     public org.drip.function.definition.RdToR1 steadyStatePDF()
  340.     {
  341.         double[][] omega = _riskenOmegaEstimator.estimateOmega (
  342.             _diffusionTensor,
  343.             _driftFunctionArray
  344.         );

  346.         final double[][] omegaInverse =
  347.             org.drip.numerical.linearalgebra.Matrix.InvertUsingGaussianElimination (
  348.                 omega
  349.             );

  351.         if (null == omegaInverse)
  352.         {
  353.             return null;
  354.         }

  356.         final int dimension = _diffusionTensor.dimension();

  358.         double rdNormalizer = java.lang.Double.NaN;

  360.         try
  361.         {
  362.             rdNormalizer = java.lang.Math.sqrt (
  363.                 java.lang.Math.pow (
  364.                     2. * java.lang.Math.PI,
  365.                     -1. * dimension
  366.                 ) / new org.drip.function.matrix.Square (
  367.                     omega
  368.                 ).determinant()
  369.             );
  370.         }
  371.         catch (java.lang.Exception e)
  372.         {
  373.             e.printStackTrace();
  374.         }

  376.         final double rdNormalizerFinal = rdNormalizer;

  378.         final org.drip.function.definition.R1ToR1 r1ToR1Exponential =
  379.             new org.drip.function.definition.R1ToR1 (
  380.                 null
  381.             )
  382.         {
  383.             @Override public double evaluate (
  384.                 final double x)
  385.                 throws java.lang.Exception
  386.             {
  387.                 return java.lang.Math.exp (
  388.                     -0.5 * x
  389.                 );
  390.             }
  391.         };

  393.         return new org.drip.function.definition.RdToR1 (
  394.             null
  395.         )
  396.         {
  397.             @Override public int dimension()
  398.             {
  399.                 return dimension;
  400.             }

  402.             @Override public double evaluate (
  403.                 final double[] xArray)
  404.                 throws java.lang.Exception
  405.             {
  406.                 return rdNormalizerFinal * r1ToR1Exponential.evaluate (
  407.                     org.drip.numerical.linearalgebra.Matrix.DotProduct (
  408.                         xArray,
  409.                         org.drip.numerical.linearalgebra.Matrix.Product (
  410.                             omegaInverse,
  411.                             xArray
  412.                         )
  413.                     )
  414.                 );
  415.             }
  416.         };
  417.     }
  418. }
Created with JaCoCo 0.7.9.201702052155