optionalsamplingtest.java

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/* WARANTY NOTICE AND COPYRIGHT
This program is free software; you can redistribute it and/or
modify it under the terms of the GNU General Public License
as published by the Free Software Foundation; either version 2
of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
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You should have received a copy of the GNU General Public License
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Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA  02111-1307, USA.

Copyright (C) Michael J. Meyer

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*/



/*
 * OptionalSamplingTest.java
 *
 * Created on January 18, 2001, 4:25 AM
 */
 
package Examples.Probability;

import Statistics.*;
import Processes.*;
import java.lang.Math.*;


 

/** <p>We set up a symmetric random walk X (a martingale) and check the Optional
 * Sampling Theorem (E(X_Tau)=X_0) for a hitting time Tau (the first exit time 
 * from the interval [-20,20]). We then load the odds unfavourably to obtain a
 * supermartingale and check the Optional Sampling Theorem 
 * (E(X_Tau)&lt=X_0) again.</p>
 *
 * @author  Michael J. Meyer
 */
public class OptionalSamplingTest{

   
    public static void main(String[] args)
    {
          
        String message;
        message="Checking the Optional Sampling theorem for a random\n"+
                "walk W starting at zero sampled at the first exit time\n"+
                "from [-20,+20] (patience, large sample):\n\n"; 
        System.out.println(message);
        
        int T=2000000;    //time steps to horizon
        double W_0=0.0;   //initial value
        
        
        /** First let us throw a fair coin (martingale):
         */
        SymmetricRandomWalk W=new SymmetricRandomWalk(T,W_0); 
        
        //the set (-oo,-20)U(20,oo) as a local class
        Region_1D I=new Region_1D(){
        
           public boolean isMember(double x)
           {  
                return ((x>20)||(x<-20));  }
                
        }; //end I
        
        
        //first exit time from [-20,20]:
        final StoppingTime tau=new HittingTime_1D(W,I);
        
        //the random walk sampled at tau
        RandomVariable W_tau=W.sampledAt(tau);
        
        //Monte Carlo mean, sample size 300000
        double E=W_tau.expectation(300000);  
        
        //cut this down to 5 decimals
        E=1.0*Math.round(10000*E)/10000;
        
        message="Fair odds (martingale case): 0=W(0)=E(W_tau)="+E+"\n";
        System.out.println(message);
        
        
        /** Now we change the odds to 0.49/0.51 (supermartingale):
         */
        BiasedRandomWalk X=new BiasedRandomWalk(T,W_0,0.49); 
        
        final StoppingTime rho=new HittingTime_1D(X,I);
        
        //the random walk sampled at tau
        RandomVariable X_rho=X.sampledAt(rho);
        
        //Monte Carlo mean, sample size 50000
        E=X_rho.expectation(50000);
        
        //cut this down to 5 decimals
        E=1.0*Math.round(10000*E)/10000;
        
        message="Subfair odds (0.49/0.51, supermartingale case): "+
                "0=W(0)>E(W_tau)="+E;

                   
        System.out.println(message);
        
        
   } //end main
   
   
   
}//end OptionalSamplingTest_1