Wed Apr 30 22:40:04 2025

sde_test():
  python version:     3.10.12
  matplotlib version: 3.5.1
  numpy version:      1.26.4
  Test sde().

bpath():
  Brownian path simulation
  Computation required  -0.0002703666687011719  seconds.
  Graphics saved as "bpath.png"

bpath1():
  Brownian path simulation
  Graphics saved as "bpath1.png"

bpath2():
  Brownian path simulation
  Graphics saved as "bpath2.png"

bpath3():
  Average 1000 Brownian path simulations.
  Graphics saved as "bpath3.png"

  Maximum error in averaged data is  0.029258237272114407

bpath_average():
  Average  1000  Brownian path simulations.
  Computation required  0.01598191261291504  seconds.
  Graphics saved as "bpath_average.png"

  Maximum error in averaged data is  0.045447938779414

bpath_vectorized():
  Brownian path simulation
  Computation required  0.00010156631469726562  seconds
  Graphics saved as "bpath_vectorized.png"

chain():
  Solve a stochastic differential equation involving a function
  of a stochastic variable X.
  We can solve for X(t), and then evaluate V(X(t)).
  Or, apply the stochastic chain rule to derive an
  an SDE for V, and solve that.

  Maximum difference =  0.021033466003910317
  Graphics saved as "chain.png"

em():
  Apply the Euler-Maruyama method to an SDE.

  Xem(Tfinal) - Xtrue(Tfinal) =  7.625759669964541
  Graphics saved as "em.png"

emstrong():
  Test the strong convergence of the Euler-Maruyama method.

  Least squares solution to Error = c * dt ^ q
  Expecting a value near 0.5
  q =  45.510951010857745
  Residual is  0.3877521021831991
  Graphics saved as "emstrong.png"

emweak():
  Test the weak convergence of the Euler-Maruyama method.

  Using standard Euler-Maruyama method.
  Least squares solution to Error = c * dt ^ q
  Expecting a q value near 1
  q =  1.0089500414640569
  Residual is  0.03365132561578807
  Graphics saved as "emweak0.png"

emweak():
  Test the weak convergence of the Euler-Maruyama method.

  Using weak Euler-Maruyama method.
  Least squares solution to Error = c * dt ^ q
  Expecting a q value near 1
  q =  0.9717449171659258
  Residual is  0.045665891071889585
  Graphics saved as "emweak1.png"

milstrong():
  Test the strong convergence of the Milstein method.

  Least squares solution to Error = c * dt ^ q
  Expecting a value near 0.5
  q =  1.0264503304455188
  Residual is  0.013910580363834588
  Graphics saved as "milstrong.png"

stab():
  Check asymptotic and mean-square stability.
  Graphics saved as "stab1.png"
  Graphics saved as "stab2.png"

stab_asymptotic():
  Investigate asymptotic stability of Euler-Maruyama
  solution with stepsize DT and MU.

  SDE is asymptotically stable if
    Real ( lambda - 1/2 mu^2 ) < 0.

  EM with DT is asymptotically stable if
    E log ( 1 + lambda dt - sqrt(dt) mu n(0,1) ) < 0.
  where n(0,1) is a normal random value.

  Lambda =  0.5
  Mu =      2.449489742783178
  SDE asymptotic test =  -2.4999999999999996
  Graphics saved as "stab_asymptotic.png"

stabmeansquare():
  Check mean-square stability.
  Graphics saved as "stab_meansquare.png"

strint():
  Approximate a stochastic integral.
                                                  Abs      Rel
              N        Exact        Estimate      Error    Error

    Ito    100000   -0.49704889   -0.50169634    0.0046   -0.0094
  Strat    100000  0.0029511142  0.0024189048   0.00053      0.18

stochastic_integral_ito_test():
  Estimate the Ito integral of W(t) dW over [0,1].

                                             Abs      Rel
         N        Exact        Estimate      Error    Error

       100  -0.019303279  0.0017154541     0.021      -1.1
       400   -0.49996658   -0.49000674      0.01     -0.02
      1600   -0.48806956   -0.46878104     0.019     -0.04
      6400  -0.092664852  -0.092921942   0.00026   -0.0028
     25600   -0.41660019   -0.41730026    0.0007   -0.0017
    102400   -0.27141849   -0.27055463   0.00086   -0.0032
    409600     3.4224037     3.4228411   0.00044   0.00013

stochastic_integral_strat_test():
  stochastic_integral_strat() estimates the Stratonovich
  integral of W(t) dW over [0,1].

                                             Abs      Rel
         N        Exact        Estimate      Error    Error

       100    0.12989271    0.18790147     0.058      0.45
       400    0.12068011    0.10257367     0.018      0.15
      1600    0.40903571    0.41844121    0.0094     0.023
      6400     1.3229228     1.3075181     0.015     0.012
     25600  0.0016105814  0.00063667537   0.00097       0.6
    102400    0.35048817    0.35276745    0.0023    0.0065
    409600    0.23537708    0.23541654   3.9e-05   0.00017

sde_test():
  Normal end of execution.
Wed Apr 30 22:40:12 2025