18-Jun-2023 08:58:03

sde_test():
  MATLAB/Octave version 5.2.0
  Test sde().

bpath():
  Brownian path simulation
Elapsed time is 0.004565 seconds.
  Graphics saved as "bpath.png"

bpath_vectorized():
  Brownian path simulation
Elapsed time is 8.79765e-05 seconds.
  Graphics saved as "bpath_vectorized.png"

bpath_average()
  Average 1000 Brownian path simulations.
Elapsed time is 0.0216219 seconds.
  Graphics saved as "bpath_average.png"

  Maximum error in averaged data is 0.029095

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 = 0.00836347
  Graphics saved as "chain.png"

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

EM:
  Xem(Tfinal) - Xtrue(Tfinal) = 0.236012
  Graphics saved as "em.png"

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

EMSTRONG:
  Least squares solution to Error = c * dt ^ q
  Expecting a value near 0.5
  q = 0.527929
  Residual is 0.0137309
  Graphics saved as "emstrong.png"

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

EMWEAK:
  Using standard Euler-Maruyama method.
  Least squares solution to Error = c * dt ^ q
  Expecting a value near 1
  q = 0.953218
  Residual is 0.0905497
  Graphics saved as "emweak0.png"

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

EMWEAK:
  Using weak Euler-Maruyama method.
  Least squares solution to Error = c * dt ^ q
  Expecting a value near 1
  q = 0.963571
  Residual is 0.0650395
  Graphics saved as "emweak1.png"

milstrong() tests the strong convergence of the Milstein method.

MILSTRONG:
  Least squares solution to Error = c * dt ^ q
  Expecting a value near 0.5
  q = 1.04379
  Residual is 0.020589
  Graphics saved as "milstrong.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 =     2.44949
  SDE asymptotic test = -2.5

  dt = 1
  EM asymptotic test = 0.44471

  dt = 0.5
  EM asymptotic test = 0.114679

  dt = 0.25
  EM asymptotic test = -0.0943688
  Graphics saved as "stab_asymptotic.png".

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

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

                                             Abs      Rel
         N        Exact        Estimate      Error    Error


stochastic_integral_ito() approximates an Ito integral.
       100   -0.10596285   -0.12630468      0.02     -0.19

stochastic_integral_ito() approximates an Ito integral.
       400   -0.38353012   -0.37481511    0.0087    -0.023

stochastic_integral_ito() approximates an Ito integral.
      1600   -0.49978914   -0.52277554     0.023    -0.046

stochastic_integral_ito() approximates an Ito integral.
      6400  -0.069911968  -0.075886632     0.006    -0.085

stochastic_integral_ito() approximates an Ito integral.
     25600   -0.13166983   -0.13159223   7.8e-05  -0.00059

stochastic_integral_ito() approximates an Ito integral.
    102400   -0.45502565    -0.4565369    0.0015   -0.0033

stochastic_integral_ito() approximates an Ito integral.
    409600   0.013855888   0.014765204   0.00091     0.066

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


stochastic_integral_strat() approximates a Stratonovich integral.
       100   0.031189692   0.018957407     0.012      0.39

stochastic_integral_strat() approximates a Stratonovich integral.
       400    0.14856278    0.14271509    0.0058     0.039

stochastic_integral_strat() approximates a Stratonovich integral.
      1600     1.5900703     1.5897263   0.00034   0.00022

stochastic_integral_strat() approximates a Stratonovich integral.
      6400    0.13984083    0.14339502    0.0036     0.025

stochastic_integral_strat() approximates a Stratonovich integral.
     25600      0.232725     0.2386596    0.0059     0.026

stochastic_integral_strat() approximates a Stratonovich integral.
    102400    0.52705585    0.52771619   0.00066    0.0013

stochastic_integral_strat() approximates a Stratonovich integral.
    409600    0.56235532     0.5611869    0.0012    0.0021

sde_test():
  Normal end of execution.

18-Jun-2023 08:58:27