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08-Oct-2025 00:25:13

sde_test():
  MATLAB/Octave version 9.11.0.2358333 (R2021b) Update 7
  Test sde().

bpath():
  Brownian path simulation
Elapsed time is 0.001262 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 0.042222

bpath_vectorized():
  Brownian path simulation
Elapsed time is 0.000166 seconds.
  Graphics saved as "bpath_vectorized.png"

bpath_average():
  Average 1000 Brownian path simulations.
Elapsed time is 0.008785 seconds.
  Graphics saved as "bpath_average.png"

  Maximum error in averaged data is 0.020009

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 = 0.009541
  Graphics saved as "chain.png"

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

EM:
  Xem(Tfinal) - Xtrue(Tfinal) = 0.905154
  Graphics saved as "em.png"

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

EMSTRONG:
  Least squares solution to Error = c * dt ^ q
  Expecting a value near 0.5
  q = 0.52539
  Residual is 0.0363483
  Graphics saved as "emstrong.png"

emweak():
  Test 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 = 0.994906
  Residual is 0.0405151
  Graphics saved as "emweak0.png"

emweak():
  Test 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 = 0.966318
  Residual is 0.0446525
  Graphics saved as "emweak1.png"

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

MILSTRONG:
  Least squares solution to Error = c * dt ^ q
  Expecting a value near 0.5
  q = 1.03189
  Residual is 0.0185899
  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
  Graphics saved as "stab_asymptotic.png".

stabmeansquare():
  Check 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():
  Approximate an Ito integral.
       100     0.7983281    0.81888718     0.021     0.026

stochastic_integral_ito():
  Approximate an Ito integral.
       400   -0.43876328   -0.39618096     0.043    -0.097

stochastic_integral_ito():
  Approximate an Ito integral.
      1600    -0.4999248   -0.51235666     0.012    -0.025

stochastic_integral_ito():
  Approximate an Ito integral.
      6400    0.10202292   0.092991463     0.009     0.089

stochastic_integral_ito():
  Approximate an Ito integral.
     25600   -0.49996454   -0.50164592    0.0017   -0.0034

stochastic_integral_ito():
  Approximate an Ito integral.
    102400    -0.4795178   -0.47984949   0.00033  -0.00069

stochastic_integral_ito():
  Approximate an Ito integral.
    409600   -0.48572133     -0.485865   0.00014   -0.0003

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():
  Approximate a Stratonovich integral.
       100    0.21280928    0.20446563    0.0083     0.039

stochastic_integral_strat():
  Approximate a Stratonovich integral.
       400    0.65703283    0.64663931      0.01     0.016

stochastic_integral_strat():
  Approximate a Stratonovich integral.
      1600   0.002366621   0.014682057     0.012       5.2

stochastic_integral_strat():
  Approximate a Stratonovich integral.
      6400   0.032183765   0.027180082     0.005      0.16

stochastic_integral_strat():
  Approximate a Stratonovich integral.
     25600    0.93499493    0.93318567    0.0018    0.0019

stochastic_integral_strat():
  Approximate a Stratonovich integral.
    102400      3.027782     3.0306685    0.0029   0.00095

stochastic_integral_strat():
  Approximate a Stratonovich integral.
    409600    0.85741889    0.85897004    0.0016    0.0018

sde_test():
  Normal end of execution.

08-Oct-2025 00:25:27