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13-May-2025 17:27:50

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

bpath():
  Brownian path simulation
Elapsed time is 0.001471 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 0.034254

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 0.008866 seconds.
  Graphics saved as "bpath_average.png"

  Maximum error in averaged data is 0.029474

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 = 0.00517923
  Graphics saved as "chain.png"

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

EM:
  Xem(Tfinal) - Xtrue(Tfinal) = 0.414262
  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 = 0.52342
  Residual is 0.050042
  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 = 0.997185
  Residual is 0.0523563
  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 = 0.946417
  Residual is 0.0628648
  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 = 1.01879
  Residual is 0.0251643
  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   -0.24427711   -0.20052111     0.044     -0.18

stochastic_integral_ito():
  Approximate an Ito integral.
       400   -0.47570981   -0.45056317     0.025    -0.053

stochastic_integral_ito():
  Approximate an Ito integral.
      1600   -0.27401097   -0.27578035    0.0018   -0.0065

stochastic_integral_ito():
  Approximate an Ito integral.
      6400   -0.17555849   -0.17506378   0.00049   -0.0028

stochastic_integral_ito():
  Approximate an Ito integral.
     25600     2.1490739      2.143929    0.0051    0.0024

stochastic_integral_ito():
  Approximate an Ito integral.
    102400      1.820913     1.8214757   0.00056   0.00031

stochastic_integral_ito():
  Approximate an Ito integral.
    409600   -0.45712393   -0.45753142   0.00041  -0.00089

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   0.054035197    0.12828743     0.074       1.4

stochastic_integral_strat():
  Approximate a Stratonovich integral.
       400   0.018922851   0.033294236     0.014      0.76

stochastic_integral_strat():
  Approximate a Stratonovich integral.
      1600  0.0036419399   0.010065358    0.0064       1.8

stochastic_integral_strat():
  Approximate a Stratonovich integral.
      6400    0.22499733    0.22774499    0.0027     0.012

stochastic_integral_strat():
  Approximate a Stratonovich integral.
     25600      2.770749     2.7701527    0.0006   0.00022

stochastic_integral_strat():
  Approximate a Stratonovich integral.
    102400   0.031073406   0.032703747    0.0016     0.052

stochastic_integral_strat():
  Approximate a Stratonovich integral.
    409600   0.028154107   0.028244884   9.1e-05    0.0032

sde_test():
  Normal end of execution.

13-May-2025 17:28:03