08-Jan-2022 09:27:49

sde_test():
  MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2
  Test sde().

BPATH
  Brownian path simulation
Elapsed time is 0.003094 seconds.
  Graphics saved as "bpath.png"

BPATH_VECTORIZED
  Brownian path simulation
Elapsed time is 0.000986 seconds.
  Graphics saved as "bpath_vectorized.png"

BPATH_AVERAGE
  Average 1000 Brownian path simulations.
Elapsed time is 0.039470 seconds.

  Graphics saved as "bpath_average.png"

  Maximum error in averaged data is 0.027278

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, we can apply the stochastic chain rule to derive an
  an SDE for V, and solve that.

  Maximum difference = 0.00480098
  Graphics saved as "chain.png"

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

EM:
  Xem(Tfinal) - Xtrue(Tfinal) = 0.119163
  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 = 0.535065
  Residual is 0.020036
  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 = 1.00404
  Residual is 0.105014
  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 = 0.941048
  Residual is 0.0341653
  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 = 1.0328
  Residual is 0.0129414
  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 = 0.418118

  dt = 0.5
  EM asymptotic test = 0.167537

  dt = 0.25
  EM asymptotic test = -0.122654
  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   0.050502081  -0.065101999      0.12       2.3

stochastic_integral_ito() approximates an Ito integral.
       400    -0.2940163   -0.25533496     0.039     -0.13

stochastic_integral_ito() approximates an Ito integral.
      1600    -0.4902263   -0.48900014    0.0012   -0.0025

stochastic_integral_ito() approximates an Ito integral.
      6400  -0.050810943  -0.062394504     0.012     -0.23

stochastic_integral_ito() approximates an Ito integral.
     25600    0.52077235    0.52012375   0.00065    0.0012

stochastic_integral_ito() approximates an Ito integral.
    102400   -0.16014824   -0.15791005    0.0022    -0.014

stochastic_integral_ito() approximates an Ito integral.
    409600   -0.45330208   -0.45417355   0.00087   -0.0019

STOCHASTIC_INTEGRAL_STRAT_TEST:
  Estimate 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     1.7569096     1.7194951     0.037     0.021

stochastic_integral_strat() approximates a Stratonovich integral.
       400    0.48749578     0.4866482   0.00085    0.0017

stochastic_integral_strat() approximates a Stratonovich integral.
      1600     1.4806852     1.5008999      0.02     0.014

stochastic_integral_strat() approximates a Stratonovich integral.
      6400   0.051390831   0.047949909    0.0034     0.067

stochastic_integral_strat() approximates a Stratonovich integral.
     25600     0.1893866    0.18928291    0.0001   0.00055

stochastic_integral_strat() approximates a Stratonovich integral.
    102400    0.35142323      0.352795    0.0014    0.0039

stochastic_integral_strat() approximates a Stratonovich integral.
    409600   0.028277635   0.028461763   0.00018    0.0065

sde_test():
  Normal end of execution.

08-Jan-2022 09:28:55