03 March 2022 06:17:40 PM

SDE_TEST
  C version
  Test the SDE library.

TEST01:
  BPATH generates a sample Brownian motion path.

  BPATH data stored in "bpath_data.txt".
  BPATH plot commands stored in "bpath_commands.txt".

TEST02:
  BPATH_AVERAGE generates many Brownian paths
  and averages them.

  BPATH_AVERAGE data stored in "bpath_average_data.txt".
  BPATH_AVERAGE plot commands stored in "bpath_average_commands.txt".

TEST03:
  CHAIN solves a stochastic differential equation for
  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 | Sqrt(X) - V | = 0.0142923

  CHAIN data stored in "chain_data.txt".
  CHAIN plot commands stored in "chain_commands.txt".

TEST04:
  EM solves a stochastic differential equation
  using the Euler-Maruyama method.

  | Exact X(T) - EM X(T) | =       0.662559

  EM data #1 stored in "em1_data.txt".

  EM data #2 stored in "em2_data.txt".
  EM plot commands stored in "em_commands.txt".

TEST05:
  EMSTRONG investigates the strong convergence
  of the Euler-Maruyama method.

EMSTRONG:
  Least squares solution to Error = c * dt ^ q
  (Expecting Q to be about 1/2.)
  Computed Q = 0.523444
  Residual is 0.132182

  EMSTRONG data stored in "emstrong_data.txt".
  EMSTRONG plot commands stored in "emstrong_commands.txt".

TEST06:
  EMWEAK investigates the weak convergence
  of the Euler-Maruyama method.

EMWEAK:
  Using standard Euler-Maruyama method.
  Least squares solution to Error = c * dt ^ q
  (Expecting Q to be about 1.)
  Computed Q = 1.01802
  Residual is 0.150786

  EMWEAK data stored in "emweak0_data.txt".
  EMWEAK plot commands stored in "emweak0_commands.txt".

EMWEAK:
  Using weak Euler-Maruyama method.
  Least squares solution to Error = c * dt ^ q
  (Expecting Q to be about 1.)
  Computed Q = 1.00968
  Residual is 0.184308

  EMWEAK data stored in "emweak1_data.txt".
  EMWEAK plot commands stored in "emweak1_commands.txt".

TEST07:
  MILSTRONG investigates the strong convergence
  of the Milstein method.

MILSTEIN:
  Least squares solution to Error = c * dt ^ q
  Expecting Q to be about 1.
  Computed Q = 1.0172
  Residual is 0.00449634

  MILSTRONG data stored in "milstrong_data.txt".
  MILSTRONG plot commands stored in "milstrong_commands.txt".

TEST08:
  STAB_ASYMPTOTIC investigates the asymptotic
  stability of the Euler-Maruyama method.

  For technical reasons, the plotting is done
  in the same routine as the computations.

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 stability test = -2.5

  dt = 0.5
  EM asymptotic test = 0.150451

  Data for DT = 0.5 stored in 'stab_asymptotic1_data.txt'

  dt = 0.25
  EM asymptotic test = -0.0378234

  Data for DT = 0.25 stored in 'stab_asymptotic2_data.txt'

  dt = 0.125
  EM asymptotic test = -0.14777

  Data for DT = 0.125 stored in 'stab_asymptotic3_data.txt'
  STAB_ASYMPTOTIC plot commands stored in "stab_asymptotic_commands.txt".

TEST09:
  STAB_MEANSQUARE investigates the mean square
  stability of the Euler-Maruyama method.

  For technical reasons, the plotting is done
  in the same routine as the computations.

STAB_MEANSQUARE:
  Investigate mean square stability of Euler-Maruyama
  solution with stepsize DT and MU.

  SDE is mean square stable if
    Real ( lambda + 1/2 |mu|^2 ) < 0.

  EM with DT is mean square stable if
    |1+dt^2| + dt * |mu|^2 - 1.0 < 0.

  Lambda = -3
  Mu =     1.73205
  SDE mean square stability test = -1.5

  dt = 1
  EM mean square stability test = 6

  Data for DT = 1 stored in 'stab_meansquare1_data.txt'.

  dt = 0.5
  EM mean square stability test = 0.75

  Data for DT = 0.5 stored in 'stab_meansquare2_data.txt'.

  dt = 0.25
  EM mean square stability test = -0.1875

  Data for DT = 0.25 stored in 'stab_meansquare3_data.txt'.
  STAB_MEANSQUARE plot commands stored in "stab_meansquare_commands.txt".

TEST10:
  Estimate the Ito integral of W(t) dW over [0,1].

                                                 Abs          Rel
         N        Exact        Estimate          Error        Error

       100       -0.30114922       -0.35566608       0.055       -0.18
       400       -0.17995973       -0.18591577       0.006      -0.033
      1600        0.41074701        0.41902712      0.0083        0.02
      6400       -0.49717412       -0.50117424       0.004      -0.008
     25600       -0.49994574       -0.50567961      0.0057      -0.011
    102400       -0.12558214       -0.12625784     0.00068     -0.0054
    409600       -0.46687285       -0.46555045      0.0013     -0.0028

TEST11:
  Estimate the Stratonovich integral of W(t) dW over [0,1].

                                                 Abs          Rel
         N        Exact        Estimate          Error        Error

       100        0.16801093        0.25183686       0.084         0.5
       400       0.031144274       0.017883905       0.013        0.43
      1600         2.7648883          2.869913        0.11       0.038
      6400       0.054520034       0.056285076      0.0018       0.032
     25600         1.1192225          1.100534       0.019       0.017
    102400        0.69741193        0.69694422     0.00047     0.00067
    409600        0.32644418        0.32389391      0.0026      0.0078

SDE_TEST
  Normal end of execution.

03 March 2022 06:17:47 PM