15 September 2021 3:23:11.966 PM SDE_TEST FORTRAN90 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 | = 0.142923E-01 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 = 1.01720 Residual is 0.449634E-02 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 = 0.500000 Mu = 2.44949 SDE asymptotic stability test = -2.50000 dt = 0.500000 EM asymptotic test = 0.150451 Data for DT = 0.500000 stored in "stab_asymptotic1_data.txt". dt = 0.250000 EM asymptotic test = -0.694878E-01 Data for DT = 0.250000 stored in "stab_asymptotic2_data.txt". dt = 0.125000 EM asymptotic test = -0.204544 Data for DT = 0.125000 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 = -3.00000 Mu = 1.73205 SDE mean square stability test = -1.50000 dt = 1.00000 EM mean square stability test = 6.00000 Data for DT = 1.00000 stored in "stab_meansquare1_data.txt". dt = 0.500000 EM mean square stability test = 0.750000 Data for DT = 0.500000 stored in "stab_meansquare2_data.txt". dt = 0.250000 EM mean square stability test = -0.187500 Data for DT = 0.250000 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 0.55E-01 -0.18 400 -0.17995973 -0.18591577 0.60E-02 -0.33E-01 1600 0.41074701 0.41902712 0.83E-02 0.20E-01 6400 -0.49717412 -0.50117424 0.40E-02 -0.80E-02 25600 -0.49994574 -0.50567961 0.57E-02 -0.11E-01 102400 -0.12558214 -0.12625784 0.68E-03 -0.54E-02 409600 -0.46687285 -0.46555045 0.13E-02 -0.28E-02 TEST11: Estimate the Stratonovich integral of W(t) dW over [0,1]. Abs Rel N Exact Estimate Error Error 100 0.19885078 0.25183686 0.53E-01 0.27 400 0.92485014E-02 0.17883905E-01 0.86E-02 0.93 1600 2.8490694 2.8699130 0.21E-01 0.73E-02 6400 0.51811392E-01 0.56285076E-01 0.45E-02 0.86E-01 25600 1.1052669 1.1005340 0.47E-02 0.43E-02 102400 0.69697521 0.69694422 0.31E-04 0.44E-04 409600 0.32536627 0.32389391 0.15E-02 0.45E-02 SDE_TEST Normal end of execution. 15 September 2021 3:23:18.207 PM