4/15/2021 HW5 E[1 – JXpWeU NoWebook localhoVW:8888/noWebookV/DoZnloadV/HomeZoUk/HW5 E[1.ipnb# 1/5 IQ|[39]: :L[[PUN! :PTSH[PVUPZHSZVZLKPU[OL UHUJPHSTHYRL[-VYL_HTWSLZPTSH[PVUJHUILZLK[VZPTSH[L Z[VJRWYPJL:PUJL[OLZ[VJRWYPJLPZZHSS`HZZTLK[VILJVU[PUVZS`JOHUNPUNWLVWSLZHSS` KPZJYL[PaL[OL[PTLPU[VZTHSSPU[LY]HSZH[ HUKZPTSH[LZ[VJRWYPJLH[LHJO[PTLWVPU[ 8B7VPU[ZD ;VWYLWHYLMVYZ[VJRWYPJLZPTSH[PVUSL[Z YZ[SVVRH[HZPTWSLYWYVJLZZ!)YV^UPHUTV[PVUWYVJLZZ ;OLZ[HUKHYK)YV^UPHUTV[PVUJHUILKL ULKHZMVSSV^Z! MVYHU`NP]LU[^V[PTLWVPU[Z HUK ^OLYL PZHUVYTHSKPZ[YPI[PVU^P[OHZ[OLTLHUHUK HZ[OL]HYPHUJL 5V[PJL[OLZPTPSHYP[`IL[^LLU)YV^UPHUTV[PVUHUKHOVTVNLULVZ7VPZZVUWYVJLZZ! >LJHUZPTSH[L)YV^UPHUTV[PVUZPUN[OLMVSSV^PUNHSNVYP[OT! :[HY[^P[O HUK MVY [V ! ZHTWSL MYVT WKH[L ZPUN )HZLKVU[OLWYL]PVZHSNVYP[OT :PTSH[L H[ #This imports all the packages ou need for the homeZork #Please run this first import QXmS as QS import VciS.VSecial as VSVS import maWSlRWlib.SSlRW as SlW %maWSlRWlib iQliQe import VmS import VciS.VWaWV as VSVW import VimS import cRS !SiS iQVWall hSeURSW -T import hSeURSW 4/15/2021 HW5 E[1 – JXpWeU NoWebook localhoVW:8888/noWebookV/DoZnloadV/HomeZoUk/HW5 E[1.ipnb# 2/5 :OV^[OL[YHQLJ[VY`VM ZPUNYVUKZVMZPTSH[PVUB@V^PSSILWYV]PKPUNSPULWSV[ZPU [OLZHTLNYHWOD IQ|[22]: IQ|[23]: IQ|[24]: 8B7VPU[ZD )YV^UPHUTV[PVUWYVJLZZPZZHSS`JVUZPKLYLK[VIL[VVZPTWSLHUKUYLHSPZ[PJMVY[OLZ[VJR THYRL[-VYL_HTWSL[OLZ[VJRWYPJLZOVSKILWVZP[P]LHUKZ[HY[MYVTHUHYIP[YHY`]HSL(SZV KP LYLU[Z[VJRZZOVSKMVSSV^KP LYLU[[YLUKZHUK]HYPH[PVUZ(ZHYLZS[WLVWSLZLH.LVTL[YPJ )YV^UPHUTV[PVU.)4WYVJLZZ[VTVKLSZ[VJRWYPJL-VY[OL.)4WYVJLZZWLVWSLHZZTL[OH[ MVYHU`NP]LU[^V[PTLWVPU[Z HUK ^OLYL HUK HYLJHSSLKKYPM[HUK]VSH[PSP[`YLZWLJ[P]LS ` WVZP[P]LS`PTWHJ[Z ^OPSL HUK IV[OWVZP[P]LS`H LJ[Z PZ[OLPUP[PHSZ[VJRWYPJL PZ[OLZ[VJRWYPJLH[[PTL (ZHYLZS[WLVWSLJHUZPTSH[L.)4WYVJLZZMYVT [V ZPUN[OLMVSSV^PUN HSNVYP[OT! ZL[ MVY [V ! ZHTWSL MYVT WKH[L ZPUN )HZLKVU[OLWYL]PVZHSNVYP[OT :PTSH[L H[ HZZTPUN :OV^[OL[YHQLJ[VY`VM ZPUNYVUKZVMZPTSH[PVUB@V^PSSILWYV]PKPUNSPULWSV[ZPU [OLZHTLNYHWOD #generate Bt, an arra that giYes Bt Yalue for each t Yalue in t_arra def B_W(W): return BW ##run the B_t(t) 5 times and plot the trajector for each case QS.UaQdRm.Veed(10) 4/15/2021 HW5 E[1 – JXpWeU NoWebook localhoVW:8888/noWebookV/DoZnloadV/HomeZoUk/HW5 E[1.ipnb# 3/5 IQ|[25]: IQ|[26]: IQ|[27]: 8B7VPU[ZD (ZZTL[OH[^L^HU[[VZLZPTSH[PVU[VLZ[PTH[L ^OLU /LYL PZ[OLH]LYHNLWYPJLVMZ[VJRWYPJLIL[^LLU HUK .LULYH[LZHTWSLZVM ZPUN -VYLHJOYVUK[OLZHTWSLVM JHUILHWWYV_PTH[LKZPUN[OLH]LYHNLVMHSS[OLZ[VJRWYPJLZ^LJHSJSH[LKIL[^LLUHUK; )HZLK[OLZLZHTWSLZYLWVY[[OLJLU[LYHUK[OL^PK[OVMH JVU KLUJLPU[LY]HSMVY (WWS`[OLHU[P[OL[PJTL[OVK[V HUKYLWVY[[OLJLU[LYHUK[
OL^PK[OVMH JVU KLUJL PU[LY]HSMVY IHZLKVUNLULYH[PUNNLULYH[LZHTWSLZVM ZPUN IQ|[28]: IQ|[29]: 8B7VPU[ZD (ZZTL^L^HU[[VLZ[PTH[L ^OLU >P[OV[ZPUNHU`]HYPHUJLYLKJ[PVU[LJOUPXLNLULYH[LZHTWSLZVM ZPUN )HZLK[OLZLZHTWSLZYLWVY[ [OLJLU[LYHUK[OL^PK[OVMH JVU KLUJLPU[LY]HSMVY ZHTWSLZVM ZPUN )HZLK[OLZL ZHTWSLZYLWVY[[OLJLU[LYHUK[OL^PK[OVMH JVU KLUJLPU[LY]HSMVY #generate S_t def S_W(W,a, mX, Vigma2): return SW ##run the S_t(t) 5 times and plot the trajector for each case QS.UaQdRm.Veed(10) #Zithout using Yariance reduction QS.UaQdRm.Veed(10) #antithetic method QS.UaQdRm.Veed(10) 4/15/2021 HW5 E[1 – JXpWeU NoWebook localhoVW:8888/noWebookV/DoZnloadV/HomeZoUk/HW5 E[1.ipnb# 4/5 IQ|[30]: IQ|[31]: 8B7VPU[ZD (ZZTL[OH[HWLYZVU^HU[Z[VTHRLHUPU]LZ[TLU[KLJPZPVU;VL]HSH[L[OPZKLJPZPVU[OPZWLYZVU PZLZWLJPHSS`PU[LYLZ[LKPUTLHZYPUN HZZTPUN HUK ;VLZ[PTH[L[OPZWYVIHIPSP[ `^LJHUKV[OLMVSSV^PUN! ZPTSH[L[OLZ[VJRWYPJLIL[^LLU HUK ZPUN ,HJO[PTL^LJVTW[L ^OLYL PZHZHTWSLVM PM PZ ;YL6[OLY^PZL -VSSV^PUN[OPZ JHUILLZ[PTH[LKZPUN -VY[OPZXLZ[PVU )HZLKVU[OPZTL[OVKLZ[PTH[L IHZLKVUZPTSH[PUN[OLZ`Z[LT[PTLZ )HZLKVUJLU[YHSSPTP[[OLVYLTJHSJSH[L[OL^PK[OVM[OL JVU KLUJLPU[LY]HS 9LWVY[IV[O]HSLZ IQ|[32]: 8B7VPU[ZD 0[JHUILKLYP]LK[OH[[OLL_WLJ[LKZ[VJRWYPJLH[[PTL OHZHJSVZLKMVYTZVS[PVUPM[OLZ[VJR WYPJLMVSSV^Z[OL.LVTL[YPJ)YV^UPHU4V[PVUKL ULKPU :WLJP JHSS ` ]HYPHISLPZ[OLZ[VJRWYPJLH[ ,Z[PTH[L IHZLKVUZPTSH[PUN[OLZ`Z[LT [PTLZ )HZLKVU[OLJLU[YHSSPTP[[OLVYLTJHSJSH[L[OL^PK[OVM[OL JVU KLUJLPU[LY]HSVM 9LWVY[IV[O]HSLZ IQ|[40]: ##Zithout Yariance reduction QS.UaQdRm.Veed(10) #Common random number QS.UaQdRm.Veed(10) QS.UaQdRm.Veed(10) #control Yariate QS.UaQdRm.Veed(10) 4/15/2021 HW5 E[1 – JXpWeU NoWebook localhoVW:8888/noWebookV/DoZnloadV/HomeZoUk/HW5 E[1.ipnb# 5/5 8B7VPU[ZD >LJHUZLLPU8[OH[P[PZH]LY`ZTHSSJOHUJL[VVIZLY]L 3L[ZZLPTWVY[HUJLZHTWSPUN[VYLKJL[OL]HYPHUJL>OLUZHTWSPUN PUZ[LHKVMZHTWSPUNMYVT SL[ZZHTWSL ZV[OH[P[PZTVYLSPRLS`MVYZ[V ZLL (ZZTL[OH[ ZLPTWVY[HUJLZHTWSPUN[VYLKJL[OL]HYPHUJLPU8,Z[PTH[L IHZLKVUZPTSH[PUN[OLZ`Z[LT[PTLZ )HZLKVU[OLJLU[YHSSPTP[[OLVYLTJHSJSH[L[OL^PK[OVM[OL JVU KLUJLPU[LY]HSVM 9LWVY[IV[O]HSLZ IQ|[34]: 8B7VPU[ZD >LJHUZLO`WLYVW[[V UK[OLILZ[WHYHTL[LY :LHYJO IL[^LLUHUKZPUN hSeURSW [VYLKJL[OL^PK[OVM[OL JVU KLUJL PU[LY]HSVM IHZLKVUZHTWSLZ>OLUKVPUN[OLZLHYJOWLYMVYT WSe HSNVYP[OTHUKZLP[LYH[PVUZ5VULLK[VZL[WLHYS`Z[VWWPUN9LWVY[[OLILZ[ 9LWVY[[OLLZ[PTH[L )HZLKVUJLU[YHSSPTP[[OLVYLTJHSJSH[L[OL^PK[OVM[OL JVU KLUJLPU[LY]HSVM 9LWVY[[OL]HSL IQ|[35]: IQ|[36]: IQ|[37]: IQ|[38]: #importance sampling QS.UaQdRm.Veed(10) ##hopt to find optimal alpha ##report the best alpha #use more samples to estimate the probabilit and Zidth of confidence inter