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MTH5210 continuous functions
Creation date：2024-05-30 15:30:53
Belonging category：数学mathematics
Tag MTH5210
continuous functions

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MTH5210 PRACTICE Exam.

You can use 1 double-sided A4 sheet of hand written notes.
There are five questions, 20 marks each.
In the following, Bt, t ≥ 0, denotes the standard Brownian motion process started
at zero.
1. Let Vt = e
∫ t
0
µ(s)ds+
∫ t
0
σ(s)dBs , where µ(t) and σ(t), 0 ≤ t ≤ T , are deterministic
continuous functions.
(a) Show that Xt = log Vt is a Gaussian process with independent increments.
(b) Find the quadratic variation of Vt. [5 marks]
(c) Give the necessary and sufficient condition on the functions µ(t) and σ(t),
so that the process Vt is a martingale. Justify your answer. [5 marks]
(d) It is given that for any u ∈ R the process eu log Vt−u24 t3 is a martingale. Find
µ(t) and σ2(t). [5 marks]
2. Consider the following market model. Let St = Bt+ t
2 be the price process of
an asset and let βt = 1, be a savings account, 0 ≤ t ≤ T .
(a) Find the change of measure dQ/dP = Λ so that St is a martingale under
Q. You should show that EΛ = 1. [10 marks]
(b) Give the formula for the price at time t < T of the contract that pays
Y = eST at time T . [5 marks]
(c) Give the self-financing replicating portfolio for the contract in (b). [5
marks]
3. Let Mt = B
2
t −Bt − t, 0 ≤ t ≤ T .
(a) Show that Mt is a martingale. [5 marks]
(b) State what is meant by the Predictable Representation Property of Brow-
nian motion process Bt. Specify this result for the martingale Mt in this
question. [5 marks]
(c) Find a random time change τt so that the process Vt =M(τt) is a Brownian
motion. Justify this by using the theorem of Levy on characterisation of
Brownian motion. [5 marks]
(d) Write Vt in terms of Brownian motion Bt. Show that the process Xt =
V ([M,M ]t) is in fact Mt. [5 marks]
34. Solve the following partial differential equations for the function f(x, t) by
using the probability method. Specify the diffusion process and give its gener-
ator. You may exchange expectation and integration without justification, and
use that the fourth moment ofN(µ, σ2) distribution is given by µ4+6µ2σ2+3σ4.
(a) t∂
2f
∂x2
(x, t) + ∂f
∂t
(x, t) = 0, for 0 ≤ t ≤ T , and f(x, T ) = x4. [10 marks].
(b) t∂
2f
∂x2
(x, t) + ∂f
∂t
(x, t) = −x4, for 0 ≤ t ≤ T , and f(x, T ) = 0. [10 marks].
5. Explain briefly (in one or two sentences and referring to mathematical theo-
rems if possible) the following concepts.
(a) A market model and a claim or option. [5 marks]
(b) Self-financing portfolios and their use in pricing theory. [5 marks]
(c) Arbitrage, and how to check if it exists. [5 marks]
(d) Equivalent martingale measure and its role in pricing options. [5 marks]
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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 EMESTER1 MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 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CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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