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Stochastic Analysis MATH4512

创建日期：2024-06-03 15:48:10

所属分类：数学mathematics

标签： MATH4512

Stochastic Analysis

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Mathematics and Statistics

Assignment 1: Stochastic Analysis
MATH4512
1. Let τ be an exponentially distributed random variable, so P(τ > t) = e−λt for some λ > 0 and
each t ∈ [0,∞).
(a) Let Ft = σ(1{τ≤s} : s ≤ t), and show that τ is an (Ft)t stopping time.
(b) Find an expression for E[τ |Ft].
(c) Are either τ2 or
√
τ also (Ft)t stopping times? Justify your answer.
2. Let (Wt,Ft) be a Brownian motion.
(a) Find the distribution of random variable 5W1 −W3 +W7.
(b) For what values of parameters a, b ∈ R, the random variable aW1 −W2 and W3 + bW5
are independent?
3. Let (Wt,Ft)t∈[0,1] be a Brownian motion and define
Ht =
(
(1 + t)W 1
1+t
−W1
)
t≥0
.
Show that (Ht,Ht)t≥0 is a Brownian motion, where the filtration (Ht)t should be specified.
4. For a stopping time τ we define a σ-algebra Fτ by
Fτ = {A ∈ F : A ∩ {τ ≤ t} ∈ Ft ∀t ≥ 0}.
(a) Let τ and σ be stopping times. Show that the sets {τ < σ}, {τ = σ} and {τ ≤ σ} are
elements of Fτ , Fσ and Fτ∧σ respectively.
(b) Let σ be a stopping time such that τ ≥ σ and τ is Fσ-measurable, then τ is a stopping
time.
5. Let (Wt) be a Brownian motion.
(a) Using Itoˆ’s formula find ⟨W 2⟩t and ⟨W, eW ⟩t.
(b) Let f(x) = |x| ∧M for some M > 1. Show that f is a difference of two convex functions
and compute its left derivative and second distributional derivative.
(c) Let Xt = f (exp(Wt)) (where f : R → R is given in (b)). Using Itoˆ-Tanaka formula find
semimartingales decomposition of X
6. Question 33 in MRLN.

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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 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