9191769-Stochastic-Calculus-Notes-3-5.pdf - 1 MSc Financial...

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MSc Financial Mathematics - SMM30213Brownian MotionsTimeline:1828the Brownian Motion is introduced by the Scottish royal botanist RobertBrown in an attempt to describe the irregular motion of pollen grains sus-pended in liquid1900Louis Bachelier considers Brownian motion as a possible model for stock mar-ket values1905Albert Einstein considers Brownian motion as a model of the motion of aparticle in suspension and uses it to estimate Avogadro’s number1923Norbert Wiener defines and constructs Brownian motion rigorously for thefirst time. The resulting stochastic process is often called the Wiener processin his honour.Definition 1 (Brownian motion)The processW:={Wt:t0}is called standardBrownian motion if:1.W0= 02. forst,WtWsis independent of the past history ofWuntil times, i.e. theBrownian motion has increments which are independent of the natural filtrationFs=σ(Wτ: 0τs)3. for0st,WtWsandWtshave the same distribution, which is Gaussianwith mean zero and variance(ts), i.e.WtWsD=WtsN(0, ts)4.Whas continuous sample paths1 and 3 imply thatWtN(0, t).Proposition 2Cov(Wt, Ws) = min(s, t) =stProof.Assume thatst. ThenCov(Ws, Wt)=E[(WtEWt) (WsEWs)]=E[WtWs]=E[Ws(WtWs)] +E(W2s)=s.Exercise 1 a)Define a process byXt:=whereξis a standard Gaussian randomvariable. Explain whyXis not a Wiener process.b)Define a process byXt:= 10Wt+100twhereWtis a standard one-dimensional Wienerprocess. Compute the probability of the event thatX1<900.0ccirclecopyrtLaura Ballotta - Do not reproduce without permission.
23BROWNIAN MOTIONS0285684112140168196224252280308336364-0.4-0.3-0.2-0.100.10.20.30.40.5Wt0285684112140168196224252280308336364-0.04-0.0200.020.040.060.080.10.120.140.16Bt= μt + σWt02856841121401681962242522803083363640.90.9511.051.11.15Xt= e(μ- σ2/2)t + σWtFigure 1:Sample trajectories of the Wiener process, the arithmetic Brownian motion and thegeometric Brownian motion. Parameter set:
c)Define a process byXt:=twhereξis a standard Gaussian random variable.Explain whyXis not a Wiener process.

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