Upcoming SlideShare
×

# Brownian motion by krzysztof burdzy(university of washington)

624
-1

Published on

Published in: Technology, Education
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total Views
624
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
18
0
Likes
0
Embeds 0
No embeds

No notes for slide

### Brownian motion by krzysztof burdzy(university of washington)

1. 1. BROWNIAN MOTIONA tutorialKrzysztof BurdzyUniversity of Washington
2. 2. A paradox∞<′′→∈|)(|sup,]1,0[:]1,0[tfRft(*)))((21exp)()10,)()((102′−≈<<+<<−∫ dttfcttfBtfP tεεε
3. 3. (*) is maximized by f(t) = 0, t>0Microsoft stock- the last 5 yearsThe most likely (?!?) shape of aBrownian path:
4. 4. Definition of Brownian motionBrownian motion is the unique processwith the following properties:(i) No memory(ii) Invariance(iii) Continuity(iv) tBVarBEB tt === )(,0)(,00
5. 5. Memoryless process0t2t1t 3t,,, 231201 tttttt BBBBBB −−−are independent
6. 6. InvarianceThe distribution ofdepends only on t.sst BB −+
7. 7. Path regularity(i) is continuous a.s.(ii) is nowhere differentiable a.s.tBt →tBt →
8. 8. Why Brownian motion?Brownian motion belongs to several familiesof well understood stochastic processes:(i) Markov processes(ii) Martingales(iii) Gaussian processes(iv) Levy processes
9. 9. Markov processes}0,|,{}|,{ suBstBBstB utst ≤≤≥=≥ LLThe theory of Markov processes usestools from several branches of analysis:(i) Functional analysis (transition semigroups)(ii) Potential theory (harmonic, Green functions)(iii) Spectral theory (eigenfunction expansion)(iv) PDE’s (heat equation)
10. 10. Martingales are the only family of processesfor which the theory of stochastic integrals isfully developed, successful and satisfactory.Martingalessst BBBEts =⇒< )|(∫tss dBX0
11. 11. Gaussian processesnttt BBB ,,, 21 is multidimensionalnormal (Gaussian)(i) Excellent bounds for tails(ii) Second moment calculations(iii) Extensions to unordered parameter(s)
12. 12. The Ito formula∑∫ =+∞→−=ntknknknkntss BBXdBX0//)1(/0)(lim∫ ∫ ′′+′+=t tssst dsBfdBBfBfBf0 00 )(21)()()(
13. 13. Random walkIndependent steps, P(up)=P(down){ } { }0,0,/ ≥ →≥ ∞→tBtWa taat(in distribution)tWt
14. 14. ScalingCentral Limit Theorem (CLT),parabolic PDE’s}10,{}10,{ / ≤≤=≤≤ tBatB atDt
15. 15. The effect is the same as replacingwithMultiply the probability of each Brownian pathbyCameron-Martin-Girsanov formula}10,{ ≤≤ tBt′−′∫ ∫10102))((21)(exp dssfdBsf s}10,{ ≤≤ tBt }10),({ ≤≤+ ttfBt
16. 16. Invariance (2)}10,{}10,{ 11 ≤≤−=≤≤ − tBBtB tDtTime reversal0 1
17. 17. Brownian motion and the heat equation),( txu – temperature at location x at time t),(21),( txutxutx∆=∂∂Heat equation:dxxudx )0,()( =µ)(),( dyBPdytyu t ∈= µForwardrepresentation)0,(),( yBEutyu t +=Backward representation(Feynman-Kac formula)0 tµy
18. 18. Multidimensional Brownian motion,,, 321ttt BBB - independent 1-dimensionalBrownian motions),,,( 21 dttt BBB  - d-dimensional Brownianmotion
19. 19. Feynman-Kac formula (2))(xf−= ∫ττ0)(exp)()( dsBVBfExu sxτB x0)()()(21=−∆ xuxVxu
20. 20. Invariance (3)The d-dimensional Brownian motion is invariantunder isometries of the d-dimensional space.It also inherits invariance properties of the1-dimensional Brownian motion.)2/)(exp(21)2/exp(21)2/exp(2122212221xxxx+−=−−πππ
21. 21. Conformal invariancef}0),()({ 0 ≥− tBfBf tanalytictB )( tBfhas the same distribution as∫ ′=≥tstc dsBftctB02)( |)(|)(},0,{
22. 22. If thenThe Ito formulaDisappearing terms (1)∫ ∫∆+∇+=t tssst dsBfdBBfBfBf0 00 )(21)()()(∫∇+=tsst dBBfBfBf00 )()()(0≡∆f
23. 23. Brownian martingalesTheorem (Martingale representation theorem).{Brownian martingales} = {stochastic integrals}},{,)|(0tsBFMMMMEdBXMsBttssttsst≤=∈== ∫σ
24. 24. The Ito formulaDisappearing terms (2)∫∫∫∂∂+∂∂+∂∂=−tststsstdsBsfxdsBsfsdBBsfxBtfBtf022000),(21),(),(),(),(∫∫ ∂∂+∂∂=−tststdsBsfxEdsBsfsEBtEfBtEf02200),(21),(),(),(
25. 25. Mild modifications of BMMild = the new process correspondsto the Laplacian(i) Killing – Dirichlet problem(ii) Reflection – Neumann problem(iii) Absorption – Robin problem
26. 26. Related models – diffusionsdtXdBXdX tttt )()( µσ +=(i) Markov property – yes(ii) Martingale – only if(iii) Gaussian – no, but Gaussian tails0≡µ
27. 27. Related models – stable processes2/1)(dtdB =α/1)(dtdX =Brownian motion –Stable processes –(i) Markov property – yes(ii) Martingale – yes and no(iii) Gaussian – noPrice to pay: jumps, heavy tails, 20 ≤<α220 ≤<
28. 28. Related models – fractional BMα/1)(dtdX =(i) Markov property – no(ii) Martingale – no(iii) Gaussian – yes(iv) Continuous∞<<α1∞<< 21
29. 29. Related models – super BMSuper Brownian motion is related to2uu =∆and to a stochastic PDE.
30. 30. Related models – SLESchramm-Loewner Evolution is a modelfor non-crossing conformally invariant2-dimensional paths.
31. 31. (i) is continuous a.s.(ii) is nowhere differentiable a.s.(iii) is Holder(iv) Local Law if Iterated LogarithmPath propertiestBt →tBt →tBt → )2/1( ε−1|log|log2suplim0=↓ ttBtt
32. 32. Exceptional points1|log|log2suplim0=↓ ttBtt1|)log(|log)(2suplim =−−−↓ ststBB ststFor any fixed s>0, a.s.,),0()(2suplim ∞∈−−↓ stBB ststThere exist s>0, a.s., such that
33. 33. Cut pointsFor any fixed t>0, a.s., the 2-dimensionalBrownian path contains a closed looparound in every intervaltB ),( ε+ttAlmost surely, there existsuch that)1,0(∈t∅=∩ ])1,(()),0([ tBtB