Investigating the connection between stochastic differential equations and partial differential equations in order to use PDEs to solve or find characteristics of the SDE and viceversa
SDEstoPDEsandbat
Recall pox t tht which is the heat kernelsolution
of process Be satisfies pix t Ige of
Pe Pax
PDEs arisefromSDEs in numerousways
The GENERATOR L plays a keyrole
TRANSITION DENSITY
Consider theSDE
axe acte t at bexe E olBt Xs y set
Define the TRANSITION PROBABILITY DENSITY Pex tly s to be the pay of
00
It Nhhimtmw Ptp's't'd
here
immune s
y
mum
www
Often helpful in problems to think of functions
Uly s LgexPt x tly s ax Ct fixedconstant
choose
y gex so that ulys is probabilityof acxteb
Whoppingb given that Xs y
Iygygygg
y y
If we choose
g g g
gex s
paying g
axe b
o otherwise
g
Pt x.tl y s pelfof XelXs y
Xan
Mildmay ptcx.tly s pelfof it here
y
Mynydd dXt acte E att b Xe E dBe
Xs y
s t
,PROPERTIES of Pt
1 MARKOV PROPERTY increments Xe Xs are statisticallyindependent of Xuforall ucs
Why INTEGRAL VERSION of SDE
a si
pfx.tly.siyaSai PtC t Yn's
i Yn.sn s sac on
Pdfof Xt Xs yesXs ya guy y i.e oldinformation is worthless
g CHAPMAN KOLMOGOROV PROPERTY
Consider the Jointpelf plx.tiz.gly s i e jointpelf of Xe x and Xq z given
that xs y
Notice that pex t zig ly s dz
ptcx.tly s for all scgct
Ttgraigoreanpossibleintermediatestates
Recall that PLANB P AIB PCB
poly version pixt.z.gl y s plx.tlzig y s Pt zigly s
Markov
ptex.tlzig Pt Zigly s
i
FOKKER PLANCK EQUATION
Consider
pix t ptcx.tly.ge ediegnstPaifofXtlXo Y
Can we find a PDEgoverning the behaviour
Consider
ofpox t
some smoothfunctionLex
ofIggy fer
of Elf Xt ft Ifa pix tax fcxlptcx.tlox
It ECLCXED EffI
of fixeldXt If Xe ext
L Xe acxe.tlelttblxt.tlaBelt Xt blxt.tt at
Lf a nonanticipatingproperty
Effin Effcxdlalxe.tl 4t
tbfttzf
blxt.ttdp
, Eff cxaalxe.tl IL Xe b Xe E
L'cxaalxe.tl IL Xe b Xt E 2 pix E at useintegrationby
parts
fence
ftwice
I fix a x e pix t x
I fix pcx.tlbex E 2
xx dx
Boundary termsvanish e
g Ifcxlacx.tlelx fffx.pl tJ.j Iffcxlacx.tlplx.tlx
P'FIJIpolysotherwiseegffx.ndxs.is impossible
also need px so as ixia forotherintegral
Now
figs p ix e ax fix cap x BP xx at
fflx
of petcap bapxx ax D
the
yardpossiblefed
By the FUNDAMENTAL LEMMA ofCALCULUS ofVARIATIONS this is onlypossible
if Petcap bap D
Of Pet ap x 12lb p xx FOKKER PLANCK EQUATION fer D SDE
T
Drift Diffusion
EXAMPLE Xe Be axe dBe a a b s
FP E Pt LPxx
The F P E is a PARABOLIC Cheatlike PDE
Diffusion b'so ensures that it is wellposed in forwardstime
pelf pix t DIFFUSES in x as t increases
EXAMPLE axe Volt FedBe Vik o constants
FPE PttVp Kpxx Advection Diffusion in D e g dye in rivervelocity V
diffusion K
plx.tl concentrationofdye
Xt individualparticlesofdye
EXAMPLE OUPROCESS OX XEat dBt
E
Pt f Px 2 22 P PE HEI EPxx
Look for a STEADY SOLUTION psex
xp
Ifp's integrate xps raps to ppg
E
InPs
Egtina
Ps Ae Nt and get A from ftp.ixiax s NORMALISATION CONDITION
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