Stochastic Differential Equations
EEmhmhythm viii
I E
roughnotdifferentiableButcontinueseverywhere a s
Stochastic differential equations SDEs arebuiltusingtheBrownianor WIENER process Bt
extendstheidea randomvariableto a continuoustimeProcess to
of
TheBrownian process hasthefollowing Properties
Bo o Cine starts at theorigin
Nco
Be Bsdistributed ts
agssian rvwith
meano variance
is ts
PdfofNimra pix
prhobability
tf exp ayy
densityfunction
of sling PCBe fpiti.it Iggexp
bgecxt z g
pelf of Be hat on
Pix t Heatkernel solution of pt
Iq exp
If tp
NONOVERLAPPING increments
ofBt are INDEPENDENT
Recall events A B are independent
if PlanB PCAPCB
Btc Bts Bts Bts for tr Eta Et E tu
i e increments donotcare about what theprocess hasdoneinthepast
specialcase
of the MARKOV PROPERTY
DEFINITION a continuous process Xt satisfies the MarkovProperty if given Xs theincremen
Xe Xs is independent of Xuforall ucs
i e PCE for all of u o canes
xslxs.EE gy Plgtgg1xscurrent
state
Wiener Nowheredifferentiable everywherecontinues as
IDEA Build Be from a sumof n NormalRandomvariables in thelimit
ofsmallvarianceand
large n Correctlytaken
LAW of ADDITION of NORMALRANDOM VARIABLES
Let Xa Xn no s be i i d independentidenticallydistributions and let da an ER be a sequence
Then Sit EgriXi NCO
Isai
LEMMA 1
If X has pdf Paix then Y x X has pelf Py y Ipx t
, Proof Pyly sting PCYEjyy
D.gggEyggt8I I Px
SPECIALCASE XNNCO s Px x é Y haspelf pyly é Y Nco 22
1a
LEMMA 2
If Y NCO Ty and Z NCO TE then Yt Zn NCO Tytre
Proof Need FT see onlinelectures
FORWARD
Jfk 1yfÉfx e ax
INVERSE
fix Itjadeikolic
YesandEx s Hz x
pay og ite Pitt I É1jÉÉÉ
East independents
i e Pytz x Py Pz x where
f g x
Ijf s g x Das convolution
CONVOLUTION
of FTSTHEOREM Fg Ck Dejager
CHARACTERISTIC FUNCTION
of RANDOMVARIABLE X X D Elem
E fix LexpCDax FITPEK
4
Pyly 8 YNNCO Ty
Eye 4 5
f Iffy
Need pyck e eindy
IT f é rjéiYdy USE FEYNMAN'STrick
Take Mart ing
q PyCk Iggy eye dy
If
4 5 icy
Use PT Cdt kphpayed igtKoa e dy
Ig
Yoo ik
Itiragy dy o
Ey
e
Éytowndedthus oat In
decays
Thenwe have an ODE Py'tk off D Koy logPy pyo Ae
Pity
KII toga
We have that pro see from integralusingthefactthat payoffintegrates to 1
IF
EEmhmhythm viii
I E
roughnotdifferentiableButcontinueseverywhere a s
Stochastic differential equations SDEs arebuiltusingtheBrownianor WIENER process Bt
extendstheidea randomvariableto a continuoustimeProcess to
of
TheBrownian process hasthefollowing Properties
Bo o Cine starts at theorigin
Nco
Be Bsdistributed ts
agssian rvwith
meano variance
is ts
PdfofNimra pix
prhobability
tf exp ayy
densityfunction
of sling PCBe fpiti.it Iggexp
bgecxt z g
pelf of Be hat on
Pix t Heatkernel solution of pt
Iq exp
If tp
NONOVERLAPPING increments
ofBt are INDEPENDENT
Recall events A B are independent
if PlanB PCAPCB
Btc Bts Bts Bts for tr Eta Et E tu
i e increments donotcare about what theprocess hasdoneinthepast
specialcase
of the MARKOV PROPERTY
DEFINITION a continuous process Xt satisfies the MarkovProperty if given Xs theincremen
Xe Xs is independent of Xuforall ucs
i e PCE for all of u o canes
xslxs.EE gy Plgtgg1xscurrent
state
Wiener Nowheredifferentiable everywherecontinues as
IDEA Build Be from a sumof n NormalRandomvariables in thelimit
ofsmallvarianceand
large n Correctlytaken
LAW of ADDITION of NORMALRANDOM VARIABLES
Let Xa Xn no s be i i d independentidenticallydistributions and let da an ER be a sequence
Then Sit EgriXi NCO
Isai
LEMMA 1
If X has pdf Paix then Y x X has pelf Py y Ipx t
, Proof Pyly sting PCYEjyy
D.gggEyggt8I I Px
SPECIALCASE XNNCO s Px x é Y haspelf pyly é Y Nco 22
1a
LEMMA 2
If Y NCO Ty and Z NCO TE then Yt Zn NCO Tytre
Proof Need FT see onlinelectures
FORWARD
Jfk 1yfÉfx e ax
INVERSE
fix Itjadeikolic
YesandEx s Hz x
pay og ite Pitt I É1jÉÉÉ
East independents
i e Pytz x Py Pz x where
f g x
Ijf s g x Das convolution
CONVOLUTION
of FTSTHEOREM Fg Ck Dejager
CHARACTERISTIC FUNCTION
of RANDOMVARIABLE X X D Elem
E fix LexpCDax FITPEK
4
Pyly 8 YNNCO Ty
Eye 4 5
f Iffy
Need pyck e eindy
IT f é rjéiYdy USE FEYNMAN'STrick
Take Mart ing
q PyCk Iggy eye dy
If
4 5 icy
Use PT Cdt kphpayed igtKoa e dy
Ig
Yoo ik
Itiragy dy o
Ey
e
Éytowndedthus oat In
decays
Thenwe have an ODE Py'tk off D Koy logPy pyo Ae
Pity
KII toga
We have that pro see from integralusingthefactthat payoffintegrates to 1
IF
