scf materials

13
<:.J{ A PTt=R. "2 - ~~17c. M~S~ 0A1.~. NQ"W) W"t' ~~ c....eJ2~ . ~~a..bt~ ~ o€~ ~ ~~l.?t p-w ~teo. 0k ~ c...lw...... .,; ~. '>"N:laM J?~ ~~ ( ", • 13 . c.e~j .~ "-e J.~~ (}- ~ t.5>tp-tW. ~~ fLebCflh fA tM CD1~ t ~G;'CfC~~0 ~ ~ ~~ck£~', . . ~~ ~ S\.-o~'-=c. ~ ua~ ... ~\- kVorw\~ ~ ~lCM'vt Vo.N:a..£.tL.. ~ ~hA.. ~~ 't RV/~ Xl) X J...) • ••. ~ ..4-t.tN\\d,. ~ C ex /1.)) c( kJ.'-) J' .. ~ ~ ,~~~~~~V~rJ,G-s- ~ .' N<!'.v t" flY, ~~ ~ r;(xtJ rr)(~ (~&-IM\"ti;~ .. ~- ~ I?f( 'K,-r \h)'- } <:: "" . ~ J2l~xD~" c2... E(X,L) . i= I~ le. f1::o .)t1Ad 1-.JI? ~t1 ~ RV~ ';') ~ PA.ct€Wn..P..L, S~lQ. ~~ ~ ~ AJ-t~S;-"CQ. . JJ- E" {'x I~ "- }-==- -<.. 'X. I) )( z.. '7 U:t.l..Q. ze- ~~p'");Ok~~ X, ~~~ ~k ~CA!fJ.Ais 0+ ~~ ~ ~\ <}</)'x L ')l<.oa. ~~~t\~r~~J ~~~~ zX)A> ':1 O.J «x) ')( >:::'0 1? K~ 6. «)(. \.J 'I.. 2..'/'::: <. x ,:j.. v. (J ~" v '7 ft.~~ -l- 1~ . L... C r-l)c a..)' ~ l. <::. '1...'2, J Xl '/ .=: C<-- ,,) J /'" z:. ) ~. < "1 +'1.'1.-) )<~.>'::' <...X() ~ ~ '7 + '-. x L) X?

Upload: meher-shiva

Post on 16-Nov-2015

36 views

Category:

Documents


0 download

DESCRIPTION

Stochastic Calculas

TRANSCRIPT

  • tp-tW. ~~fLebCflh fA tM CD1~ t ~G;'CfC~~0 ~ ~

    ~~ck~', . .~~ ~ S\.-o~'-=c. ~ ua~...

    ~\- kVorw\~ ~ ~lCM'vt Vo.N:a...tL.. ~ ~hA.. ~ ~ 't RV/~Xl) X J...) . ~ ..4-t.tN\\d,. ~ C ex /1.)) c( kJ.'-) J' ..~ ~ ,~~~~~~V~rJ,G-s-

    ~ .'N~k ~CA!fJ.Ais 0+ ~~ ~ ~ \

  • (2.)

    w~~D ~ I~ ~~ )J?t}} ~ L. ')()~/ y,~~~W~~.

    n x JJ 'li 0) /J x IJ ~ W- X~ (:;~J} ~AIJ ~ lcl(jxJl ~ ~~ 4- fWv-fe1\ xr-r-'x,JJ ~ !lX,II-rf\xLII.

    UVext-~ ~~~ ~~\~}{.z. ~ ~0J~(~lJ 'X~ :::'/i 'X I-X2..\)")

    ok:, jAJ,~Clz41b ~ IW-I('XI).lo~2.) ~6 /

  • (3)

    ~~~ ~~X/t-).;J t- E:J ~ ~ ~ ... 8t-':'~ ~~~ ~r~ "'fr~ ~ f:l) ~~ ... {,l-t-~ R.vs )(l~)~ XI6.))o. X(f")~~l-~s~~

    ,II. .~wr",f~ h In(Jf-) Co ~~ ~

    r;;: [)(~~

  • (ft),

    ~~J.~.k ~ tv\~ s~ ~- =--"J .. ~t.c.'-A M~ rrrh..A 'r x In) I::- G-1" r)., """.,g ..e.uwl:cy"' ..."'f!f~.

    r(l-J1)

  • (S)

    ~,rr~~~ AM~ ~,.g,pX{~).) t-ET4~~ ll~~~ oY ')V)"A.d.eMV~ xr,\) cJ:''t

    ~ t. WI Ix (~+c)- 'r-(~:>l ::: X. (I:-) (j)~-)b ~ ~

    tvk~ ~~ ev: ~ -f-w ~~...

    A~JoJ"" ~ ..~.X(0) t-f:T ~ ~W\.,g. ~o-k-t-' tf It:4Z~J. 1M~

    ~ . J [r-(b+-r: ~+~I)_ r(I:--t"t:)..! _"r(b/~+L0"C) ""(;!..;l~ zx' ~ -t- r( t-jh)J

    ~ ~~ ls- oJ- ~-}t-) v-J. ~ ~ ( ~

    ~~ ~ ~~~\;c. ~UL.., X(~)~ ~r LjL-~ It ~ rA. ~~J. ~ RV ~ ~ ~ VctM~lQ er-

    r( \;-).&) -::, t:[~1:--0~~-o)]= t= ~'A~ ,at-lr"~

  • (6)~~

    ~ J-. [~ l"(f-~.J..$-f-r)- i{f+t1A)L~l"'c: -)b -r-c' J

    -j I1t} ..:HC') - 1(t-;..1}] -eY. .t;f-{(;..I!:?Naw '(~)~).::. a-2-b~.

    v~~ ~l.&

    ~ -L [o-J.. (r-+9 (~T?) - o-J-({--+?)~J-c-,-c'-7b Lt' -..cr2....~ 0+z') + a-'2.t--,s

    J ~ t ~ I ~j=- ~vJv. r 2... [ b~ +r-c +-...& 1:. +t:e/_ t-..J!. -1: ~ -d - ~'t4- ~. -- ')(~.., -, t '

    t/-c' -=> () L1: ' ,_~ cr '2.. _ "C-c J ~ \) 2-. ~. (4)--r \~'-\1 --Cc: '

    ---------------------~ A ~Urn~ ~ ~r X('=-) l:-f:-T ~ "YY\. t8. ~

    ~ ~r E- [~))t-J cT ~ X(l-)~ n,.)3. ~.~OV)~ ~ [t- t \\) '1-J.

  • S..'~~~4-H~S'~ ~~

  • :::- cr-2.... j (1-t1:-~) 'A-v.., .O-t-s)

    ~ fi ll-) ~ ~ Q, tl P'Y1?tQ.&> ' ~-'\ -J:, ~~ p~~. ~ I'M .L.4J ')...

    J).. r?. ( t--+1:) - \1,.(~~ 2.:::.lJr Cel. P.,(f-~ ::: .[)..... ~--c~c:> t ~ J 'l"-.:,0 L ~ 'L":;>o ~ 1-

    .:::::-U.L :::.ob'G-')c L

  • .,~_S~~ -

    Cbv,1~ G. ~-k.I,. -t5'V)11O-N. ~ [qJ~~(:-~iN e.41A..t~. d- ?S ~ ~ f ~h~) ~ ~ 0 J 1\ L) .. n. ~L.t..d..'n::..1. .

    C\.:: ~o L.. f-J'~" . ..

  • @l\)~ 9t-~ ~4~~ f 3d~ ~ ~~1kii P~d .fj
  • (p~1-~~~.

    Q) ')Y)'..f{ ~~ -:"i ... ~ ~ (f-) fN'J (0.1 fJ ~ ~ 'h1S. ~~"''Itv~~ ~ d )(-0 \r)'\ Lq)~ )~~.A, ~ ~~~)L. '\.o\.;' ~ lA./vv\/r-

    (~ ~ ~(f:-)~ ~~~ ~~Y\M.C)1A..C. crn[~J ~ ~

    \\I ,eXt\-\ M: {( ~ l.e II :Xr~HtH- 6 H(-t ~'9~ H.:- ~x J) Xlf:-)J/

    t-~[q/~~ k itJ

    \.4) f (I',. )(!:-)-+-.(,'1I~.u-:: ""S xrH 1k + 4J y(f:) tU-~ ~ ~

    IJ ~(.,;) J X(I:-) oU- ==J X/r).u-:f -tl ~) J.b-. 0..~ 4. e,.. .~ ~ c

    .u.tt"~ ~ p~_h} Xft-)te 'Y\I).. ~~ ~J~~IN

    ~ ovv-a- fu--~ .f1t-) '-9 # ~.1\M.O\Ao6 f>\ TXT ~lk ~ LMlloJN..t~Ht;,p.~~ ~~~~~. ~d~ ~

    '1II:-J.::j (-:f-rb )

  • @M~ ~ ~ ~~. t rt:e -rn .;6,;- R4~/It~

    I~I. ~,f,No-w ~ 'J (lAJ ::: S4- J i-J'.I) X (~) Jt- .

    tv\.e-.- ~ ~W\~h\- S~~ ~~it4-.. V, ~.Jf({-) q X(Y

    ~ g.V2. ~ J \((f-) ~:f

  • J3

    W~ V, ~U.2. ~~\o;- )~~ ~~t.fJ- eJL,v-~ dt L t.,(,l X/H4f(.f),:: L!ff-) )(rl-l- J-frt-I ~~ rc4M l' 1>0. ~ 1- ~ ~~.J,..., 'flU=- J X ('t.)J.r '} Xff-) ~ -l?'P~

    {) _ oll t) -t2-\.~ 0J'01~l{ ~ ~ Ix I{-, J~2. ).:: j) -e