Exchageable Sequeces, Laws of Large Numbers, ad the Mortgage Crss. Myug Joo Sog Advsor: Prof. Ja Madel May 2009
Itroducto The law of large umbers for..d. sequece gves covergece of sample meas to a costat,.e., a determstc quatty. The yeld from a mortgage ca be uderstood as a radom varable. If a bak ca make a large umber of mortgages such ther yelds are..d., the average yeld wll coverge to a determstc quatty. But evets that appear to be depedet may fact be oly exchageable. I a real facal market, there s always a fudametal factor whch ca affect all the evets at the same tme, such as a war, or a ecoomc crss whch rus the assumpto of depedece.
Idepedece of radom varables Defto: ( Ω Y P) Let,, be a probablty space. The, the fucto : Ω R s a (real - valued) radom varable f ( ω) { ω } : r Y r R. Two radom varables ad Y are depedet ff ( ) ( ) ( ) ( Y) = Y P a, Y b = P a P Y b a, b { } { } { } the also, E E E ( Y) ote: depedet ucorrelated (cov, = 0) but the coverse s ot true.
Laws of Large Numbers Weak Law of Large Numbers: ( 2 L) ( ) µ Gve,, a fte sequece of.. d. r.v.s wth E = <, Ν, P = ( + L+ ) µ as ( ) That s, lm P µ < ε = for ay ε.
Laws of Large Numbers Strog Law of Large Numbers: ( 2 L) ( ) µ Gve,, a fte sequece of.. d. r.v.s wth E = <, Ν, a. s. = ( + L+ ) µ as ( ) µ That s, P lm = =.
Exchageablty D efto: A fte sequece of,,, L of radom L varables s sad to be exchageable f 2, D ( ) ( ) L () L ( ),, =,, S ( ), π π π { L } where S ( ) s the group of permutatos of,,.
Exchageablty Example: Polya s ur A ur has tally r red ad b black balls. Draw a ball at radom ad ote ts color, ad replace the ball back ad add aother ball of the same color. th Let = f the draw yelds a red ball ad =0 otherwse. The, r r+ b r+ 2 P(,,0,) = b+ r b+ r+ b+ r+ 2 b+ r+ 3 r b r+ r+ 2 = = P b+ r b+ r+ b+ r+ 2 b+ r+ 3 Smlarly for other cases. The sequece, L,, L s exchageable. ( ),0,,.
Codtoal Expectato ( ω) { ω : } ( Ω S P) Let be r.v. o probablty space,,. Gve aother σ -algebra Y S, a r.v. Y s called codtoal expectato of gve Y f Y s Y -measurable, that s, Ω Y a Y a R ad YdP= dp A. Y ( Y ) ( Y ) I ths case, deote Y = E. A A Roughly speakg, E s averagg of to the graularty of Y (f Y s fte, averagg o the atoms of Y ).
LLN for exchageable sequeces ( ) ( Ω P) 0 ( ) σ + Let Y be a sequece of -algebras o, Y,. Y Y 0. The, a sequece of r.v.s s called a martgale f, { } ( ) E <, each ; ( ) s Y -measurable, each ; { Y } ( )E = a. s., each m. m m 0 Martgale Covergece Theorem: ( ) { } Let be a martgale s.t. sup E <. The lm = exsts a. s. ( ad s fte a. s.). Moreover, s L.
LLN for exchageable sequeces Recall { }..d., N N = Ε <+ Ε [ ] a.s., a determstc umber. Theorem: { } exchageable, N Ε <+ N Ε[ Y ] a.s., a radom varable for some Y =
LLN for exchageable sequeces proof : Let a fte sequece = (,, L) of radom varables + 2 be exchageable ad Let Y be the σ -algebra geerated by all the - symmetrc fuctos of. Y Y If f s a measurable fucto for whch E <+, ad f Y = g s bouded -symmetrc r.v., the for j, { f ( ) ( )} ( ) ( ) j g = f g j 2 L j j+ L { } E E,,,,,, = ( ) ( ) { f g } E, so that E f ( ) E { ( ) } j Y = f Y. j= ( cotued) ( )
LLN for exchageable sequeces proof : ( cotued) The, take Y f ( j) dp f ( ) dp ( A Y ) A as the dcator of A,, so that = = j The, by the defto of codtoal expectato, f = f Y = j A ( j) E { ( ) } Sce partal sums form a martgale, by the Martgale covergece theorem, lm f ( j) = E { f ( ) Y } a. s. where Y = I Y. A j= =..
De Fett s Theorem Defto : Let be r.v.s ad let Y be a σ -feld. { } Say s codtoally.. d. gve Y f for A { } R P A, Y = P A Y, ad ( Y ) ( Y ) ( ) ( ) Y j Y P A = P A a.s., for each A, j.
De Fett s Theorem De Fett's Theorem : { } { } If s a exchageable sequece the s codtoally..d. gve Y Proof : By the LLN of exchageable sequece, lm f ( j) = E { f ( ) Y } a. s. where Y = I Y. j= = y x Let f y f j x 0 y> x j= ad lm f ( j) = E { f ( ) Y } = P ( x Y ) = F ( x), ( ) =. The, ( j) = #{ ; j } j= ( ) { Y } where F x =P x s a radom dstrbuto fucto. ( cot.)
De Fett s Theorem Proof : ( cot.), L are..d. F dstrbuto fucto s.t. ( ) ( ) ( ) ( x ) = I ( ) P x L x = F x L F x x, L, x. { } x Sce P E, (, ] ( x L k xk I ) = I(, x ( ) ] L I(, xk ]( k) = F( x) L F( xk) ( ) ( ) Y P E k { Y } x, L, x : ω a F ω, x L F ω, x s -measurable. { } { { ( ) ( ) } } (, x ( ) ( ) ] L (, xk ] k Y = L k F Y ( x) F urable =,2, Lk. { I ( ) ( ) } (, x ( ) ( ) ] L I(, xk ] k F = x L xk ( x L x F) = ( x ) L ( x ) E E I I F E F x F x F where, ad F s -meas The, E F F, ad thus, P F F. k k k k
Applcato to the mortgage mess Let a r.v. be the payoff from mortgage ad bak wats to spread the rsk by makg a large umber of such mortgages ad create a mortgage pool wth determstc payoff. However, f are ot..d but oly exchageable, there s some otrval σ -algebra Y that uderles them all,.e. are codtoally..d. o Y. The the payoff seems to be (asymtotcally) determstc but s actually a radom varable, Y -measurable, so ts value chages depedg o whch set Y, the evet ω s. Thus, there are oly exchageable sequeces realty, there s o such a thg as..d. sequece sce there are always some uderlyg assumptos for whch set S Y we have ω S that ca chage.
Refereces Aldous, D. (985). Exchageablty ad related topcs. I: École d'été de Probabltés de Sat-Flour II Heequ P. L., ed. (985) Berl: Sprger. 98. Lecture Notes Mathematcs 7 Doob, J.L. The developmet of rgor mathematcal probablty (900-950). Amer. Math. Mothly, 03(7):586-595, 996. Reprted from Developmet of mathematcs 900-950, edted by J.P.Per, pp.57-70, Brkhauser, Basel, 994. Kgma, J.F.C. Uses of exchageablty. A. Probablty, 6(2):83-97, 978 Jacod, J. ad Protter, P. Probablty essetals. Uverstext. Sprger- Verlag, Berl, secod edto, 2003 Shryaev, A.N. Probablty, Vol 95 of Graduate Texts Mathematcs. Sprger-Verlag, New York, secod edto, 996.