Discrete time approximation for continuously and discretely reflected BSDE's

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(1)Discrete time approximation for continuously and discretely reflected BSDE’s Bruno Bouchard, Jean-François Chassagneux. To cite this version: Bruno Bouchard, Jean-François Chassagneux. Discrete time approximation for continuously and discretely reflected BSDE’s. Stochastic Processes and their Applications, Elsevier, 2008, 118, pp.22692293. �hal-00020697�. HAL Id: hal-00020697 https://hal.archives-ouvertes.fr/hal-00020697 Submitted on 14 Mar 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(13). F"$ hk"YfR.4"Y4XY"g.+-#4I&+- !./ hk"Gq+J)N -)Yh -13k+-#4U T >0 (Ω, F, P) :=/4XY"$.+- !2c<S#+]'>\!uXY+-+- MGop"G-3XY"Y !1."7R.24#N!+- d W F = (Ft )t≤T UZ"$."$#N!",/ehKQ !R ",>." 3.3 !2-3Xg+-.!./ ! M W. {. FT = F.

(14) F"$. X. hk"C."1+-243+-e+-. [0, T ]. Xt = X 0 +. +!5." +;) -)1/k"$#"$J\!2 ",PJ3 !+-. Z. t. b(Xu )du +. 0. Z. t. σ(Xu )dWu 0. '>."$#" %!./H% !./ !#"0-3XY",/ +1hk" FH4.)N4IZ% X0 ∈ R d b : Rd 7→ Rd σ : Rd 7→ Md L− rM "VM |b(x) − b(y)| + |σ(x) − σ(y)| ≤ L|x − y|. 5L+-#!242. Ý. ~M . x, y ∈ R d .. "$#" ."1 -),"1+!5 :=/4XY"$.+- !2 XG!#),",&% /"$.+-",." B 3.)$24/\!e.+-#X +- Md d |·| +-# !./!242"$2"$XY"$J0+!5 d !#" v;"$'S",/j-),+-243XgvZ",)$+-#&M Rd Md R <(Q),+-KvZ"$J+-% '("-3XY" ! MT."C5t+-242+&'>4U.%.'(" |X0 | + T + |b(0)| + |σ(0)| ≤ L GUZ"$."$#)k+V44vZ"7),+-.N!J'>)Np/"$ "$./C+-24Qe+- h3XG,Q !242[/"$.+-" hJQ CL L N!?Z"*/ k"$#"$JvV!243.", M[op" '>#4" p 54/"$k"$./c5L#+-X !"Es#N7 !#N!XY"$"$# M . CL. p>0. Ý. 8 +-#2\!"$#3."V%'(" #",)&!242." '("$242:r?J.+]'>j),+-.",PJ3."$.),"1+!5 ~M k sup |Xt | kLp t≤T. '>."$#"V%.5L+-# #N!./+-X. v-!#\!h2". ξ. %'(" .+-". ~Mm~ . ≤ CLp . 1. kξkLp := E [|ξ|p ] p. M. bc."C/)$#"$"E:r4XY" !#+&s;4XG!+-e+!5 -(hk","$ '>/"$24QG3./",/ 4f."0244"$#N!3#"V%"," X "VM U.M^ a=Myo|."$ )&!.+-h "Ck"$#5L",)$24Qq4X32\!",/H% '(" 3."C."1N!./.!#/ B 32"$# (Xti )i≤N N)."$XY" /"ER..",/u5L+-#7 !#4+- +!5 % Xπ π := {0 = t0 < t1 < . . . < tN = T } [0, T ] %hKQ. Ý. N ≥1 ( X0π Xtπi+1. = X0 = Xtπi + b(Xtπi )(ti+1 − ti ) + σ(Xtπi )(Wti+1 − Wti ) , i ≤ N − 1 .. Y."N",PK3."$2r%K'S" !242..+-" ."XY+;/3243.6+!5 !./G-N3XY" |π| := maxi≤N −1 (ti+1 − ti ) π ! N |π| ≤ L. '>)N .+-2/7'>4 '>."$x."fUV#/ #"$UV32\!#&%SrM "VM 5t+-#g!242 L ≥ 1 π (ti+1 − ti ) = |π| M. i≤N −1. W>3.3 !2r%.'S"1/"ER.."*7),+-J4K3.+-3.:r4XY" vZ"$#+-j+!5. Xπ. hKQ"$4U. Xtπ = Xtπi + b(Xtπi )(t − ti ) + σ(Xtπi )(Wt − Wti ) , t ∈ [ti , ti+1 ) , i ≤ N − 1 .

(15) . t≤T. Ý. >>'S"$242?;.+&'>e !3./"$#. k sup |Xt − Xtπ | kLp + max k i<N. sup t∈[ti ,ti+1 ]. ~Mm{ . ~M . 1. |Xt − Xtπi | kLp ≤ CLp |π| 2 , p ≥ 1 .. . ~M .

(16) 4UGN!./.!#/!#UV3XY"$K+-."1)&!j!2+g+-hN!49g),+-./4+- !2vZ"$#+-j+!5>#",324 i h Ei |Xti+1 − Xtπi+1 |2 ≤ eCL |π| |Xti − Xtπi |2 + CL |π|2 Ei (XT∗ )2 ∀ i ≤ N − 1 ,. '>."$#". Ei [·] M maxt≤T |Xt |. /"$.+-",9."O),+-./4+- !2Y"Esk",)$N!+-. %. E [· | Fti ] i ≤ N. %*!./. ~M } . XT∗ :=. =5[.+-4UG"$2"C>k",)$R ",/H% π '>4242 !24'(&Q;0/"$.+-"C."*!hk+&vZ"1/"ER..",/#+;),",&M[o|."$ X )&!h "Ck"$#5t",)$24Qf4X32\!",/e!>." 4XY", %."$#"C>.+7.",",/+74K#+;/3.)," 4 X (ti )i≤N B 32"$#7N)."$XY"VMd=w7)&-"V%'S"qN&Qp ! i @ .+-2/g!./w/"ER.." -4 ~Mm{ +- Xπ h3'>4 42\-),"1+!5 π %rM "VM [ti , ti+1 ). X ti. X ti. Xtπ = Xti + b(Xti )(t − ti ) + σ(Xti )(Wt − Wti ) , t ∈ [ti , ti+1 ) , i ≤ N − 1 .. . . . k1 . cN . . . . . ,> . .

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(18) y y . . !#"$&%'. 9N",)$+-%'("/"ER.."g /)$#"$"$24Q#"ED ",)$",/d< @;ACB Mbc." #"ED ",)$+- +-k"$#N!",+-24Q !."1/.!",. 5t+-#7+-XY" rκ := T. 0 < r1 < · · · < rκ−1 < T. M op"f"$ ' ."$#" hKQO),+-JvZ"$K+- > !./ < = {rj , 0 ≤ j ≤ κ} r0 := 0 A B [),+-32" M bc."c+-243+-g+!5 ."(/N)$#"$"$24Q*#"ED ",)$",/g< @;C !5 Q;4U b b N κ ≥ 1. (Y , Z ). YTb = Y˜Tb := g(XT ). !./H%5L+-#. "$#" . (Z b )0. !./. %. j ≤κ−1 t ∈ [rj , rj+1 ) ( b Rr Rr Y˜t = Yrbj+1 + t j+1 f (Θbs )ds − t j+1 (Zsb )0 dWs , Ytb = R t , Xt , Y˜tb .. Ý. {M . !5 Q -+ % % % g≥h Rd f : Rd × R × Rd 7→ R Θb := (X, Y˜ b , Z b ) >." #N!.k+V",/vZ",)$+-#0+!5 b . % !./. h, g : Rd 7→ R. Z. R(t, x, y) := y + [h(x) − y]+ 1{t∈<\{0,T }} , (t, x, y) ∈ [0, T ] × Rd+1 .. < QG+-243+-%;'S"0XY"&!q!-/.!",/f#+J),",N ( '>."$#"V%J5L+-# % (Y b , Z b ) ∈ S 2 × H2 p ≥ 1 Sp ."1"$+!5[#"&!2v-!243.",/#+-UV#",4vZ"$24QqXY"&-3#N!h2" 3.) ! U. ||U ||S p. !./. Hp. := k sup |Ut | kLp < ∞ , t≤T. >." N"$+!5#+-UV#",4vZ"$24QfXY"&-3#N!h2" ||V ||Hp. := k. Z. T. Rd 2. |Vr | dr. 0. }. :rvV!243.",/#+;),",",. 21. k Lp < ∞ .. V. N!5LQJ4U.

(19) p."75t+-242+&'>4U.% 'S" ! 242T"Es"$./ ."Y/"ER.4+- +!5 !./ +#+;),",", || · ||S p || · ||Hp % .","*"Es;"$.+-.h "$4UG/"ER..",/4j7#N!4UVK5t+-#'(!#/j'(&QZM '>4ev-!243.",4 d +-# d R. . h. Ý. M. h.N"$#vZ" !C."7+-243+- +!5 !./ !#" :=FH4.N)4I f. {M )&!9h " ),+-.#3.)$",/d",),"$'>"VM W3Xg4Uq !. g. L. |g(x1 ) − g(x2 )| + |h(x1 ) − h(x2 )| + |f (θ1 ) − f (θ2 )| ≤ L (|x1 − x2 | + |θ1 − θ2 |). 5t+-#!242 ! ./ %."Y"Es;"$.),"G!./ 3PJ3."$.",*+!5>." x1 , x 2 ∈ R d θ1 , θ 2 ∈ R d × R × R d N+-243+-Y5t+-242+&'x5L#+-X ^ {!a=M[<(Q ),+-JvZ"$K+-%;'S"-N3XY"> ! M. .

(20) . Ý. . '>4. Ý. 8.+-#2\!"$#3."V%+-h."$#vZ" !. Y˜tb = g(XT ) +. Z. T. t. {M )&!h "C'>#4"$d- Z. f (Xu , Y˜ub , Zub )du −. ˜ tb := K. κ−1 h X j=1. |g(0)| + |h(0)| + |f (0)| ≤ L. T t. ˜ Tb − K ˜ tb , t ≤ T (Zub )0 dWu + K. h(Xrj ) − Y˜rbj. i+. {Mm~ . 1{rj ≤t} .. <(Qp#"$k"&!4Ud."f!#UV3XY"$Jg+!5." #+J+!5+!50iT#+- +V4+- {M } 4|^mlna=%['("f."$w"&-424Q )N.",)? !. 0",)&!242 !. ||Y˜ b ||S 2 + ||Y b ||S 2 + ||Z b ||H2 + kKTb kL2 CL > 0. 7),+-.N!J4./"$k"$./"$K0+!5. <. M. ≤ CL .. {Mm{ . op"),+-.)$243./"10",)$+-'>49g#"$UV32\!#4=Q#",324+- '>.+V"*#+J+!5[UV4vZ"$d!." Yb "$./e+!5 @ ",)$+- } Mm{M.

(21)

(22) p ≥ 2 max E. i≤N −1. . ". sup t∈(ti ,ti+1 ]. |Ytbi+1. !$" !" "$# % $'&(&( % )#*$ "$&%. −. Ytb |2. #. ≤ CL |π| .. . 8.#+-X .+&' +-%'S"Y-3XY"7 ! %HrM "VM."7#"ED ",)$+-p4XY",*!#"74.)$243./",/p4 ." <⊂π !#4+-9/"ER.4UG." B 32"$#)."$XY"*+!5."C5t+-#'(!#/#+J),", M X op"d!#+,s4XG!" hJQ ."j",),"$'>"j),+-.N!Kq#+J),",N ¯ π ¯ π /"ER..",/|hKQ (Y b , Z b ) (Y , Z ) 4./3.)$+-jhKQ  h i ¯tπ = (ti+1 − ti )−1 Ei Y¯tπ (Wt − Wt )  Z  i+1 i i+1   i h i π π π π Y˜ti = Ei Y¯ti+1 + (ti+1 − ti )f (Xti , Y˜ti , Z¯tπi )    ¯π  Yti = R ti , Xtπi , Y˜tπi , i ≤ N − 1 ,. . {M . %.

(23) !./hKQq."C"$#Xg4 !2 ),+-./4+- Y¯Tπ = Y˜Tπ := g(XTπ ) .. 0",)&!242 !. Ei [·]. N!./c5t+-#. E [· | Fti ]. (Y¯tπ , Z¯tπ ) = (Y¯tπi , Z¯tπi ). MT8.+-#"&-" +!5[.+-N!+-.&%'("1"$ 5t+-#. t ∈ [ti , ti+1 ) i ≤ N − 1 .. {M } . 4Uq!j4./3.)$+-p!#UV3XY"$K1!./e."FH )N4I$:=),+-K4K34Qp-3Xg+-d+- % !./ g h % +-.""&-424Q).",)?C !." !hk+&vZ" #+;),",",C!#"PJ3 !#"14J"$UV#N!h2"VM =5L+-242+]' ! f ."*),+-./4+- !2 "Es ",)$N!+-.!#" 'S"$242/"ER..",/j!"&-)e"$e+!5."1!24UZ+-#4XM. . h."$#vZ" ! ˜ π /"ER..",/4Xg24)$424Qe-."1+-243+-e+!56Rs",/ +-4J Y #+-h2"$XM @ 4.)," 6FH4.)N4I$:=),+-K4J3.+-3.&%4TT/"ER..",/G'>4G.+*!Xh4UV34=QZMT9+-#",+&vZ"$#&% f 41)&! hk"g",4XG!",/dK3XY"$#)&!2424Q 4O vZ"$#Qj5t-1!./O-),)$3#N!" 5t+-#1XG!242yvV!243.",1+!5 |π| 'c,QZ%J5.+-y"Es24)$4&M op"c#"E5t"$#y+7^m~!a5L+-#T/)$3.+-Y+-g."(/ H"$#"$.),">hk"$='(","$Y4Xg24)$4 !./e"Es;24)$4N)."$XY",&M

(24) . . . 8 +-# 2\!"$# 3."V%H2"$C3.C4K#+;/3.),"g."g),+-K4J3.+-3.14XY"g)N."$XY"Y-+;)$\!",/d+ ¯ π ¯ π M (Y , Z ) <(Qf." XG!#4UK!2"1#"$#",N"$KN!+-9.",+-#"$X%."$#"1"Es π 2 3.)e ! Z ∈H. i Z h π π ¯ ¯ Yti+1 = Ei Yti+1 +. ti+1. ti. (Zuπ )0 dWu , i ≤ N − 1 .. op"*)&!."$e/"ER.." ˜ π +- hKQ [ti , ti+1 ) Y Y˜tπ = Y¯tπi+1 + (ti+1 − t)f (Xtπi , Y˜tπi , Z¯tπi+1 ) −. !./e"$. Z. ti+1 t. (Zuπ )0 dWu ,. {Mm . 5L+-# Ytπ := R(t, Xtπ , Y˜tπ ) t≤T ,. N+7 !. .

(25) . . +- Y π = Y¯ π π c5L+-242+]'c5 #+-X. ."1=. Z¯tπ = (ti+1 − ti )−1 Ei. #",)&!242 {M } M. !./. Z. . {M`_ . N+-XY"$#Q !. ti+1. ti. +- Y π = Y˜ π [0, T ] \ < .. Zuπ du. _. . , ∀ t ∈ [ti , ti+1 ) , i ≤ N − 1 ,. {M .

(26) % ( $ &". . . +-#/"$# +9N!"f+-3#*R.##",324&%T'S" .",",/u+j4K#+;/3.),"f."f#+J),", ¯ b /"ER..",/ +- Z "&-)Ne4J"$#v-!2 hKQ [ti , ti+1 ). .

(27) . Z¯tb := (ti+1 − ti )−1 Ei. . 8.+-#2\!"$#3."V%+-h."$#vZ" !&%hJQ h. E |Z¯tb − Z¯tπ |2. '>)Ne4Xg24",. i. ≤ (ti+1 − ti ). ||Z¯ b − Z¯ π ||H2. −1. Z. ti+1. ti. . Zub du. {Mml . .. {M !./

(28) Z"$."$ 4.",PK3 !244QZ%. Z. ti+1. ti. i h E |Zub − Zuπ |2 du ,. Ý. {M . ÝÝ. {M . ≤ ||Z b − Z π ||H2 .. bc."05t+-242+&'>4Ug#",324c.+]'S !c." !#+,s4XG!+-"$##+-#c64J4XG!"$24Q #"$2\!",/f+ ." .+-#X +!5 b ¯ b MyW 4Xg42\!#>#+- "$#Q .+-2/>4."C.+-:r#"ED ",)$",/e)&-"V%","g^m~na=%H^m{na=% H2 Z −Z ^ $ a !./p^ &} a=M. Ý. Ý.

(29)

(30) !#" max k. j≤κ−1. $%. &'(*),+-. !. 1 ≤ CL |π| 2 + ||Z b − Z¯ b ||H2 ,. sup |Ytπ − Ytb | kL2. t∈[rj ,rj+1 ]. ||Z π − Z b ||H2. / 01. ||Z π − Z b ||H2. 1 1 ≤ CL κ 2 |π| 2 + ||Z b − Z¯ b ||H2 . 1 ≤ CL |π| 2 + ||Z b − Z¯ b ||H2 .. bc."1#+J+!5y","$J\!2424Q 5t+-242+&'."*!#UV3XY"$K+!5>^m{na !./>#+&v;/",/4."! "$./sHM. . 32. h "$#v;4UY ! ¯ b >." hk", 2 . :!#+,s4XG!+-p+!5 b hKQ Z L (Ω × [0, T ]) Z %y'S"f/",/3.),"f ! -/.!",/ #+;),",",'>) !#"f),+-.N!J7+-x"&-)Nw4J"$#v-!242" [ti , ti+1 ) UZ+;",(+ - UZ+J",(+ MTbcK3.]%."C!hk+&vZ"#+-k+V4+-e-)$3 !2424Qq.+&' b b 2 ¯ ||Z − Z ||H2 0 |π| 0 !0+-3#/)$#"$"E:r4XY"*N)."$XY" ),+-JvZ"$#UZ"$K&M(bc0!2+74Xg24", !

(31) . . ||Z − Z¯ b ||2H2 b. ≤. N −1 X. E. i=0. Z. ti+1 ti. |Ztb. − Ztbi |2 dt. . .. q+-#/"$#S+*UZ"$c1h +-3./f+- ."),+-KvZ"$#UZ"$.),"#N!"V%;46#"$XG!4.(+ ),+-K#+-2 M ||Z b − Z¯ b ||2H2 @ 3.)N97),+-J#+-2'>4242hk" +-hN!4.",/3./"$#+-." +!5[."C5t+-242+&'>4Uq-//4+- !2-3Xg+-.]M . h ∈ Cb1. (H1). +-#. (H2). . h ∈ Cb2. '>4 '>4. L. L. :=F4.)4I /"$#4v-!4vZ"VM :=F4.)4I*R.#0!./",),+-.//"$#4vV!4vZ",&M .

(32)

(33)

(34) / . 1. α(κ) = κ 4. (H1). . . .0 . . 1 ||Z b − Z¯ b ||H2 ≤ CL α(κ) |π| 2 ,. (H1). $%. α(κ) = 1. bc."1#+J+!5['>4242hk"C#+&v;/",/4 @ ",)$+- } M. % . (H2). . +-Xh44U ."!h +]vZ" #+- +V4+-.&%'S"*+-hN!4." XG!4#",324+!5[N",)$+-M / (H1) 0 max k. j≤κ−1. $%. . 1. % . 1. (αY (κ), αZ (κ)) = (κ 4 , κ 2 ). % . (H2). % . () +- 0 /. (H2). 0",)&!24244U. ≤ CL αY (κ) |π| 2. t∈[rj ,rj+1 ]. ||Z π − Z b ||H2. /. 1. sup |Ytπ − Ytb | kL2.

(35) %! . /. 1. ≤ CL αZ (κ) |π| 2 (H1). %. 1. (αY (κ), αZ (κ)) = (1, κ 2 ) 1. αZ (κ) = κ 4. ÝÝ. % . (H1). %. Ý. . αZ (κ) = 1. Ý. { M`_ % {M !./j),+-X*h44Uiy#+-k+V4+- {M '>4jbc.",+-#"$X {M %.'("1R. !2424Q +-hN!4 7hk+-3./+-e.""$##+-#/3."1+G."*!#+,s4XG!+- +!5 hKQq."1",),"$'>N" (Y b , Z b ) ),+-.N!K>#+;),", ¯ π ¯ π '>)N)&!-)$3 !2424QGhk"",4XG!",/ J3XY"$#)&!2424QZ%.","."C"$./ (Y , Z ) +!5[." 4J#+J/3.)$+-M.

(36) / (H1) max k |Y¯tπi − Ytbi | +. i≤N −1. $%. 1. 1. sup |Y¯tπi+1 − Ytb | kL2 ≤ CL αY (κ) |π| 2. t∈(ti ,ti+1 ]. ||Z¯ π − Z b ||H2. /. . 1. 1. (αY (κ), αZ (κ)) = (κ 4 , κ 2 ). % . (H2). () +- 0 /. % .

(37) %! . /. 1. ≤ CL αZ (κ) |π| 2 (H1). %. 1. (αY (κ), αZ (κ)) = (1, κ 2 ). % . %. . α (κ) = 1 (H2)

(38) 'c-Y.+]'> 4 ^ ÝÝ a !g."q#",324Y+!5Ciy#+- +V4+-z{Mm{ !./xbc.",+!: #"$X {M Ý .+-2/e'>49."1hk+-3./ '>."$ 0#"$2\-),",/jhJQ."*N+-243+- C |π| (Y , Z , K ) +!50),+-J4K3.+-3.24Qu#"ED ",)$",/u< @;ACB % "," M Ý h "$2+]'1Mbc."$4#*#+;+!5ch -",/ (Y, Z, K). % . . L. 1. αZ (κ) = κ 4. 1 4. l. b. b. b. (H1). Z.

(39) -+ p !#)$32\!# #"$#","$JN!+-O+!5 +-hN!4.",/dhJQ9!d4K"$UV#N!+- hJQe !#1!#UV3XY"$K&M Z M 3#g!#+Z-) ),+-X7: +&'("$vZ"$#&%T4#",PK34#",g!w35L+-#X "$24244)$4=Qw),+-./4+- +- σ 2"$"$24Qg/ H"$#"$K&M[y[h -",/g+-G0#"$#","$KN!+-G5L+-# b 4g"$#XYT+!5."c."Esy#"ED ",)$+- Z 4XY"V% "," @ ",)$+- } h "$2+]'1M6bc!242+]'>3.>+7UZ"$>#/+!5[."C4KvZ"$#4h4244Q),+-./4+-j+- Mbc." !h +]vZ"*#",324'>4242 h ""Es;"$./",/e+Y." ),+-K4K3.+-3.24Qe#"ED ",)$",/9)&-"14 @ ",)$+- σ h "$2+]'1M. . !$" . $! "

(40). . % "( " . #" "

(41) .,&. . # . . 3h.",)$+-%-'S"(4K#+;/3.),"c."(+-243+- +!5 0/)$#"$"$24Q#"ED ",)$",/ (Y b,e , Z b,e , K b,e ) < @;ACB /"ER..",/9- "&-/e+!5 %.rM "VM b b b h3>'>4 π 4. (Y , Z , K ). X. X. YTb,e = Y˜Tb,e := g(XTπ ). !./H%5L+-#. !./. %. j ≤κ−1 t ∈ [rj , rj+1 ) ( b,e R rj+1 R rj+1 b,e 0 Y˜t = Yrb,e f (Θb,e (Zs ) dWs , u )ds − t j+1 + t b,e b,e π . Yt = R t , Xt , Y˜t. Ý. {M ~ . '>4 M Θb,e := (X π , Y˜ b,e , Z b,e ) bc0),+-.#3.)$+-9'>4242h "C3.N"E5 32+G"Es;"$./."*#",324+!5y." #"$v;+-3.",)$+-j+g." ),+-J4K3.+-3.24Qe#"ED ",)$",/)&-"VM . h.N"$#vZ" ! Y˜tb,e = g(XTπ ) +. '>4. Z. t. T. b,e. f (Θ )du −. t. κ−1 h X. ˜ b,e := K t. T. Z. j=1. ˜ b,e − K ˜ b,e , t ≤ T , (Zub,e )0 dWu + K t T. h(Xrπj ) − Y˜rb,e j. i+. 1rj ≤t .. 9+-#",+&vZ"$#&%465L+-242+]'S5 #+-X ."0N!XY"C!#UV3XY"$K>-S4 ."#+J+!5+!5iT#+- +V4+-{Mm~%;"," "$ } +!5."1#+J+!5[4."1W0 "$./sH% ! 1 ≤ CL |π| 2 + ||Z b,e − Z¯ b,e ||H2 ,. ||Z π − Z b,e ||H2. Ý. {M { . '>."$#" ¯ b,e /"ER..",/4Xg42\!#24Q- ¯ b %rM "VM Z Z Z¯tb,e := (ti+1 − ti )−1 Ei. Z. ti+1 ti. Zsb,e ds. . , t ∈ [ti , ti+1 ), i ≤ N − 1 .. op" !242!2N+g#+&vZ"4 @ ",)$+- } !."*#",324+!56iy#+-k+V4+-d{Mm{Y)&!ehk"1"Es"$./",/ + b,e M Z.

(42)

(43) . /. (H1). . .0 . . ||Z b,e − Z¯ b,e ||H2 ≤ CL. . Ý. 1. 1. 1. κ 4 |π| 2 + |π| 4. . ..

(44) C y C

(45)

(46)

(47) y y. F"$. hk"C.". (Y, Z, K). F. :r#+-UV#",N4vZ"$24Q XY"&-3#N!h2"1#+;),",N!5LQ;4U. Yt = g(XT ) +. T. Z. f (Xs , Ys , Zs )ds −. t. Yt ≥ h(Xt ) , 0 ≤ t ≤ T. '>4. ),+-J4K3.+-3.&%.+-:=/",)$#"&-4U.% 3.) !. K. . Z. Z. T t. (Zs )0 dWs + KT − Kt ,. K0 = 0. Ý. M . !./ Mm~ . T. 0. (Yt − h(Xt ))dKt = 0 .. B ;s "$.),"u!./3PK3."$.",Ne+!5Y +-243+- 5t+-242+&'5L#+-X (Y, Z, K) ∈ S 2 × H2 × S 2 bc.",+-#"$X } Mm~74p^ml!a=%#",)&!242 ! % !./ !#" H F 4.N)4I$:=),+-K4J3.+-3.&M g h. f. Ý. W 4 @ ",)$+-x{M %'("f!2+j/"ER.." -1."f+-243+-w+!5 (Y e , Z e , K e ) 2\-),"*+!5 %.rM "VM. M '>4. Xπ. 4. X. Yte = g(XTπ ) +. Z. T t. f (Xsπ , Yse , Zse )ds −. Yte ≥ h(Xtπ ) , 0 ≤ t ≤ T ,. '>."$#". c),+-K4J3.+-3.0!./f.+-:=/",)$#"&-4U.%. Ke. Z. T t. (Zse )0 dWs + KTe − Kte ,. K0e = 0. !./ R T 0. (Yte − h(Xtπ ))dKte = 0. 3 # R.# #",324 #N!."$# N!./.!#/HM .+&' ! !./ )&!wh "q!: (Y, Z) (Y e , Z e ) 1 3./"$#(." #+&s;4XG!",/fhJQg."+-243+-.(+!5/)$#"$"$24Qg#"ED ",)$",/f< @;ACB c!(*k",",/ |<| 2 -N3Xg+- . (H3). bc."$#"1"Es. ρ1 : Rd 7→ Rd. !./. 3.) !. ρ2 : Rd 7→ R+. |ρ1 (x)| + |ρ2 (x)| ≤ CL (1 + |x|CL )

(48) . h(x) − h(y) ≤ ρ1 (x)0 (y − x) + ρ2 (x)|x − y|2 , ∀ x, y ∈ Rd .. . Ý. +!5 A "ER.4+-. bcc),+-./4+-S244UVK24QG'("&!?Z"$#> !q."0"$Xg:=),+-JvZ"Es;4Qq-N3Xg+- 4O^ a'>)Ne !R ",/'>."$."$vZ"$# +-# .+-2/HM. Ý.

(49)

(50) . (H3). !. . 01. t∈[0,T ]. max. j≤κ−1. sup kYte − Ytb,e kL2 + ||Z e − Z b,e ||H2. !. t∈[0,T ]. (H1) k. / 01. sup |Yt − Ytb | kL2 + k. t∈[rj ,rj+1 ]. (H2). . sup kYt − Ytb kL2 + ||Z − Z b ||H2. $% &'. / /. (H1). 1. ≤ CL |<| 2. sup |Yte − Ytb,e | kL2. t∈[rj ,rj+1 ]. ÝÝ. 1. ≤ CL |<| 2 .. !. 1. ≤ CL |<| 2 .. M.

(51) bc."1#+J+!5[>#+&v;/",/4."*Wk"$./skM op"7)&!9.+&'"Es"$./e."7),+-KvZ"$#UZ"$.),"7#",324+!5T." #"$v;+-3.C",)$+-j+ ."7),+-K4J3: +-3.24Q #"ED ",)$",/j)&-"VM. . / (H1) / 0 . . max k. . i≤N −1. / &'. . 1. sup |Y¯tπi+1 − Yt | +. t∈(ti ,ti+1 ]. sup |Ytπ − Yt | kL2 ≤ CL α(π). t∈[ti ,ti+1 ]. 1. ||Z¯ π − Z||H2 + ||Z π − Z||H2 ≤ CL |π| 4 ,. % . α(π) = |π| 4 % (*),+-. (H2). (H1). / 01. $%. % . 1. α(π) = |π| 2. (H2). . 1. ||Z¯ π − Z||H2 + ||Z π − Z||H2 ≤ CL |π| 2 ,. Ý M bc.""$##+-#+-. ! ./."*",4XG!"*+- 3./"$# !./ i @ 5t+-242+&' 5 #+-X Z (H2) M iT#+- +V4+- M % +-#+-242\!#Q{M !./bc.",+-#"$X {M !24",/'>4 <=π ~M bc."j",4XG!"5t+-# 4|."UZ"$."$#N!2 )&-"q h4fXY+-#"j4KvZ+-24vZ",/HM op"R.#!: Z #+&s;4XG!" hKQ M05t+-242+&'05L#+-X iT#+- +V4+-p{MmG4w^mlna=% +-3#CFH4.)4I$: (Y, Z) (Y e , Z e ) p M bc."$%c'S" ),+-J4K34Q|-3Xg+-.]% ~Mm~ !./ ~M ! ||Z − Z e ||2H2 ≤ CL |π| !#+&s;4XG!" hKQ /"ER..",/z4 @ ",)$+-{Mm{M <(Q iy#+-k+V4+- M % (Y e , Z e ) (Y b,e , Z b,e ) M 84 !2424QZ%>4G5t+-242+&' 5L#+-X {M { ! ||Z e − Z b,e ||2H2 ≤ CL |π| ||Z π − Z b,e ||2H2 ≤ J % > ' . $ " # " . 0 " \ 2 6 $ " # X S , ) + J # + 4 2 2 , " G / J h G Q y i #+-k+V4+-{M Mb + CL |π| + ||Z b,e − Z¯ b,e ||2H2 ),+-.)$243./"V%K'S">/",/3.),"c5L#+-X Z"$."$ y4.",PJ3 !244=Q ! ¯ π ||Z −Z b,e ||H2 ≤ ||Z π −Z b,e ||H2 + % # , " & ) ! 4 2 2 { M M b,e ¯ b,e. Ý . Y. Ý. Ý. Ý. Ý. ||Z. W>4. −Z. ||H2. 2. {Mml % 'S" .+]' /"ER.." Z¯t := (ti+1 − ti )−1 Ei Z¯te := (ti+1 − ti )−1 Ei. Z. ti+1. t Z iti+1 ti. h.N"$#vZ" !&%hKQ Z"$.N"$ 4.",PJ3 !244=QZ% . ¯ H2 ||Z¯ b − Z||. . ≤ ||Z b − Z||H2. . Zu du , 5L+-# e Zu du t ∈ [ti , ti+1 ) , i ≤ N − 1 .. !./. Mm{ ||Z¯ b,e − Z¯ e ||H2 ≤ ||Z b,e − Z e ||H2 .. Ý. -+ Xh44U Mm{ %iT#+- +V4+- M %[iy#+- +V4+-w{Mm{e!./uiy#+-k+V4+-w{M 5L+-# % <=π '("*+-hN!4."5t+-242+&'>4U #"$UV32\!#4=Q#",3245L+-# !./ e M Z.

(52) / (H1) ! / 01. Z. 1. &'. . ¯ H2 + ||Z e − Z¯ e ||H2 ≤ CL |π| 4 . ||Z − Z||. (H2). !. / 01. ¯ H2 ≤ CL |π| 21 . ||Z − Z||. Ý~.

(53) . W("Es2\!4.",/4q."C#"$v;+-3.>",)$+-%.4Xg42\!#c#",324>'("$#" +-hN!4.",/4 g35t+-#Xg24Qx"$24244)VM "$#"V%6'("/+ ^ a=M 0+]'S"$vZ"$#&%c."$4# !#+Z-)N #",PJ34#",Y ! σ .+-7.",",/wg),+-./4+- +- Mdop"f!2+d+-hN!4 hk"$"$# hk+-3./ 5t+-# ¯ H2 !./ σ ||Z − Z|| 3./"$# Mqbc*2\--3Xg+-w*244UVJ24Qp#+-UZ"$#* ! supt∈[0,T ] kYtπ − Yt kL2 (H2) ." 2 #"$UV32\!#4=Q4Xgk+V",/+- hKQd^ a=M

(54) . ÝÝ. . Cb. ÝÝ. h. . . y0c . . . . .

(55) >k .

(56). $ # ($' & . . q."C",PK3."$2r%'("C/"$.+-"ChKQ 1,2 ."C -)," +!5#N!./+-X D \!h2"*4."1d!2424\,v;4e"$."1!./3.) ! kF k2L2 +. Z. 0. T. Zb. . . vV!#\!h2". Z b,e. F. '>)e!#"C/H"$#"$:. kDt F k2L2 dt < ∞ .. Ý. "$#"V% /"$.+-",."19!2424\&vJ4j/"$#4v-!4vZ"1+!5 !>4XY" %"," "VM U.M^ ~!a=M Dt F F t≤T op" !2+4K#+;/3.),"g."Y -)," 1,2 +!5>-/.!",/O#+J),",N", 3.)p !&%n5 "$#1k+V4h24Q L V !./ -N4UY+Yg34N!h2"1vZ"$#+-% 1,2 5t+-#!242 . Vs ∈ D s≤T Z T ||Dt V ||H2 dt < ∞ . ||V ||H2 + 0. ."C5t+-242+&'>4U.% 'S"1 !242!24'(,Q),+-./"$#0734N!h2" vZ"$#+-e5.",),", !#QZM N",)$+-%'S" '(+-#?q3./"$#."1#+-UZ"$#0-N3Xg+-. . %. %. (H0 ) b σ g. !./. !#". f. Cb1. M. bc."1UZ"$."$#N!2 )&-"C'>4242hk" +-hN!4.",/hJQf3.4U !e!#+,s4XG!+-d!#UV3XY"$J&M

(57) . Ý. 2 (c'("$242?;.+&'> !>3./"$#."1!hk+&vZ"1-N3Xg+-.. ^ ~-a=%.!./N!R ",c5t+-#. p≥2. s≤T. Z. ∇Xt = Id +. ∇X. +!5. X. t. ∇b(Xr )∇Xr dr +. 0. %"," "VM U.M. 1. ≤ CLp |t − u| 2 , t, u ≤ T .. sup kDs Xt − Ds Xu kLp + ||Dt X − Du X||S p. 9+-#",+&vZ"$#&% ."CR.#>vV!#\!+-e#+;),",. X ∈ L1,2. 'S"$242 /"ER..",/j!./+-24vZ",+-. Z tX d 0 j=1. Ý. } M . [0, T ]. ∇σ j (Xr )∇Xr dWrj. '>."$#" >." /"$J4=QXG!#s+!5 d % j ." :rj),+-243Xg9+!5 %!./ % ." Id M σ j σ ∇b ∇σj J-),+-h\!eXG!#s+!5 !./ Myc4KvZ"$#" +!5 −1 ."1+-243+-e+- b. (∇X)−1 t. σj. = Id − −. Z. 0. (∇X). t. (∇X)−1 r. Z tX d 0 j=1. . ∇b(Xr ) −. d X j=1. ∇σ j (Xr )∇σ j (Xr ) dr. j j (∇X)−1 r ∇σ (Xr )dWr ,. Ý{. [0, T ] .

(58) !./."C5L+-242+]'>4U N!./.!#/e",4XG!",.+-2/ } Mm~ . ≤ CLp .. ||∇X||S p + ||(∇X)−1 ||S p. 84 !2424QZ% 'S" #",)&!242."1'S"$242:r?;.+&'>e#"$2\!+-jh "$'S","$. ∇X. !./. 5t+-#!242. Dt Xs = ∇Xs (∇Xt )−1 σ(Xt )1t≤s. DX. s≤T. . h."$#vZ" !. Ds Xtπ = σ(Xφπs ) +. '>."$#"   Y . !2N+gh "$2+-UZ+. Xπ. Z. t s. k∈Ns,t. L1,2. ∇b(Xφπr )Ds Xφπr dr +. φt = max{u ∈ π : u ≤ t} . MybcJ3.&%. Id + ∇b(Xtπ )(tk+1 ∧ t − tk ) + k. Ds Xtπ d X j=1. σ. %'S"1/",/3.),"1 ! } M . ≤ CLp .. || sup |Ds X| ||S p. 2 . } Mm{ . t, s ≤ T .. 4UY."1!hk+&vZ"*",4XG!",&% ~Mm~ !./."1FH4.)N4I$:=),+-K4K34Q9+!5.

(59) . . 3./"$#. Z tX d s j=1. (H0 ). !./N!R ",. ∇σ j (Xφπr )Ds Xφπr dWrj. >UV4vZ"$hJQ.   j j  j π σ(Xφπs ) ∇σ (Xtk )(Wtk+1 ∧t − Wtk ) . '>4 M 4Uq." h +-3./d+- !./ % % 'S" Ns,t := {k ≤ N : s ≤ tk < t} ∇b ∇σ j j ≤ d +-hN!4. E. ". sup |Ds Xtπ |p. s,t≤T. #. ≤ CLp 1 + CLp |π|. '>)Ne2"&-/+ E. ". sup |Ds Xtπ |p. s≤t≤T. #. 2p N. ". #! 1 2. 1 + E sup |Xtπ |2p t≤T. } M} . ≤ CLp , p ≥ 1 .. Ý. <(Qj3.4UN!./.!#/p!#UV3XY"$K]%+-."Y!2+q"&-424Qj)N.",)? ! ."7hk+-3./ "Es"$./",/+ π %35t+-#Xg24Qq4 X. } M )&!9hk". π. 1. . sup kDs Xtπ − Ds Xuπ kLp + ||Dt X π − Du X π ||S p ≤ CLp |t − u| 2 , t, u ≤ T .. s≤T. } Mm . &($ '" $ $" &% . +-#/"$#>+7#+&v;/"*734N!h2"C#"$#",N"$KN!+-j+!5 b %'("1 !242!k"&!2+ ."C5L+-242+]'>4U Z "&-Q 2"$XgXGM . .

(60). 2 &'. . F ∈ D1,2 /. 0 . [F ]+ ∈ D1,2. Ý$. %. Dt [F ]+ = (Dt F )1{F >0}. .

(61) 0<(Q0#N!4UVJ5L+-#'c!#/G-/.!N!+-Y+!5iy#+-k+V4+- Ý Mm~Mm{04^ Ý ~na=%-'S"c+-h."$#vZ"( !. h "$2+-UZ+ 1,2 !./ '>."$#" 0g#N!./+-X Vv !#\!h2" h +-3./",/ D Dt [F ]+ = α(Dt F ) α hJQ N!5LQ;4U Mpbc."q#+J+!507."$ ),+-.)$243./",/ Kh Qw!k"&!244U + 1{F >0} α = 1{F >0} iT#+- +V4+- Mm{M`_ 4p^ ~!a=M. Ý. [F ]+. Ý. Ý. 0",)&!24244Uq !. Ý. . 2. Ý. % 3.4U "$XG!#? } M %kF"$XgXG } M %Hiy#+-k+V4+- } Mm{Y4x^naT!./ !. 4./3.)$+-d!#UV3XY"$K&%'S"1"&-424Qf/",/3.),"C5L#+-X {M ! ˜ b b h "$2+-UZ>+ 1,2 M g≥h. Ý. (Y , Z ).

(62)

(63) 2 / (H ) ! 1 / 0 (Y˜ , Z ) ! / t ≤ T D (Y˜ , Z ) ! $ [r , r ) j ≤ κ − 1 0. t. b. b. b. j. b. L. L1,2. % . j+1. } M`_ . Dt Y˜sb = (Dt h(Xrj+1 ) − Dt Y˜rbj+1 )1{h(Xr )>Y˜ b } rj+1 j+1 Z rj+1 Z rj+1 b b b ˜ + Dt Yrj+1 + ∇f (Θu )Dt Θu du − Dt Zsb dWs . s. d+-#/"$#0+GUZ"$0#/9+!5T."*4./)&!+-#5L3.)$+-. ! "&!#4Uf4 5t+-242+&'>4Uf",PJ3."$.),"1+!5[+-4UY4XY",. Ý. s. } M`_ % '("*.+]' /"ER.." .". τj = inf{t ∈ < | t ≥ rj+1 , h(Xt ) > Y˜tb } ∧ T , j ≤ κ − 1 .. 8 +-242+&'>4Ud^ $ a6%'S"*!2N+g/"ER.." Λst. := exp. t. Z. s. ∇z f (Θbu )0 dWu. −. Z t s. 1 b 2 b |∇z f (Θu )| + ∇y f (Θu ) du , s ≤ t ≤ T , 2. '>."$#" / "$.+-"." !#\!2[/"$#4v-!4vZ" +!5 '>49#",k",)$+ 40",),+-./jvV!#\!h2" % ∇y f f y !./ ./ '>4e#", ",)$>+Y4cR.#!./2\-v-!#\!h2"VM 0 ! 0 ." UV#N-/"$J0+!5 (∇x f ). (∇z f ). 2.

(64) . f. bc."5t+-242+&'>4U ",4XG!",0!#"1N!./.!#/ k sup Λst kLp s≤t≤T. 4U. k sup |Λut − Λus | kLp. Ý. u≤t∧s. } M %.'S"1/",/3.)," ! ||. sup u∨t≤s≤T. } M . ≤ CLp ,. } Mml . 1. ≤ CLp |t − s| 2 , t, s ≤ T .. |Λts Dt Xs − Λus Du Xs | ||S p. 1. ≤ CLp |t − u| 2 , u, t ≤ T .. Ý. } M . op")&!G.+]'|N!"."XG!4f#",324(+!5S",)$+- '>)f#+&v;/",> #"$#","$JN!+- 5t+-# bM. Z.

(65) 2 /. ! . . 01. .

(66) '. ! * . 1 . (H0 ) Zb / / / % " j ≤κ−1 t ∈ [rj , rj+1 ) h i (Ztb )0 = E ∇g(XT )(Λt Dt X)T 1{τj =T } + ∇h(Xτj )(Λt Dt X)τj 1{τj <T } | Ft Z τj b t ∇x f (Θu )(Λ Dt X)u du | Ft . + E. . t. Ý&}.

(67) Ý M7=C5t+-242+&' 5 #+-X %. t≤T j ≤κ−1 Dt Y˜sb. Ý. iy#+-k+V4+- } M !./d." -3Xg+- !&%H5t+-#*!242 g ≥h %'S"* ,vZ". !./. s ∈ [rj , rj+1 ) = ∇h(Xrj+1 )Dt Xrj+1 − Dt Y˜rbj+1 1{h(Xr )>Y˜ b } rj+1 j+1 Z rj+1 Z rj+1 + Dt Y˜rbj+1 + ∇f (Θbu )Dt Θbu du − Dt Zub dWu . s. !#)$32\!#&%. s. ∇h(Xrj+1 )Dt Xrj+1 − Dt Y˜rbj+1 1{h(Xr )>Y˜ b } rj+1 j+1 Z rj+1 Z rj+1 Dt Zub dWu . ∇f (Θbu )Dt Θbu du − + Dt Y˜rbj+1 +. Dt Y˜rbj. =. . (. rj. rj. @ 4.)," ˜ b % 4>5t+-242+&' ! M 0",)&!24244UG ! %.4 Dt Y˜rbκ = ∇g(XT )Dt XT g≥h Yrκ = g(XT ) ."$e#",324>5L#+-X 74Xg2" 4./3.)$+-j !>5L+-# s ∈ [rj , rj+1 ). Dt Y˜sb = ∇g(XT )Dt XT 1{τj =T } + ∇h(Xτj )(Dt X)τj 1{τj <T } Z τj Z τj b b Dt Zsb dWs . ∇f (Θu )Dt Θu du − + s. s. <(Qq."1N!XY" !#UV3XY"$KC-4eiT#+- +V4+- } Mm{g4O^na=% 'S" &vZ" Dt Y˜tb = Dt Ytb = (Ztb )0 +- Mybc." #",324>."$5t+-242+&'c5L#+-X ." #"$v;+-3.",PJ3 !+-%.= (5t+-#X*32\ !./ (rj , rj+1 ) hJQ),+-./"$#4Uf734N!h2"1vZ"$#+-M. 2 W3XY"> !.

(68) . Ý. 2\!#Q } M !. 232 F"$. M. .+-2/&Mybc."$%J4T5t+-242+&'S5L#+-X. } M % } M !./. . +-#+-2:. +-2/HMop"/",/3.),">5 #+-X." !XY"!#UV3XY"$Kc-y4Y."c#+;+!5k+!5 . +-#+-242\!#Q } M ! ."$#"(yvZ"$#+- +!5 b,e 3.)N ! 5t+-#"&-)N % .

(69) . . ||Z b ||S p ≤ CLp. (H0 ). 2. Ý. (Ztb,e )0. '>."$#". !./. (H0 ). Z. t ∈ [rj , rj+1 ) j ≤ κ−1 i = E ∇g(XT )Dt XTπ 1{τje =T } + ∇h(Xτπje )(Λe,t Dt X π )τj 1{τje <T } | Ft Z τ e j e,t π + E ∇x f (Θb,e )(Λ D X ) du | F t u t , t≤T , u h. t. τje = inf{t ∈ < | t ≥ rj+1 , h(Xtπ ) > Y˜tb,e } ∧ T , j ≤ κ − 1 .. Λe,s t. /"ER..",/H%5L+-#. Λe,s := exp t. Z. t s. s≤t≤T. %hJQ. 0 ∇z f (Θb,e u ) dWu −. bc." 5L+-242+]'>4U ",4XG!",!#" N!./.!#/ . k sup Λe,s t k Lp s≤t≤T. k sup. u≤t∧s. |Λe,u t. − Λe,u s | k Lp. Z t s. 1 2 b,e |∇z f (Θb,e )| + ∇ f (Θ ) du . y u u 2. 1. ≤ CLp |t − s| 2 , t, s ≤ T .. Ý. ÝÝ. } M . ≤ CLp ,. Ý. } M ~ .

(70) 4U. } Mm %.'S"1/",/3.)," ! k. sup t∨u≤s≤T. $ #". . π e,u π |Λe,t s D t X s − Λ s D u X s | k Lp. . h ∈ Cb3. Ý. } M { . . N",)$+-%'S" #"$2\-)," (H20 ). 1. ≤ CLp |t − u| 2 , u, t ≤ T .. (H2). hKQq."1#+-UZ"$#-3Xg+- . %'>4e/"$#4vV!4vZ",3+Y+-#/"$##","1h +-3./",/hJQ. L. M. b ." "Es"$.+-+!5 ."05t+-242+&'>4UY#",324c+ c '>4242khk"C+-hN!4.",/qhKQG3.4UG!!#+&sJ: (H2) 4XG!+-d!#UV3XY"$K&M.

(71)

(72) . 2. . . 1. α(κ) = κ 4. /. (H1). . . .0 . 1 ||Z b − Z¯ b ||H2 ≤ CL α(κ) |π| 2 ,. (H1). $%. α(κ) = 1. bc." 5L+-242+]'>4UG#"$XG!#?q#"$ !#",>5t+-#>." #+;+!5M

(73) . . 2. % . (H20 ). . @ "$ β :=. 1 + sup |Ds Xt | + sup |Xt | + sup s≤t≤T. !./e+-h."$#vZ" !&%hJQ. t≤T. ~Mm~ % } M !./. s≤t≤T. |Λst |. } M %. !4. Ý. } M $ . ≤ CLp , p ≥ 2 .. ||β||S p. ,. 8 s !./Y2"$ !./ h ">='(+*+-4U4XY",63.)NG ! k M &M t≤T τ1 τ2 t ≤ τ 1 ≤ τ2 ≤ T P − <(Qf."1FH4.)N4I$:=),+-K4J34=Q - 3Xg+-e+- !./ %'(" ,Zv " b. 0./"$#. E |Xτ1 − Xτ2 |2 | Fτ1. (H20 ). . %'("1/",/3.),"C5L#+-X. . σ. ≤ CL E [β(τ2 − τ1 ) | Fτ1 ] .. Ý. } M &} . = F"$XgXGg !. Ý.

(74)

(75)

(76) E ∇h(Xτ )Λtτ (Dt X)τ − ∇h(Xτ )Λtτ (Dt X)τ | Fτ

(77) ≤ CL E [β(τ2 − τ1 ) | Fτ ] . 1 1 1 1 2 2 1 2. } M . o|."$. (H1). .+-2/]%'S"1)&!e3."C." hk+-3./.

(78)

(79) ∇h(Xτ2 )Λtτ (Dt X)τ2 2. +G+-hN!4. |∇h| ≤ L

(80) − ∇h(Xτ1 )Λtτ1 (Dt X)τ1

(81) ≤ β |∇h(Xτ2 ) − ∇h(Xτ1 )|

(82)

(83) + CL

(84) Λtτ2 (Dt X)τ2 − Λtτ1 (Dt X)τ1

(85) ,. '>)N%chKQxFH4.)N4I$:=),+-K4J34=Q +!5 %S GF"$XgXGO!./x." ∇h ",PJ3 !244=QZ% 4Xg24",. . !3.)JQK: @ )NK'c!#Ip4:. Ý.

(86) 1

(87) E

(88) ∇h(Xτ2 )Λtτ2 (Dt X)τ2 − ∇h(Xτ1 )Λtτ1 (Dt X)τ1

(89) | Fτ1 ≤ CL β¯ E [β(τ2 − τ1 ) | Fτ1 ] 2. } M _ . Ý_.

(90) '>."$#" β¯ := sup E β 2 | Ft. Ý. t≤T. #",)&!242 } M $ M.

(91)

(92) . Ý M6=c5t+-242+&'5 #+-X . '>."$#"V%.5L+-# Vtj,s. 8 +-#. 2. . . %. j ≤κ−1 i h := E ∇g(XT )(Λs Ds X)T 1{τj =T } + ∇h(Xτj )(Λs Ds X)τj 1{τj <T } | Ft Z τj b s ∇x f (Θu )(Λ Ds X)u du | Ft , s ≤ t . + E. Ý. } M %. %'S" ."$ &vZ". i |Ztb − Ztbi | ≤ |Vtj,t − Vtj,ti | + |Vtj,ti − Vtj,t |, i. ~M 04Ug."1XG!#4UK!2"1#+-k"$#=Q+!5 h E |Vtj,ti. !./. Ý. Ý. kVtj,t − Vtj,ti k2L2. Ajt. } M. (Ztb )0 = Vtj,t , rj ≤ t < rj+1 , j ≤ κ − 1 ,. t ∈ [ti , ti+1 ) ⊂ [rj , rj+1 ). '>."$#". ¯ Sp ≤ C p , p ≥ 2 , ||β|| L. +-#+-242\!#Q } M !&% n5L"$# -4Ug+Yg34N!h2" vZ"$#+-%. s. '>."$#"V% hKQ. N!R ",. +-. V j,ti. ≤ CL |π| .. !./. Ý. } M l . } Mm~ } Mm{ %'S"1/",/3.),"1 !. [ti , ti+1 ] i h i h i j,ti 2 j,ti 2 j j 2 2 2 i 2 − Vtj,t | ≤ E |V | − |V | ≤ E (|A | − |A | )|η | ti ti+1 ti ti+1 ti i. h i := E ∇g(XT )Λ0T ∇XT 1{τj =T } + ∇h(Xτj )(Λ0 ∇X)τj 1{τj <T } | Ft Z τj b 0 ∇x f (Θu )(Λ ∇X)u du | Ft , t ≤ T +E t. ηt := (Λ0t ∇Xt )−1 σ(Xt ) , t ≤ T .. 4UY." F4.)4I$:=),+-J4K34=Qd+!5. <(Q. σ. %.'S"1+-h."$#vZ"1 !. 1 E |ηti+1 − ηti |4 2. ~Mm~ % } Mm~ % } Mm{ % } M !./. . ≤ CL |π| .. !3.)JQK: @ )NK'(!#Ig4.",PJ3 !244=QZ% 4."$5t+-242+&' !. h i h i j j j 2 2 2 2 i 2 E |Vtj,ti − Vtj,t | ≤ E |A η | − |A η | + |A | |η − η | ti+1 ti ti+1 ti+1 ti ti ti+1 i h i } Mm~ j,t i 2 ≤ E |Vti+1i+1 |2 − |Vtj,t | + CL |π| . i. Ý. Ý. .

(93) Ý. } Mm~ M A "ER..". { M =g#"$XG!4.G+p3./Qw."fR.#Y"$#X 4x."#4UVJ:r !./z/"+!5 #+-3UV % !./+-h.N"$#vZ" !. ij. t ij = r j j ≤ κ. κ−1 ij+1 X−1 X. Σ :=. j=0 k=ij. κ−1 X. =. j=0. h i j,t k 2 E |Vtk+1k+1 |2 − |Vtj,t | k. h i j,r j,r E |Vrj+1j+1 |2 − |Vrj j |2. κ−1 h i X j−1,rj 2 j,rj 2 0,r0 2 κ 2 E |V | − |V | E |Vrκ−1,r | − |V | + rj rj r0 κ. ≤. j=1. . ≤. CL 1 +. κ−1 X j=1. h. i. j−1,rj 2. E |Vrj. Ý. . j,r | − |Vrj j |2 . '>."$#"*."C2\-4.",PJ3 !244=Qf5t+-242+&'>5L#+-X } M $ M Mg<(Q 5t+-# Mg8 +-#*"&-"g+!5S.+-N!+-.]%'("g.+&' '>#4" Erj [·] E · | F rj 4.",PJ3 !244=QZ% j−1,rj 2. j,r. j−1,rj. | − |Vrj j |2 ≤ |Vrj. |Vrj. j,r. j−1,rj. ≤ CL Erj [β] |Vrj. . j−1,rj. − Vrj j | |Vrj. . } Mm~V~ . !3.)NKQJ: @ )J'(!#I. j,r. + V rj j |. } Mm~V{ . j,r. − V rj j | ,. '>."$#" /"ER..",/4 0"$XG!#? } MmM β % !#"1hk+-3./",/jhKQ !./ ! %.'("+-h.N"$#vZ" MmM ",)&!24244Uf ! ∇g ∇h L τj−1 ≤ τj ≤ T !. . CL 1{τj−1 <τj =T } + ∇h(Xτj )(Λt Dt X)τj − ∇h(Xτj−1 )(Λt Dt X)τj−1 1{τj−1 <T }. ≥ ∇g(XT )Dt XT 1{τj =T } + ∇h(Xτj )(Λt Dt X)τj 1{τj <T }. o|."$. −∇g(XT )Dt XT 1{τj−1 =T } − ∇h(Xτj−1 )(Λt Dt X)τj−1 1{τj−1 <T } . (H1). .+-2/]%4>."$5t+-242+&'>5L#+-X. j−1,rj. |Vrj. } M % } M !./. Ý. } M _ !. h i j,r − Vrj j | ≤ CL Erj 1{τj−1 <τj =T } 1 1 + CL Erj [β(τj − τj−1 )] + β¯ 2 Erj [β(τj+1 − τj )] 2 .. @ 4.)," Pκ−1 %;." !h +]vZ"4.",PK3 !244Q ),+-X*h4.",/q'>4 j=1 1{τj−1 <τj =T } ≤ 1 4Xg24", . Σ ≤ C L E 1 + ≤ CL. κ−1 X j=1. } Mm~V~ !./. . 1. . 1 Erj [β] Erj [β(τj − τj−1 )] + β¯ 2 Erj [β(τj − τj−1 )] 2 .   κ−1   X ¯ j − τj−1 ) + E [β(τj − τj−1 )] 21 E ββ(τ 1+   j=1. } Mm~V{ . Ýl.

(94) '>."$#"'S"*3.",/ !. . !3.)NKQJ: @ )K'c!#IY4.",PJ3 !244=Qe!./. Σ ≤ CL ≤ CL. } M. Ý . Mc<(Q. } M. Ý . !UK!4% 0.+]'. o √ ¯ κ−1 − τ0 ) + κ E [β(τκ−1 − τ0 )] 12 1 + E ββ(τ √ 1+ κ .. n. } mM ~ . %S'S"3.""Es.-)$24Qw."N!XY"!#UV3XY"$Kf"Es),"$g !G'S"!k"&!2+ M hM 0./"$# (H20 ) "&-/+!5 } M _ M6bc>2"&-/+ } M 4.. Ý. } M6<SQ. Ý. Ý.  . Σ ≤ CL. Ý. 1+. . κ−1 X j=1.   ¯ j − τj−1 ) E ββ(τ ≤ CL .  . } M l % } Mm~ % } Mm~ !./q."1/"ER.4+-j+!5 n−1 X Z ti+1 i=0. ti. Σ. 4. } Mm~V~ . i h E |Ztb − Ztbi |2 dt ≤ CL |π| (1 + Σ) .. bc."(#+;+!5."$g),+-.)$243./",/7hKQ!k"&!244U + 0 %!./hKQf3.4U 0"$XG!#?q{M } M.

(95)

(96) F"$. } Mm~ } . } Mm~ %V3./"$#. (H1). %Z!./ +. } Mm~ } %V3./"$#. (H2 ). fn (x, y, z) =. Z. R2d+1. fn. 2. hk" /"ER..",/hKQ . φn (x − ξ, y − υ, z − ζ)f (ξ, υ, ζ)dξdυdζ ,. ' 4 > !./ ),+-Xg -)$24Qp3k+-#",/OXY+J+-p#+-h n: φn (x, y, z) = n2d+1 φ(n(x, y, z)) φ H F :r244.)4I 'S" &vZ" h4244Q/"$.4=Q 5L3.)$+-j+- 2d+1 M @ 4.)," f. R. ||f − fn ||∞ ≤. CL , n. 5t+-#*+-XY" MqF"$ % % % hk"Y/"ER..",/O4Xg42\!#24Qd5L+-# % % % N+q !*'S" C >0 σ n bn gn hn σ b g h &vZ" ||σ − σn ||∞ + ||b − bn ||∞ + ||g − gn ||∞ + ||h − hn ||∞ ≤. CL . n. F "$ hk" ." 5L+-#'c!#/ / 3.+-O-+;)$\!",/d+ n !./ n !./92"$ Xn b σ (Y b,n , Z b,n , K b,n ) hk"C."1+-243+-j+!5."1/N)$#"$"$24Q #"ED ",)$",/j< @;ACB { M -;+ )$\!",/q+ % !./ n M Xn f n g W0#UV34U ->4jiy#+- +V4+-j{Mmg+!5>^mlna=%'S" UZ"$. Ý. ||Z b − Z b,n ||2H2 ≤. @ 4.),"V% hKQ Z"$."$ 4.",PK3 !244QZ% ||Z b − Z¯ b ||H2. } Mm~V . CL . n. ≤ ||Z¯ b − Z¯ b,n ||H2 + ||Z b − Z b,n ||H2 + ||Z b,n − Z¯ b,n ||H2 ≤ 2 ||Z b − Z b,n ||H2 + ||Z b,n − Z¯ b,n ||H2 ,. ."1#+J+!5[>),+-.)$243./",/hJQq!24Q;4UGiy#+-k+V4+- } Mm~ + b,n %3.4U Z UZ+g+74R.4QZM. n. ~ . } Mm~V !./q2"$4U 2.

(97) o "S.+]' ),+-./"$#[."c)&-"('>."$#"S."S5t+-#'(!#/g/k3.+-7T!#+,s4XG!",/ghKQ*4 B 32"$# p N)."$XY"VM.

(98)

(99) 2. &' . =fv;"$'. . (H1). !. / 01. 1 1 1 ||Z b,e − Z¯ b,e ||H2 ≤ CL κ 4 |π| 2 + |π| 4 .. . !+ 5 "$XG!#? } Mm~7!./ "$XG!#? } M } %'S"C)&!f5L+-242+]'244."0hJQG244."." !#UV3: XY"$J*+!5(."#+;+!5(+!5ciy#+-k+V4+- } Mm~%n5L"$# #"$2\-)$4U."7),+-##", +-./4UjPK3 !J4", 4e."1/"ER.4+-.+!5 !./ ¯ % !./q#"E:=/"ER.4U.% 5t+-# % β. j ≤κ−1. β. i h Vtj,s := E ∇g(XTπ )(Λe,s Ds X π )T 1{τje =T } + ∇h(Xτje )(Λe,s Ds X π )τje 1{τje <T } | Ft Z τ e j } Mm~Z_ b,e e,s π ∇x f (Θu )(Λ Ds X )u du | Ft , s ≤ t . + E s. bc."7+-24Qe/H"$#"$.),"7! "&!#4d"$d~M."&-/j+!5T3.4UY#"$2\!+-d244?Z" '>)Nf/+;",T.+-S.+-2/ %K'("3.".">XG!#4UK!2"#+- "$#QY+!5 j,ti +- V. } mM { 5t+-# Xπ !./G'>#4". [ti , ti+1 ). i i h h j,ti 2 j,ti 2 i 2 ≤ E |V | − |V | E |Vtj,ti − Vtj,t | ti+1 ti i i h j,t j,ti+1 j,ti+1 j,ti j,ti i 2 ≤ E |Vti+1i+1 |2 − |Vtj,t | + |V − V | |V + V | , ti+1 ti+1 ti+1 ti+1 i. '>."$#"*hKQ. ÝÝ. Ý. } M } % } M % } M { !./. bc."14.",PK3 !244Q. . !3.)KQJ: @ )J'(!#IY4.",PJ3 !244=Q. h i p j,t j,ti+1 j,ti i E |Vti+1i+1 − Vtj,t |π| . | |V + V | ≤ C L t t i+1 i+1 i+1. Ý. } Mm~ ."$hk",),+-XY",. h i h i p j,ti+1 2 j,ti 2 i 2 E |Vtj,ti − Vtj,t | ≤ E |V | − |V | + CL |π| . ti+1 ti i. (

(100)

(101) . fbc."#",PJ34#",/Y#",324T5t+-242+&'65 #+-X. iy#+- +V4+- } Mm{!./YhKQ. !#UV34Uq->4." #+;+!5[+!5yiy#+-k+V4+-j{Mm{M. Ý. 2. 2. op"*),+-.)$243./"1",)$+-e'>4."C#+J+!5T+!5yiy#+-k+V4+-j{M M cW3XY"C ! 0 .+-2/&M6<SQ "$XG!#? } M %'(" ,vZ".

(102)

(103) . (H ) Z ti+1 b 2 |Zs | ds ≤ CL |π| . E. . ti. W #UV34Uj-C4 ."7#+;+!5c+!5ciy#+-k+V4+-u{Mm{%H'("Y+-hN!4p ! ."Y!hk+&vZ"ghk+-3./d.+-2/ 0 '>4.+-3 M bc."f#",PJ34#",/w#",3247."$u5t+-242+&' 5L#+-X = 7F"$XgXG%6."qFH4.)4I$: (H0 ) ),+-J4K34QY+!5 % ~Mm~ %Z."Shk+-3./7+- b UV4vZ"$74 {Mm{ !./7<(3#?;.+-2/"$#: A &vJ:3./Q f Y 4.",PJ3 !244=QZ% #",)&!242 {M M. Ý. ~. 2. Ý.

(104) . . y . $%

(105) % % $%& % #"$&%

(106) . ÝM. @ "$. % % % δY = Y b − Y π δ Y˜ = Y˜ b − Y˜ π δZ = Z b − Z π δfs = f (Xs , Y˜sb , Zsb ) − 5L+-# M ",)&!24244U {mM ~ % {Mm % {M`_ % ."Y5t-)$ ! s ∈ [ti , ti+1 ) <⊂π f (Xtπi , Y˜tπi , Z¯tπi ) !./3.4UG F"$XgXG%'("1),+-Xg3"1 !>5t+-# t ∈ [ti , ti+1 ) Z ti+1 Z ti+1

(107) 2

(108) 2 i 2 ˜

(109) ˜

(110) = Ei At := Ei |δ Yt | + 2δ Ys δfs ds , |δZs | ds − δYti+1. . Ý. t. # ",)&!2426 ! N!./ 5L+-# M7<(Q Ei [·] E [· | Fti ] ."14.",PK3 !244Q 2 −1 2 %5t+-# Ait. t. {M % ."gFH4.)N4I$:=),+-K4K34Qw+!5 !./ f !./ %'(" ."$#"E5L+-#"*+-hN!4. xy ≤ cx + c y x, y ∈ R+ c>0 Z ti+1 Z ti+1 CL 2 2 2 ˜ ˜ ≤ Ei α|δ Ys | ds + |δZs | ds |π| |δ Yti | + α t ti Z ti+1 CL π 2 b b 2 b b 2 ˜ ˜ ¯ + |Xs − Xti | + |Ys − Yti | + |Zs − Zti | ds Ei α t. '>."$#" (Ck+V44vZ"> !#N!XY"$"$#6+1h ")N.+V"$Y2\!"$#(+-M 4U C#+-K'(!242 (F"$XgXG !./ α N!?;4U 2\!#UZ"g"$.+-3UV%'S"g/",/3.),"g !&% 5t+-# XG!242T"$.+-3UV%H."$#"7 +-XY" % α |π| η>0 4./"$k"$./"$KC+!5 %3.) ! π. . Ei |δ Y˜ti | + η 2. Z. } mM ~ |δZs | ds ≤ eCL |π| Ei |δYti+1 |2 + CL Bi } mM ~Vl h i Ei |δ Y˜t |2 ≤ CL Ei |δYti+1 |2 + |π| |δ Y˜ti |2 + Bi. ti+1 ti. sup t∈[ti ,ti+1 ]. '>."$#" Bi := Ei. Z. 2. . ti+1 ti. |Xs −. Xtπi |2. b b 2 b b 2 ˜ ˜ ¯ + |Ys − Yti | + |Zs − Zti | ds .. Ý. ~M @ 4 .)," 5L+-# %"," {M % {M !./ |δYti | ≤ max{|δ Y˜ti |; |h(Xti ) − h(Xtπi )|1ti ∈< } i<N {M`_ . % 4c5L+-242+]'c5 #+-X } Mm~ !24",/j! !./q." FH4.)4I$:=),+-J4K34Qj+!5 !&% t = ti h 5t+-# XG!242"$.+-3UV% |π|. o n |δYti | ≤ max eCL |π| Ei |δYti+1 |2 + CL Bi ; L|Xti − Xtπi |1ti ∈< .. @ 4 .)," %KhKQ7."FH4.N)4I$:=),+-K4J34=Qq+!5 %Z'S"/",/3.),">5L#+-X |δYtN | ≤ L|XtN − XtπN | g / !4./3.)$4vZ"!#UV3XY"$J ! } Mm{ !. ¯ max E |δYti |2 ≤ CL N |π|2 + B i≤N. '>4. ¯ := E B. "N −1 X i=0. ~V~. Bi. #. .. } Mm{ ~M } %.

(111) @ 4.),"*hKQ-3Xg+-. N |π| ≤ L. %>4Xg24",. Ý. } Mm{ . ¯ . max E |δYti |2 ≤ CL |π| + B i≤N. {M. h."$#vJ4UG !>5L+-# . s ∈ [ti , ti+1 )

(112) Z

(113)

(114) ˜ b ˜ b

(115) 2 ≤ CL E

(116) Y s − Y ti