Error estimates for a stochastic impulse control problem
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Error estimates for a stochastic impulse control problem J. Frédéric BONNANS — Stefania MAROSO — Housnaa ZIDANI. N° 5606 Juin 2005. ISSN 0249-6399. ISRN INRIA/RR--5606--FR+ENG. Thème NUM. apport de recherche.
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(13) . . Unité de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France) Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30. .
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(164) ( ! . !*. . . . . . . . B \( % B B ¦ ' (&B J¶v®s{_bz}qq{d8ïsuz¬5v®¹\dª¹ z°°'y{d3qKd8v5ss{_deIbbyuG«rz¬csuz¬5vq+I¨ 2 ~ 3 v 2 m 3 ºEv®¹%d»¹ z°¬°/qKs{wbo"s{_bd8z¬y cIz¬vBbyuIMdy{s{zd8q8± . . . . !" $#&%(')+*-,/.&%1032546%87:9; <,=)?>. ¤ ed ¹ z¬°°1by{6I_<d3|5w¯s{zIv 2 ~43j§po<BqKd3|5wdvdªI¨%I§¯qtsI °d y{5§b°¬d8cq± ¤ de¹ z¬°°wq{dªsu_bdq{Iced ceds{_brbq5q zvQ¿(9ºã~/yup¨jI¨+s{_dIyudc 9¯± Â:º/s{byuÃId¦su_sesu_bd?qK5°¬wrsuz¬5vqª¨hsu_bd?qKd3|6wbdvdI¨ d3|6wsuz¬5vq` IvpÃ5dyuIdjs{»s{_bd(q{I°wrsuz¬5vI¨ 2 ~43ï± ·\oUyudcIy{¾Ub± 9¯º6¹\d_GÃ5dhsu_s u ≡ 0 z}q+»Ãpzqu 6qKz¬sto¦q{wb§rµ¶qK5°¬wrsuz¬5vI¨ 2 ~43ï± %5vqKz}rd8y`s{_bd¨©5°¬°¹ z¬vªbyuI§°¬d8Bc : 2 ~ã< 3 sup L (x, Du(x)) = 0, x ∈ R . ghvrd8y5q{q{wbcersuz¬5vq 2 A(µ A. 3ïºWsu_bz}qjd3|6wsuz¬5v®_5qjwbvbz}|6wbdªÃpz}q{5q{z´sto?q{I°wrs{zIv u z¬v C (R ) ± mrz¬vd z}qhÃpz}q{5q{z´stoBqKwb§bµNq{I°wrs{zIv?I¨ ~ã 3ïºMs{_bd» 5c»¯yuzq{IvBbyuz¬vz¬°¬d 2 qKd8dU¿( 9bºW^`_bd8Iyudc ±  3%zceu b≡°¬zd80q 0 ≤ u ±%%Iv¯qKz}rdy\su_bd¨©I°°¬¹ zvb2 »byuI§b°d;c : max{sup L (x, Du(x)); u(x) − Mu (x)} = 0, x ∈ R . 2 ~+3 mrz¬vd Mu zqIv6s{zvpwbIw¯qº3wbvrd8yEIquq{wbcers{zIvq 2 A(µ A. 3 ºsu_bdyud/d «rz}qtsq`wbvbz}|6wbd/Ãpz}q{5q{z´stoqK5°¬wrsuz¬5v I¨ ~+ 3/z¬v ±/mpz¬cez°Iy{°oIºI¨©Iy n = 2, 3, · · · º5°d s u ∈ C (R ) §Md+s{_dhwbvbz}|6wbd+Ãpz}q{5q{z´sto u q{I°wrsuz¬52 vI¨ C (R ) max{sup L (x, Du(x)); u(x) − Mu (x)} = 0, x ∈ R . 2 ~/v*3 JNs'zqd85qKojs{_bd8¾hs{_s u z}q Ãpz}q{5q{z´stoq{wb§rµ¶qK5°¬wrsuz¬5v¨ (P 0) ±·\os{_d% 5ceyuzq{Ivbyuzv zb°dIº ±<i?IyudÃ5dy3º¹\d¦¾pvb¹ su_s u ≡ 0 zqª<qKw§rµNq{I°wrsuz¬5vÀI¨ (P 1) z¬v R º1v¯Às{_bd8v 0 ≤ u ≤u αi. N. αi. 0. b,l. N. 0. αi. αi. 0. 1. b,l. N. n. αi. αi. 1. 1. 0. N. 0. n−1. b,l. N. N. N. æ©Ù'ìæèÚ.
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(168) "!$#&%' ( )* +%,!$.-(%/(. zv ± ~/yuIM5q{z´suz¬5v?b±¬5º 3 ºzc»°¬zd8qhs{_s º¯s{_bd8v?¹%dªv<quGo¦su_s zq »Ãpz}q{5q{z´sto¦q{Rwb§rµ¶qK5°¬wbs{zIvB¨ 2 ~+3 º¯I2 vI°q{ u ≤ u Mu z v R ≤±/·\Mu o¦zvrw s{zIv"ÃId8y n ºb¹%d5§rsuIuz¬v : 2 r3 0 ≤ ··· ≤ u ≤ ··· ≤ u ≤ u ≤ u . ¤ dUIv¥qKd8dªsu_s3ºz¬¨ ºEs{_bd8v z}q("Ãpz}q{5q{z´sto®q{I°wrsuz¬5vI¨ 4~ 3v¯®su_bdv¥¹\dyud ¨©dy su®¿ GÂNº'¿Á3Â'¨©Iydyuy{5y+d8qKs{zc|usud8| q8±j≤mpwbkbM5q{d vu¹Qsu_s |u | > k º
(169) v¯"°¬ds µ2 ∈ (0, 1) q{w_?su_sqKds ± µ|u | < k ÆpÊ Æ 1
(170) ( %% n 2 I.3 u −u ≤ (1 − µ) |u | . Ê Ê <:6d s n ∈ N ºbIv θ ∈ (0, 1] §Md(q{w_Usu_s zv R . 2 <3 u −u ≤θ u , ·\o r 3ïºMs{_zq_b5°qjs°d85qts¨©Iy θ = 1 '± 3À d¹ yuz¬s{zvb 2 3h5q (1 − θ )u ≤ u , v®wq{z¬vb 2 y{5¯62 qKz¬s{zIv"r±IºrIds 2 9 3 (1 − θ )Mu + θ k ≤ (1 − θ )Mu + θ M0 ≤ M[(1 − θ )u ] ≤ Mu . ¤ dvb¹Qby{Ã5dhsu_s (1 − θ + µθ )u ≤u , 2 95 3 ¹ _dyud u z}qsu_bdÃpz}q{5q{z´stoZqK5°¬wbs{zIv ¨ 2 ~/v < 3ï± mrz¬vd u z}qUs{_bdÃ6z}qu 5q{z¬stoZq{I°wrs{zIv I¨ ~/v ª 3ïºãIv º1¨©5y»I°¬° v)¨©Iye°° º/¹\dU_GÃ5d¦s{_¯s z}qª Ãp2 z}q{5q{z´sto¼qKwb§bµNq{fI°wr(x) s{zIvZ≥¨ 0 sup L x(x, Dv(x)) =α0 ± i?IyudÃ5dy3º/§po)(1su_b−d<θ 5v+qtsuy{µθw¯ïs{)uzIvZ¨su_bd q{d8|6wbd8v d 2 r 3 ºbv§po 2 9 3 ºr¹%d_GÃ5d (1 − θ + µθ )u ≤ (1 − θ )u + µθ |u | , 2 5 3 Mu ≥ (1 − θ )Mu + θ k. 2 5§*3 ^'¾pzvb¥su_bdrz¬ÍWdyudvd®§Md st¹\ddv 2 II< 3Uv 2 I*§ 3ïºhIv ¾pvb¹ zvbs{_¯s u z}q¦su_bd®Ãpz}q{5q{z´sto q{I°wrsuz¬5vI¨ 2 ~ã*v 3ﺯ¹%d_GÃ5d N. u1 ≤ u 0. 2. 1 N. 1. n. 0 0. 2. 0. 1. 2. 0. 0. 0 0. 0 0. . . n. n+1. n. n+1. n. 0 0. n. N. n n. n. n. n. n. n. n. n. n. n+2. n. n. n. n+1. n. n+1. n. n+1. n+2. n+1. αi. αi. n. i. αi. n. n+1. n. n+1. n. n+1 n. n. n. n. n+1. 0 0 n. n+1. (1 − θn + µθn )un+1 (x0 ) − Mun+1 (x0 ) ≤ (1 − θn )un+1 (x0 ) + µθn |u0 |0 − (1 − θn )Mun (x0 ) − θn k. rm ¹\d Iv»quGos{_¯s z}q1Ãpzqu 6qKz¬sto(qKw§rµNq{I°wrsuz¬5vªI¨ 2 ~/v.3 ±'^`_d+ IceIy{z}q{Iv y{zv zb°dz¬ceb°z¬d3q 2 (1−θ 95 3ïºrIy`+µθ d3|5wz¬Ã)u°dv6su°¬o u −u ≤ θ (1 − µ)u . 2 <3 ≤ (1 − θn )un+1 (x0 ) + θn k − (1 − θn )Mun (x0 ) − θn k ≤ 0. n. n. n+1. n+1. ìì çKöLïõL. n+2. n. n+1.
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(176) . Ajq1zv?¿¬H9bº5~ãy{p¨W¨Ms{_dIyudc 9±ÁGÂNºI§po(su_bd+zvbd8|6wI°¬z¬s{zd8q. ¬z v ±/^`_bd8v¹\dvsuI¾Id (1 − µ)u zvrw¯ïs{zIv"¹\dR_GÃId N. 1. θ1. Iv¦u¹\d−Iu§rs≤zv uu = 1−µ 0. 1. 0. zv. #, ) <!#"$%". º5¹%dhI§rszv. ºbIv¦§po. RN u1 − u 2 ≤ 2 2 − u3 ≤ (1 − µ) u2. un+1 − un+2 ≤ (1 − µ)n+1 un+1 ≤ (1 − µ)n+1 |u0 |0 .. 2 <@3. 2. ·\o 2 r 3v 2 I.3 ºE¹\d¦IvÌvB¨©wbv s{zIv u ∈ C(R ) º1q{w_Às{_s |u − u| → 0 º'¹ _bdv ±/~/yuIM5q{z´suz¬5v<r±Ivs{_bd(qKsuI§bz°¬z¬stoUI¨q{I°wrs{zIv¯q`z¬ceb°o¦su_s z}q+eÃpz}q{5q{z´stoUq{I°wrsuz¬5v n → +∞ I¨ 2 4~ 3ï±\^`_bdv"¹\d(IvBquGos{_¯s u IvpÃId8y{5d8qãs{ u ºrs{_bdwvbz|6wbd Ã6z}qu 5q{z¬stuo¦qK5°¬wrsuz¬5v"¨ 2 ~43 º¹ _bd8v ± ¤ d(¹`v6s sud8qKs{zces{d Iv"wbbMdyh§MIwv"¨ u − u ¨©5yhv?yu§bz¬s{yyuo n ±\·\o 2 I<3`v n q{zv→ d +∞ ºb¹%d5§rsuIz¬vs{_s8ºr¨©5y+°° n ≥ 0 º (1 − µ) < 1 N. n. 0. n. n. un − u ≤ . +∞ X i=n. (1 − µ)i |u0 |0 =. !" $#&%(')+*-,/.&%1>59. %. *. = " . 2 G<3. (1 − µ)n (1 − µ)n |u0 |0 = |u0 |0 . 1 − (1 − µ) µ ! .;%. %. Ajq`¹%d_¯GÃIdrIvbd¨©5y`s{_bdd3|5w¯s{zIv 2 ~ 3ﺹ\d¹ z¬°°EIbbyu5I_ 2 m 3\§poUeq{d8|6wbd8v d¨d8|6ws{zIvq8± 6Ed s u ∈ C (RN ) §¯dsu_bdwbvbz}|6wbd(qK5°¬wbs{zIvB¨ h0. mrz¬vd ¸Iy. 2 mr<3. b. S(h, x, uh (x), uh ) = 0,. Muh0. x ∈ RN .. z}q Iv6suz¬vpwb5wqº6su_bdyudd «rz}qtsq ªwvbz|6wbd(q{I°wrsuz¬5v. uh1 ∈ Cb (RN ). max{S(h, x, uh (x), uh ); uh (x) − Muh0 (x)} = 0,. n = 2, 3, · · ·. ºr¹\dvbIs{d. uhn. ¨. 2 I<3. x ∈ RN .. s{_dwbvbz}|5wd( Iv6suz¬vpwb5wq v§MIwbvbd8"qK5°¬wbs{zIvB¨. 2 mpv*3 zv ±1ghq{z¬vb»yudcyu¾¦r± 9 ^`_bdj¨©wbv s{zIv zq`»qKw§rµNq{I°wrsuz¬5v¦¨ mr<3 ºbv¦s{_bd8v IvÀIquqKwbcersuz¬5v u2 mbh1.3 ºE¹\deÃIdyuz¬¨©o"s{_s uh 2 ≡ 0 zq"q{wb§rµ¶qK5u°¬h1wbs{z≤Iv®uIh0¨ 2 I Rh3 Nzv RN º'v®s{_dv¹%d _¯GÃId 0 ≤ uh1 ≤ uh0 zv RN ±`~/yuIM5q{z´suz¬5v<r±zceb°z¬d3q s{_¯s 0 ≤ Muh1 ≤ Muh0 ºbs{_dv uh2 zq+ q{wb§rµ¶q{I°wrs{zIv¨ 2 I ï3 ºbIvB_bdvd u ≤ u z¬v RN ±/·\o¦z¬v¯rwïsuz¬5v"Iv n ºb¹%d5§rsuIz¬v max{S(h, x, uh (x), uh ); uh (x) − Muh(n−1) (x)} = 0,. h2. x ∈ RN .. 2 I<3. h1. 0 ≤ · · · ≤ uhn ≤ · · · ≤ uh2 ≤ uh1 ≤ uh0 .. Ajqz¬v)q{wb§q{d8 s{zIvb±Iº¹%dq{wbbM5q{des{_s. ±U^`_bd8vºqKzv d u |u | > k I v % ¹ ( d 8 " v b _ p 5 { q d K q w _ { s ¯ _ s ºbIv µ|u |u | > k µ ∈ (0, 1) µ|u | < k ÆpÊ Æ ' %% n h0 0. . 0 0. 0 0. º¹\de_GÃId°}q{ ±. → u0 h0 |0 < k. h0. . uhn − uh(n+1) ≤ (1 − µ)n |uh0 |0 .. 2 b3. æ©Ù'ìæèÚ.
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(179)
(180) "!$#&%' ( )* +%,!$.-(%/(. Ê Ê <: ¤ djw¯qKd su_bdq{Ic»dhc»ds{_brbq\Iqãz¬v¦s{_bd8Iyudcb±IºIsu¾pzvbªq{Iced ±1^`_bdjwbvbz}|5wdjrz¬ÍMd8y{d8v d z}q su_s¹%d _GÃIdsuUqK_¹Qsu_s (1 − θ − µθ )u zqj¦qKw§rµNq{I°wrsuθz¬5v?¨ 2 mpv .3ïºM¹ _bz}_<Iv §Md¹ yuz´s{s{dv n. n. n. h(n+1). x ∈ RN .. max{S(h, x, uh (x), uh ); uh (x) − Muh(n+1) (x)} = 0,. ¤ z¬s{_Bsu_bdceIvbIs{Ivzz´sto¦I¨ S º¹%dI§bsuzvs{_bdyud8q{wb°¬s8± 2 ¤ dj_¯GÃId byuÃId3»s{_s u IvpÃId8y{5d8q's{ s{_bdqK5°¬wbs{zIv _¯GÃId. uh. hn. uhn − uh ≤. +∞ X i=n. (1 − µ)i |uh0 |0 =. I¨ 2 m 3 º5¨©5y. n → +∞. (1 − µ)n |uh0 |0 . µ. ±'i?5y{d8ÃId8y1¹%d 2 5.3. B ! B `Ä*À(!B&
(181) ( B %( % B'*"# + J¶v¦s{_bz}q`qKd3ïs{zIvU¹%dj¹ z¬°°WwqKdhs{_bdced su_brbqãI¨ã¿ GÂ:ºW¿ÁGÂNº6s{ 5§rsuIz¬vvUwb¯d8y%§MIwv¦I¨ u − u º6¨©Iy I°¬° ± ¤ dqKsuIyKs`¹ z¬s{_s{_bd5qKd ºbIvsu_bdvB¹%d¹ z°¬°qKs{wbos{_bd5dvbd8yuI°M8Iq{d n ≥ 1 ±1¸'zv°°¬o5º ¹\dn¹ z¬°°wq{ds{_bd3qKdd3qtsuz¬cs{d8q\suenI=§rs0zvsu_bdwbbMdy §MIwvB¨ u − u ± . n. hn. h. ;. . . */). %. -= ,/.&)?9&,. . = . 9. != )?>. . %5vq{zrd8y\s{_bdbyuI§°¬d8c 2 ~ã 3`vBz¬suq Ãpz}q{5q{z´sto¦q{I°wrsuz¬5v Lu0 := sup. u0 ∈ Cb,l (RN ). ±6d s. [cαi ]1 |u0 |0 + [f αi ]1 . 1 − [σ αi ]21 − [bαi ]1. ¤ d yud8I°¬°W_bd8y{dsu_bdy{d3qKw°´s+I¨%¿ @6º 6EdcecA(± ÂN± %Æ )¡ 2 1
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(197) . #, ) <!#"$%". ÊÊ <:/ghq{z¬vb 2 8<3 ºrIds . Q(g ) = Kc |g|0. X. 1−i |h|ki = Kc |g|0. X. |h|γ¯ (1−i)+ki .. mrz¬vd γ¯(1 − i) + k ≥ γ¯ ºb¨©Iy+°° i ∈ J ºb¹%d5§rsuIz¬vs{_bdyud8q{wb°¬s8± 2 ¤ dy{d3I°¬°
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(203) $ () 0 b,l. . N. h0. . u0 (x) − uh0 (x) ≤ C¯0 |h|γ¯ ,. . +. ) )*% 5 )F ) ) . N. ). . ¯ 3 2 E0 2 .9 3. ∀ x ∈ RN ,. C¯0 := |J|Kc |u0 |0 + R,. .!. b,l. #& !
(204) ) - . Ê Ê <: J¶v¥¿ ÂEsu_bd Iwrs{_b5yuq+ÃIdyuz¬¨©o¦s{_¯s z}qh°5q{q{z}°q{wb§rµ¶q{I°wrs{zIv"¨ 2 ~ã<3 ±`·\os{_bdª Iv¯qKz}qtµ sudvo¦_pop¯Is{_bd3qKz}q 2 mr 3ïº 2 8 3`v°¬d8cece u 9¯± bº R. K. 0. S(h, x, u0 (x), u0 ) ≤ Q(u0 ) ≤ |J|Kc |u0 |0 |h|γ¯ ,. x ∈ RN .. i<IvbIs{Ivzz´stoz¬ceb°z¬d3qsu_s u − |J|K |u | |h| ≤ u ±?·\o¼¿ rº 6d8c»c A ±¬Â:º'¹\d¦_GÃIdesu_s ºr¹ _bdyud R rd8¯d8vbq 5vb°¬oUIv K bd Ìvbd3Bz¬v 2 A 3 ±ãmpe¹%d_GÃ5dhsu_bdyud8q{wb°´s3± 2 |u − u | ≤ R 0. 0. ;. c. 0 0. γ ¯. h0. 0. . . */). %. -= ,/.. n. . = . 9 =)?>; . n≥1. %5vq{zrd8yªv¹s{_bdBbyuI§b°dc$¹ z¬s{_ n zcebwb°}qKzIvq 2 ~/v*3ïº%¨©Iy n ≥ 1 º%vPz´sqeÃ6z}qu 5q{z¬stoq{I°wrsuz¬5v ± ¤ d Id8vbdy°/z 88d%_dyud%su_bd+ced su_br»¨¿ GÂNºI§poªz¬v6s{yurrwz¬vbsu_bd ¯d8yKsuwbyu§¯d3ed8|6wsuz¬5v u ∈ C (R ) 2 ~/v4~ 3 max sup L (x + e), Du (x)); u (x) − Mu (x) = 0, su_s/_¯Iq1wbvbz}|6wbd`Ã6z}qu 5q{z¬sto qK5°¬wbs{zIv u ∈ C (R ) ±'^`_bd vd «ps/y{d3qKw°´s3ºbyuÃId8 z¬v»su_bd+bMdv¯rz´«
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