Consistency of Generalized Finite Difference Schemes for the Stochastic HJB Equation

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(1)Consistency of Generalized Finite Difference Schemes for the Stochastic HJB Equation J. Frederic Bonnans, Hasnaa Zidani. To cite this version: J. Frederic Bonnans, Hasnaa Zidani. Consistency of Generalized Finite Difference Schemes for the Stochastic HJB Equation. [Research Report] RR-4162, INRIA. 2001. �inria-00072460�. HAL Id: inria-00072460 https://hal.inria.fr/inria-00072460 Submitted on 24 May 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..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Consistency of generalized finite difference schemes for the stochastic HJB equation J.F. Bonnans — H. Zidani. N° 4162 Avril 2001. ISSN 0249-6399. ISRN INRIA/RR--4162--FR+ENG. THÈME 4. apport de recherche.

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(97) . W4]<gV]<W4aj`&hYXgVYrvdancmejv ejcmTVY o _an`:s ejgY]<lj]<jYrg¢^pcmTVY=TVejW4ehjYXgYXej_l]fgVY {_an`<]¦cd]fYlX£] ¨ Y ¨ £cmTVY o _an` sXejgVY7]bl % v e ¯ Y ¯Y ? v & ª ]fcmT g %U ? De ¯ gs & ¨ ~y_Vv ajsÆcd_an`s ejW4iV_Vcda+cd]fehgl o Yran`ª ]¦cdTµlqiahs Yl?en lqp^W4W4Y cdvm]bs W/a+cdvm]bs Yl?en lq]<¤XY ¨ ahs"T/enEcmTVYrW srang ¢Y4vdYXiVvdYrlmYXg{cdY o ¢^p]fcdl_ViViEYXv7cmvd]bangVh_V`<ajv7ianvmcr£&ajg o cmT^_l7]bl,^]fYrªY o ajl,ajg%Yr`fYrW4YXg{cen \ £ g > k UC ,k]<g!ianvmcm]bs _V`banv£ g £VV£Vajg o r£êjyv g 2 £ / £ang o Z£kvmYlqiEYrs cm]<jYX`<p ¨ ~ygsXYa7jYrgVYXv"a+cdejv?ejEcdTVY o _an`sXejgVY TajlL¢EYXYrg/ej¢kc"an]<gVY o £h]fcvdYXW/an]<glcde] o YXg{cm]fpcdTVYsX`<ahlmlmYrl?ej Y {_V]f+aj`fYrgs Y]<gJehv o YXv cdeej¢Vcdan]<gJsXejW4iajs cY tkiVvdYrldlm]fehg ¨ STV]blªajl o ejgY¢^p!lmejvmcm]<gV/cdTVYYr`fYrW4YXg{cdl eh`f`<e+ª ]<gVAcmTVY 3¼ejv o YXvdY o < ªYX]<jT{c"l en o ]banjehgan` Yr`fYrW4YXg{cdrl 3cmTVY`<ancqcmYrv)¢EYX]<gV£ahl ªYan`<vmYa o pµ©^gVe+ª,£ gejgVgVYrhancm]<jY < ¨ ~ãcyajiViYanv"lcmTanc)cmT]<l)lm_ µs Ylejv] o Yrg{cm]fp{]<gV/cdTVYY h_]f+an`<YXgs Y,s `bajldlqYlX£ajl)srang!¢Y s"TYrs"©jY o ¢^pµhYXgVYrvdancm]<gVcdTVYXW _lm]fg=cdTVY7ejvdW_`<ahlh]fhYXg¢Yr`fe+ª ang o s ehW4ianvd]fg=¢EencdT!lmY c"l ¨

(98) ¯ ¼ ³ ~ gcdTVYsXahlqY g 2 £VªY,s ejW4iV_VcmY o s"Tanv"ajsÆcdYXvd]f¤a+cd]fehglej&cdTVYlmY cdl > H C £ rg U cde r ¨ Y o ]<lmiV`baHp o YXcdan]<`<Y o vmYlq_V`fcdlAehv ng U cdeá ¨ STVYlmY c > H C ]<l/s"Tanv"ajsÆcdYXvd]f¤rY o ¢^páZ sXejglqcmv"an]<g{cdlang o 7Yr {_V]<+an`<YXgsXY,s `bajld l ' . . . STY7lmY c ehv. >H C. ¯¯ ¯ ± Y. *. 2. U ?g . *. 2. EFFF FFF FFF FFFH. ? g. *. 2 2. ¯M¯ ¯± ¯M¯ k ±&± . . ¯± }. ¯M¯. FFH. EFF. ¯± 2 ±&± ¯± } k ¯± } M¯ ¯ ±&± k 2 ¯± } ¯M¯ ±&± . ¯M¯ . k. ¨ STVY lmY c > H C ]<l?s"Tanv"ajsÆcdYXvd]f¤rY o ¢^p4+Z7s ejglxcdvdaj]fg{cdlajg o }Y h_]f+an`<YXgs Y s `bajldlqYl ' G. ehv U sX`<ahlmlmYrl('. 2 }. ¨ STVY lmY c > H C ]<l?s"Tanv"ajsÆcdYXvd]f¤rY o ¢^p r}s ejglxcdvdaj]fg{cdlajg o Z7Y h_]f+an`<YXgs Y s `bajldlqYl ' G. ehv. *. ]bl s"TajvdahsÆcmYrvm]<¤XY o ¢^p L s ehglqcmv"an]<ghc"lang o =Yr {_V]<Haj`fYrgs YsX`<ahlmlmYrl(' f. U ?g . U ? g . 2. ¯M¯ ¯± ¯M¯ k ±&± ¯± ¯M¯ k ±&± ¯ ± ¯M¯ k 2 ±&± ¯± U 2 ¯¯ k ±&± ¯± U 2 ¯¯ k ±&± U ¯ ± . ¨ STVYlmY c > H C ]bl=s"TajvdahsÆcdYXvd]f¤rY o ¢^p Z{JsXejglqcmv"an]<g{cdlajg o rYr {_V]<Haj`fYrgs Y.

(99) 'V .

(100) 7/ . ¯M¯ EFFF. .

(101)

(102) . FFF FFF FF G FFF FFF FFF FFF FFH.

(103) 9. . ¯M¯ k ¯ ±&± ± ¯ ± Ubs ¯¯ k 2 ±&± ¯± Ubs ¯¯ k ±&± UU ¯± U 2 ¯¯ k ±&± ¯ ± U 2 ¯¯ k ±&± U ¯± U ¯¯ k 2 ±&± UU ¯± U ¯¯ k ±&± U ¯± 2 s ¯¯ k ±&± ¯ ± 2 s ¯¯ k U 2 ±&± U ¯ ± . FFF. . STY7lmY c > H C ]bl s"TajvdahsÆcmYrvm]<¤XY o ¢^pZ L sXejglqcmv"an]<g{cdlang o HY h_]f+an`<YXgs Ys `bajldlmYrl(' ¯M¯. EFFF FFF FFF FFF FFF. G. FFF. FFFH. FFF. FFF. FFF. FFF. FFF. . ¯M¯ k ¯ ± ±&± U U ¯± Ubs ¯¯ k 2 ±&± ¯± Ubs ¯¯ k ±&± UU ¯± U 2 ¯¯ k ±&± ¯ ± U 2 ¯¯ k ±&± U ¯± U ¯¯ k 2 ±&± UU ¯± U ¯¯ k ±&± U ¯± 2 s ¯¯ k ±&± ¯ ± 2 s k U 2 U ds ¯¯¯¯ k ±&± ±& ± U U ¯± ¯ ± ds ¯¯ k 2 s ±&± ¯ ± . h .

(104) H STY7lmY c.

(105)

(106)

(107)

(108) . > H C. ]bl s"TajvdahsÆcmYrvm]<¤XY o ¢^pjsXejglqcmv"an]<g{cdlang o L Y h_]f+an`<YXgs Ys `bajldlmYrl(' EFFF FFF FFF FFF FFF FFF FFF FFF FFF G. FFF FFF. FFF FFF FFF FFF FFF FFF FFF FFF FFFH.

(109) ¯ ¼ ³ #. STY7lmY c. >H C. ¯M¯ ¯ ± ¯M¯ k ±&± U ¯± U ¯¯ k ±&± U ¯± U ¯¯ k ±&± U ¯± U ¯¯ k 2 ±&± UU ¯± U ¯¯ k ±&± U ¯± 2 s ¯¯ k ±&± ¯ ± 2 s ¯¯ k U 2 ±&± U ¯ ± 2 U ¯¯ k 2 ±&± U ¯± 2 U ¯¯ k Ubs ±&± 2 ¯ ± 2 ¯¯ k 2 ±&± U ¯± 2 k U U ds ¯¯¯¯ k ±&± ±& ± U U ¯± ¯ ± ds ¯¯ k 2 s ±&± ¯ ± ¯¯ k ±&± 2 ¯± k U 2 U 2 ¯¯¯¯ k ±&± ±& ± U ¯± ¯ ± 2 ¯¯ kd s ±&± U ¯ ± . TVYrg g £kªY7sXejW4iV_kcdY o s"Tanv"ajs cmYrvm]<¤rancm]<ejglencmTVY,lmY cdl ]bl s"TajvdahsÆcmYrvm]<¤XY o ¢^p f M¯ ¯ ¯ ± ¯M¯ k ±&± > 1 U9C

(110) ¯ " k> 1 U9C ± " k. 2 > 1 U9C

(111) . .. ¯±. £. > H C g U. cmeµ ¨.

(112) 'V .

(113) 7/ . ehv6? g * g ajg o EFFF. 2. FFF. 2. FFF 2. FFF FFF. FFF. . 2. FFF FFF. FFF FFF. . FF. G FFF FFF FFF FFF U FFF FFF U FFF U FFF U FFF U FFF FFH U U.

(114) ¯ ¼ ³ . §Y Qv. . ¨ STVYlmY c. % UdY 2 &. >H C.

(115)

(116) . . /.

(117) 9. ]bl s"Tanv"ajs cmYXvd]<¤XY o ¢^p. ¯M¯ ¯± ¯M¯ k ±&± ¯ ± . ¯M¯ k 2 ±&± > 1 U9C

(118) ¯± k> 1 UC ± " 1 > 1 UC

(119) ¯ " . ¯M¯ k 2 ±&± k " " > 1 U9C

(120) ¯ ± k> 1 UC ± " 1 > 1 U9C

(121) . ¯ " ¯M¯ k 2 ±&± k 2 "-" > 1 U9C

(122) ¯± k > 1 U9C ± " 1 > 1 U9. C

(123) ¯ " ¯M¯ k ±&± k " " > 1 UC

(124) ¯± k 2 > 1 U9C ± " 1 > 1 UC

(125) . ¯ " ¯M¯ k 2 ±&± k " " > 1 U9C

(126) ¯ ± k> 1 UC ± " 1 > 1 U9C

(127) . ¯ " ¯M¯ k 2 ±&± k 2 "-" > 1 U9C

(128) ¯± k > 1 U9C ± " 1 . > 1 U9C

(129) ¯ " ¯M¯ k 2 ±&± > 1 U9C

(130) ¯± k> 1 UC ± " 132 > 1 U9C

(131) ¯ " . ¯M¯ k 2 ±&± k " " > 1 U9C

(132) ¯ ± k> 1 UC ± " 1 > 1 U9C

(133) . ¯ " ¯M¯ k ±&± k 2 "-" Ubs$> 1 UC

(134) ¯± k > 1 U9C ± " 1 > 1 UC

(135) . ¯ " ¯M¯ k ±&± k 2 "-" U > 1 UC

(136) ¯± k > 1 U9C ± " 1 > 1 UC

(137) . ¯ " 2 ¯¯ k 2 ±&± k "-" Ubs$> 1 UC

(138) ¯± k $> 1 U9C ± " 1 > 1 UC

(139) ¯ " 2 ¯¯ k ±&± k "-" U > 1 UC

(140) ¯± k > 1 U9C ± " 1 > 1 UC

(141) . ¯ " 2 ¯¯ k ±&± k 2 " " U > 1 U9C

(142) ¯ ± k > 1 UC ± " 1 Ubs> 1 U9C

(143) . ¯ " . 2 ¯¯ k ±&± k " " U > 1 U9C

(144) ¯ ± k Ubs> 1 U9C ± " 1 U > 1 UC

(145) ¯ " . ¯¯ k 2 ±&± k "-" U 2 > 1 UC

(146) ¯± k $> 1 U9C ± " 1 > 1 UC

(147) . ¯ " ¯¯ k ±&± k "-" 2 > 1 UC

(148) ¯± k > 1 U9C ± " 1 > 1 UC

(149) . ¯ " ¯¯ k Ubs ±&± k 2 " " 2 > 1 U9C

(150) ¯ ± k > 1 U9C ± " 1 U 2 > 1 UC

(151) ¯ ". TYXg g £kcdTVYlmY c. EFFF M¯ ¯ FFF k FFF ¯M¯ k FFF ¯M¯ FF G 2 ¯M¯ k FFF FFF k ¯M¯ FFF FFF FFH ¯M¯ k. ±&± &± ± k " ±&± k ". k . &± ± k " 2 ±&± k . " ". . " " ". k . . >H C. ]<l s"Tanv"ajsÆcdYXvd]f¤rY o ¢{p. ¯± > 1 U9C

(152) ¯ " k> 1 UC ± " 132 > 1 U9C

(153) . ¯ ± > 1 U9C

(154) . ¯ k> 1 UC ± k. > 1 UC " 12 > 1 U9C

(155) ¯± 1 2 > 1 U9C

(156) ¯ " 132 > 1 U9C . $> 1 U9C

(157) . ¯± k $> 1 U9C ¯ " . k> 1 U9C ¯ . 12 > 1 U9C

(158) ± " 132 > 1 U9C

(159) ± 132 > 1 U9C. 2 > 1 U9C

(160) . ¯ k> 1 U9C ± . k> 1 UC " . 1 > 1 U9C

(161) ¯± 1 > 1 U9C

(162) ¯ " 132 > 1 U9C. > 1 U9C

(163) . ¯± k > 1 U9C ¯ "y. k > 1 U9C ¯ . 1 > 1 U9C

(164) ± " 1 > 1 U9C

(165) ± 132 > 1 U9C. ±" " . ±" . ". . ehv6? g * g g ang o §lY DY[v % UdY 2 & ¨ ák °³ ¯

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(167) rZ.

(168)

(169)

(170)

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(172)

(173)

(174)

(175) . 2 2 2. . 2. . 2. 2. Ubs U 2 U. 2. . . 2 2. . 2. 2. . U. 2. . U. .

(176) !#"$ . . 2. . s 2. . U. . H.

(177)

(178) !#"$ %. . . . . U 2. . U U 2 U 2 U. 2. . . U 2. . . s. U . s. U 2. . s 2. Us. U. . . .

(179) &(' )

(180) . . 2. . . . . U 2. U 22. . U U 2 U 2 2. 2. 2. . 2. 2. U. 2. 6& ¬

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(186) ³,$9 @6'* p%- !$ @ £^hej`<_VW4Y ej 6³ p6Q[+-. %,$ )- @6) ¨ \îVvd]fgVhYXv£ YXª zLejvd©C£ N n ¨ *. +. ,. -. .. 0/. 1/. 2. 3. 3. 54.

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(190) . /. /. $3. 7N 8.

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(193) Unité de recherche INRIA Rocquencourt Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France) Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot-St-Martin (France) Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France). Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France).

(194). . ISSN 0249-6399.

(195)