Strong Convergence OF An Adaptive Euler-Maruyama ... - CiteSeerX

1 downloads 0 Views 396KB Size Report
keywords stochastic differential equations, Euler-Maruyama, adaptivity, finite-time ... step of a forward Euler drift step so that it deviates only slightly from aÂ¥# ac k ...
          "!$#%&('  !) *+ ,-$.*/ 01  $- 234 56789,: # "; !84 2 A@CBD>EF=(GH@I=KJMLINPO(QC=SRUTC=(Q"V+=SRARUBWQYX[Z]\Y^_=(Qa`cb$Qa`Y>Adfe ghRjiC=S>ARk l m?n:oqp7rfsMo tuwvMxyvMz|{j}~uvMx€vMyv4‚yƒ…„|†Mu‡ƒ…„‰ˆŠu:‹…ƒŒu~‚y‚f„‰xŽFvMz|MM‘„|ƒ…’fˆ”“•MI–˜—‡™K‹~š˜›fvM‹…u:œy‚Ax8ƒ…’Žu™Kœfz|u~ŒžDŸ_v4‘œŽ{AvMˆ v ‹…¡‘’yu:ˆŠuMš¢vMxy£‚y‘q†Mu‹ŒƒŒ…AxŽ ¡AxU†Mu~…Mu:xy¡u,7†Mu~¤fxy„|ƒŒuƒ…„‰ˆŠu„‰xUƒŒu~…†4vMz‰‹8vM‹€ƒ…’yuœy‹…u~€„¥xŽ‚fœŽƒ ƒŒAz‰u~‘vMxy¡u ¦ ƒŒu:xyy‹ŠƒŒ}~u~…Ž§©¨C’yu.xUœyˆ u~‘„‰¡~vMzaˆŠu~ƒ…’Ž˜y‹ v4‘u›fvM‹Œu: Ax©‹Œ‚fz‰„|ƒŒƒ…„¥xŽ_ƒ…’yuªŽ‘„|“•ƒ%vMxyy„¬«­œy‹…„‰Ax­šKvMxy ¡Ax®ƒŒ‘Az‰z‰„‰xŽvMxu:‹Œƒ…„‰ˆ v4ƒŒuM“Iƒ…’Žuz|j¡~vMz3u~…‘M „‰xy¡~œŽ…‘u:„‰xvMŽ†4vMxy¡~„‰xŽƒ…’ŽuŽ‘„‰“¯ƒ~°KvMxU{$ˆ u~ƒ…’Žj ±I’f„‰¡‘’ ¡Ax®ƒŒ‘Az‰‹Iv ²jœyvMxUƒ…„|ƒ³{€›´Aœyxyyu:€›u:z‰q±µ›j{€ƒ…’y„‰‹aŽ‘„‰“¯ƒYu~……M¶„‰‹CvMz‰‹… ¡q†Mu~…u:›j{ƒ…’yu·vMxyvMz|{˜‹…„‰‹~§[tuFˆ v4¸Mu z|˜¡~vMz(¹?„|‚f‹…¡º’y„|ƒŒ}%vM‹…‹…œyˆ ‚yƒ…„|Axy‹¢Axƒ…’ŽuŠŽ‘„|“•ƒ¢vMxfy„¬«­œy‹…„|AxªƒŒu~‘ˆ ‹vMxyvM‹…‹…œfˆŠu%›AœfxyŽu:ªˆ AˆŠu:x®ƒ…‹wM“ ƒ…’Žuwu»ŽvM¡ƒ¶vMxfªxjœyˆŠu~‘„‰¡~vMzS‹ŒAz‰œŽƒ…„|Axf‹~§Y¼‡œyIvMxyvMz|{˜‹…„‰‹a‘u:z‰„|u:‹CœŽ‚´Ax€ƒ…’Žu½“•vM¡ƒ¢ƒ…’yv4ƒ~šf“•M‡‹…œŽ¾9¡~„|u:xUƒ…z|{€‹…ˆ vMz‰z ƒŒAz|u~‘vMxy¡u:‹:š­ƒ…’Žu½ƒ…„¥ˆŠu:‹ŒƒŒu~‚f‹Š¿ÀvMxy’Žu:xy¡u%ƒ…’Žu%z|˜¡~vMzSu~…‘M‘‹ºÁIv4…uŠvMŽu:²Uœfv4ƒŒu:z|{ª¡AxUƒŒ…Az‰z|u:„‰xˆŠA‹Œƒ‡M“hƒ…’Žu ‚f’yvM‹ŒuC‹…‚fvM¡uIvMxyŠƒ…’fv4ƒhƒ…’ŽuC¡AxUƒŒ‘„|›fœŽƒ…„‰Ax%ƒŒwƒ…’ŽuIˆŠu:vMxŽž³‹…²Uœfv4…uau~……Mh“•…AˆÂƒ…’yuC…u:ˆ vM„‰xfŽu~PM“ƒ…’yua‚f’yvM‹Œu ‹Œ‚fvM¡u·„¥‹Yƒ…’Žu:x.xŽu~Az‰„‰A„|›fz|uM§ ÃÅĎÆPǀÈÅÉ®ÊÌˠͺÎÐÏÒÑ~Ó´ÔjͺÎÐՕѠִկ×?ØÚÙºØÚێÎՕÔjÜPØ7ÝfÞ´ÔUÎկϘ۴ÍÚßÅàa޴ܯØÚٌáâ ÔjٺޝãŽÔjä_ԝ߭ÔUÖ­ÔjåÎÐÕ¯æfկΌãjߝç­Û´Õ¥ÎØqèWÎÕ¯ä_ؽÑÚϘێæ˜Ø7ÙÐé˜Ø7Û´ÑÚØUê ë

hì íîYïað,ñ ò ówîYôÚðªí õ Û,ÎÓ´Õ¯Íå­ÔjåSØÚÙö¶Ø Í‘Î޴֝ãÎÐÓ´Ø ÛyÞ´ä_Ø7ÙÐÕ¯Ñ4ÔjÜPͺϘܕÞÒÎՕÏjÛÏj÷ÌÎÐÓ´Ø8øùŠà úŽû(ü]ýºþY ÿ Yü³û(ü]ýºþºþºú˜ý (ü³û(ü]ýºþºþºú  ü]ýº þ  Âý  û(ü ˜þaÿ û ü˜êMþ ŽãäªØ4ÔjÛ´ÍwÏj÷3ÔjÖ­ÔUåÎÕ¥æ˜ØŠÎÕ¯ä_Ø:èŒÍºÎÐØÚå´åÕ•Û´é.Ôjܯé˜Ï˜ÙºÕ¯Îӝä_Í7ê  Ø7ÙÐØû(ü]ýºþ  ÷]ϘٽØ4ÔjÑ~Óý½Ôj۝Ö  ü ýºþ½Õ•ÍFÔ!ú ½ÖÕ•äªØÚ۴ͺՕϘۭÔU#Ü "¶ÙºÏMöwÛ´ÕÀÔjÛ ä_ÏjÎÐՕϘÛ(ê$ÓyÞ´Í%'&()+*  ÔjÛ´,Ö -&.)/*0213465­Ï˜ÙÌÍÐՕäªå´Ü¯Õ•ÑÚեΌã ö¶ØwÔjͺÍÐÞ´äªØÎÓ­Ô®ÎYÎÓ´ØÕ¯Û´Õ¯ÎÐÕÀÔjÜyÑ7Ϙ۴ִեÎՕÏjÛՕÍYÖ´ØqÎØ7ÙÐäªÕ•Û´Õ¯ÍºÎÕ¯Ñjê7$Ó´ÙÐÏjÞ´é˜Ó´Ï˜ÞÒÎ 8:9;8 Õ¯ÍF޴ͺØÚÖÎÏÖ´Ø7Û´ÏjÎØØ7Õ¯ÎÐÓ´ØÚÙFÎÐÓ´Ø.àaÞ´ÑÚܯՕִØÚÔjÛæ˜Ø7Ñ7ÎÐϘي۴Ϙٺä6ϘÙFÎÐÓ´,Ø 5­ÙÐÏ ?Ø7۴Օ޴Íü³Ï˜ÙFÎÐÙÔjÑ7ØMþFä_ÔUÎÙº=Õ ©ØÔjͺÍÐÞ´äªØ_ÎÐÓ­ÔU@Î ABÔjÙºØܯÏÒÑÚÔjܕܥD ã C[Օå´ÍºÑ~Ó´Õ¯BÎ EÔUÛ´Ö£ÎÓ´ÔUÎ8ÎÐÓ´ØFØ