Two Implicit Runge-Kutta Methods for Stochastic Differential Equation

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1 Alied Mahemaic, 0, 3, h://dx.doi.org/0.436/am Publihed Olie Ocober 0 (h:// wo mlici Ruge-Kua Mehod for Sochaic Differeial quaio Fuwe Lu, Zhiyog Wag * Dearme of Mahemaic, Uiveriy of lecroic Sciece ad echology of Chia, Chegdu, Chia mail: zhywag@uec.edu.c Received Augu 0, 0; revied Seember 0, 0; acceed Seember 7, 0 ABSRAC hi aer, he ô-aylor exaio of ochaic differeial equaio i briefly iroduced. he colored rooed ree heory i alied o derive rog order.0 imlici ochaic Ruge-Kua mehod (SRK). wo fully imlici cheme are reeed ad heir abiliy qualiie are dicued. Ad he umerical reor illurae he beer umerical behavior. Keyword: Sochaic Differeial quaio; mlici Sochaic Ruge-Kua Mehod; Order Codiio. roducio hi aer, we wa o obai umerical mehod for rog oluio of Sochaic Differeial quaio of ô ye. d y f y dg y d W, y (.) oe ha f i a lowly varyig coiuou comoe fucio, which i called drif coefficie, g i he raidly varyig coiuou fucio called he diffuio coefficie. W i a wieer roce. Recely, may cholar have uccefully derived ome mehod for SD for boh ô ad Sraoovich form. Burrage ad Burrage [-3] eablihed he colored rooed ree heory ad Sochaic B-erie exaio. ia ad Burrage [,4,5] derived ome rog order.0 -age Sochaic Ruge-Kua mehod, icludig emiimlici ad imlici mehod. Wag P. [6] derived ome rog order.0 3-age emi-imlici mehod. Wag ZY [7] maily coidered he rog order SRK for he SD of ô form. hi PhD hei he offered u he Colored Rooed ree heory for ô ye, ad coruced ome -age ad 3-age exlici mehod. Alog hi lie, will coruc ome imlici SRK for SD of ô ye. Secio, he colored rooed ree heory for derivig SRK for SD of ô ye i briefly iroduced ad he -age fully imlici SRK are obaied. Secio 3 we will dicu heir abiliy roery. Ad i Secio 4, we will reor he umerical exerime.. -Sage mlici SRK ad Order Codiio May cholar, icludig Burrage [], offered he defi- * Correodig auhor. iio of he order of umerical mehod i heir hei. Defiiio.. Le y be he umerical aroximaio o y afer e wih coa eize 0 ; he y i aid o be coverge rogly o y wih order if y y Ch, h 0, (.) oe ha C i a coa ha ideede of h ad >0. Bucher reeed he Rooed ree heory, afer which hi heory wa exeded io ochaic area. Burrage [] reeed Colored Rooed ree heory i her PhD hei, ad Wag [7] did he reearch eecially for ô SD. Similar o he deermiiic codiio, he defiiio of he elemeary differeial ca be aociaed wih m F y y F y f y F y g y, F y f F y,, F m y,, m, F y g F y,, F m y,, Here ad for he ree havig order 0. Wag [7] deduced he ô-aylor erie for SD. Firly le iroduce wo oeraor 0 L f g x x L g x m m Coyrigh 0 SciRe.

2 04 F. W. LU, Z. Y. WAG ow we iroduce a very imora rooiio from Kloede ad Plae [8]. Prooiio.. if A M, h : i ufficiely derivaive, ad le X() be he oluio of he equaio he 0 X0 d X f X g X d W, > 0 X 0 (.) h X h X h X Leig X aa ara X h X, he X LX LX LLX LLX LLX LLLXL X f g gg gf fg g g g g gg Ad from he defiiio of he elemeary differeial we ca kow X FX0F X0 0 FX0 FX0 FX00 F X0 F, 0 F F, FX0 FX00 F X0 0 FX0 F, X F ().5 F X F X F X Like he cocluio of Burrage [], he aylor-erie of he acual oluio of he SD i X F (.3) he rucure of Sraoovich-aylor erie i imilar o he ô-aylor exaio, however, he ochaic calculaio of hee wo ye are differe. able ree he ree ad he correodig elemeary differeial. ecially, i order o illurae he differece bewee ô ye ad raoovich ye, we li all he ochaic calculaio of ree havig order. ow we how geeral form of Ruge-Kua mehod for SD of ô form. Le he eize of he mehod i a coa h, h,,, y i he ume- rical oluio of X, he i i i 0 Y y Z f Y Z g Y 0 y y z f Y z g Y oe ha 0 i i 0 l i i l i l i l i he i,, Z h, i,,, z h,,, Z b, i,,, z,,, i i radom variable. Uig he Bucher able, SRK ca be wrie a A B B B (.4) Wag [7] deduced he aylor erie for he SRK of ô form. Ad offered he defiiio of lemeary Weigh, which ha he ame form of Burrage cocluio []. Defiiio.. e, 0 l z i,,, i l z i i 0 l z i,,, i l z i,,, i,,, A he defiiio of lemeary Weigh ha we obaied, we ca gai he ochaic Ruge-Kua erie exaio Y Fy0 l! (.5) able offer he ree ad heir lemeary Weigh. From he quaio (.4) ad (.5) we ca obai he rucaio error a. Coyrigh 0 SciRe.

3 F. W. LU, Z. Y. WAG 05 able. ree ad he correodig elemeary differeial. 0, 0, ,, , 0, 00, 0 00, able. ree ad he correodig elemeary weigh. 0 0, e.5 0 z 0.5 z e ( ) l! L F y e F y Prooiio., give by Burrage ad Burrage [3], give he eceary codiio of he mehod. Prooiio.. L i he local rucaio error of he umerical mehod a, i he global error a, if f ad g i ufficiely derivaive, ad,, he L O h z e.5 z Z e.5, L O h amely Oh Z e z Z e 6z Z Z e 3z Z e obviouly,, e e, hu i ) We u eed o coider he codiio whe.5. ow we iroduce he radom variable, h. Ad we oe c Ae, b B e, d B e, b d h ow le ar o coruc he mehod of rog order.0. ) For ree z e e h e h e From he Prooiio., he Ruge-Kua mehod of he rog order.0 have o aify e ) ha e ) ha.5 e 0 (.6) e h 0 ) For ree e, e z e e h e 3) For ree Coyrigh 0 SciRe.

4 06 F. W. LU, Z. Y. WAG amely ad amely z Z e b d b h d X DX,, X b d b d 3 0 D X b, d b 0, d 4) For ree z Z e h b d h ) For ree d 0 0 z Z e 0 h c c h 0 c 6) For ree h h B B h b d B d B b B b B d 7) For ree, 0 z Z e bd b d 0 hu, he -age imlici SRK hould aify he yem e e e b d b (.7) d d c B d B b B b B d bd b d 0 Here we gaied he codiio for he mehod wih rog order.0, heoreically we ca coruc ay-age mehod, boh exlici ad imlici. Ad ow we coider he -age imlici mehod. a 0 b b d d 0 a b b d d Brigig he able io he yem.7, ad leig he a a,,, we ca obai he fir cheme m m Furhermore if we coiue o le b b d d 0, we ca obai aoher cheme m. m Sabiliy Saio ad Mizui [9] iroduced he defiiio of meaquare(ms) abiliy, ad he cholar uch a Burrage [] ad ia [4,5] reearched i ad gave ome imrove- Coyrigh 0 SciRe.

5 F. W. LU, Z. Y. WAG 07 me. Coider he liear e equaio of ô ye of SD. ad we ue oe-e cheme d y yd ydw (3.) y,,, R h y h i he eize, i he radom variable i he umerical cheme. Saio ad Mizui [9] iroduced he defiiio Defiiio 3.. f for,,h, R h,, R h,,, < he he umerical cheme i aid o be MS able, ad he Rh,, i aid o be he MS-abiliy fucio. ) For m, we ca obai he MS-abiliy fucio y R h,,, y Rh,,, R RRq RRq oe ha R 5q 3q6 R3 R q 68q 3R 3 R3 4q 4q 0q 3 5q 4 q ad h, q h, i he adard Gauia variable 0, Figure decribe he able regio of m. ) For he mehod m, we obai ha y Rh,,, y ad oe ha R h,,, R qr R R qr R q Figure. Sable regio of m. h, q h, i he adard Gauia variable 0, Figure reree he able regio of m. 4. umerical Reul ow we reor he umerical reul of he cheme derived i hi aer. A fir we will ue he oi of umerical imulaio i a igle raecory o comare he abolue error M of five differe cheme exlici uler-maruyama cheme, exlici milei cheme, exlici wo-age cheme which i deiged by Wag [7], m ad m for a ame o-liear yem 0. Afer which we will imulae 00 raecorie of each cheme ad he comare heir abolue error M. rror for he (4.) i give by k M xi yi k i oe ha x i i he exac value a e oi i ad y i i he umerical imulaio a ha oi, k i he umber of he oi choe i he raecorie. Ad he o-liear yem (4.) i give by dx X X d X d w. 0,5 (4.) X 0 0 Ad he aalyical oluio of he yem 0 i X ih w (4.) Firly, we comare he error M i a igle raecory. From he able 3, we ca kow ha i a radom raecory(acually we chooe he fir oe), he m i obviouly beer ha all he oher cheme, ad alo, Coyrigh 0 SciRe.

6 08 F. W. LU, Z. Y. WAG obviouly beer ha all he oher cheme, eecially whe h,,. Sill, m alway ha a ame accuracy wih cheme ad milei cheme. how ha m i beer ha oher cheme, ad m i alo a roer cheme for olvig ochaic differeial equaio. Figure. Sable regio of m. able 3. he abolue error M i a igle raecory. Seize uler Milei m m able 4. Mea of he abolue error M i 00 raecorie. eize uler milei m m m ha a ame accuracy wih cheme ad milei cheme. ow le cora he abolue error M of 00 raecorie. From he able 4, we ca coclude ha m i RFRCS [] K. Burrage ad P. M. Burrage, High Srog Order xlici Ruge-Kua Mehod for Sochaic Ordiary Differeial quaio, Alied umerical Mahemaic, Vol., 996, doi:0.06/s (96)0007-x [] P. M. Burrage, Ruge-Kua Mehod for Sochaic Differeial quaio, Ph.D. hei, he Uiveriy of Queelad, Queelad, 999. [3] K. Burrage ad P. M. Burrage, Order Codiio of Sochaic Ruge-Kua Mehod by B-Serie, S Joural o umerical Aalyi, Vol. 38, o. 5, 000, doi:0.37/s [4]. H. ia, mlici umerical Mehod for Siff Sochaic Differeial quaio ad umerical Simulaio of Sochaic Model, Ph.D. hei, he Uiveriy of Queelad, Queelad, 00. [5]. H. ia ad K. Burrage, wo Sage Ruge-Kua Mehod for Sochaic Differeial quaio, B, Vol. 4, o. 3, 00, doi:0.03/a: [6] P. Wag, hree-sage Sochaic Ruge-Kua Mehod for Sochaic Differeial quaio, Joural of Comuaioal ad Alied Mahemaic, Vol., o., 008, doi:0.06/.cam [7] Z. Y. Wag, he Sable Sudy of Sochaic Fucioal Differeial quaio, Ph.D. hei, Huazhog Uiveriy of Sciece ad echology, Wuha, 008 [8] P.. Kloede ad. Plae, umerical Soluio of Sochaic Differeial quaio, Sriger-Verlag, Beli, 99. [9] Y. Saio ad. Miui, Sabiliy Aalyi of umerical Scheme for Sochaic Differeial quaio, S Joural o umerical Aalyi, Vol. 33, o. 6, 996, doi:0.37/s Coyrigh 0 SciRe.