Maximum Principle and the Applications of Mean-Field Backward Doubly Stochastic System

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1 Pre and Applied Mahemaic Jornal 5; 4(3: -8 Pblihed online Jne 8 5 (hp://wwwciencepblihinropcom/j/pamj doi: 648/jpamj5437 ISSN: (Prin; ISSN: (Online Maximm Principle and he Applicaion o Mean-Field Backward Dobly Sochaic Syem Hon Zhan Jinyi Wan eny Zhao 3 Li Zho School o Inormaion Beijin Wzi Univeriy Beijin China School o Bankin and Finance Univeriy o Inernaional Bine and Economic Beijin China 3 School o Manaemen Science and Enineerin Cenral Univeriy o Finance and Economic Beijin China addre drywenjnxian@mailcom (Hon Zhan o cie hi aricle: Hon Zhan Jinyi Wan eny Zhao Li Zho Maximm Principle and he Applicaion o Mean-Field Backward Dobly Sochaic Syem Pre and Applied Mahemaic Jornal Vol 4 No 3 5 pp -8 doi: 648/jpamj5437 Abrac: Since Pardox and Pen irly died he ollowin nonlinear backward ochaic dierenial eqaion in 99 he heory o BSDE ha been widely died and applied epecially in he ochaic conrol ochaic dierenial ame inancial mahemaic and parial dierenial eqaion In 994 Pardox and Pen came p wih backward dobly ochaic dierenial eqaion o ive he probabiliic inerpreaion or ochaic parial dierenial eqaion Backward dobly ochaic dierenial eqaion heory ha been widely died becae o i imporance in ochaic parial dierenial eqaion and ochaic conrol problem In hi aricle we will dy he heory o dobly ochaic yem and relaed opic rher Keyword: Mean-Field Backward Dobly Sochaic Syem Sochaic Conrol Inrodcion eron and Djehiche Bckdahn Djehiche and Li Meyer Brandi kendal and Zho and Lihave died he opimal conrol problem abo Mean-ield ochaic dierenial yem Inpired by he above problem in he paper we dy he opimal conrol problem abo Mean-ield backward dobly ochaic yem In he iaion ha conrol ield o he convex and coeicien conain conrol variable Uin convex variaional and dal echnoloy we preen he local and lobal ochaic maximm principle proved a icien condiion o opimaliy (veriicaion heorem and a neceary condiion[-4] he Conrol Problem o Mean-Field Backward Dobly Sochaic Syem For imple markin make m = n = d = l = k = k = Given convex be deined a k U R allowin he conrol e i ad = { :[ ] U i F - mearable E ( d < } o For any ad ξ L ( F P;R conider he ollowin MF - BDSDE: i = ( = ξ Γ ( ( Z ( ( d Z ( d W( Γ ( ( Z ( ( d B( i i Γ ( ( Z ( ( = θ ( ( Z ( v( ( Z ( ( P( d θ : [ ] R R U R R U R θ : [ ] R R U R R U R

2 Pre and Applied Mahemaic Jornal 5; 4(3: -8 Perormance indicaor i l ( J ( ( = E Γ ( ( Z ( ( d E[E h ( ( ( ] h: R R R l Γ ( ( Z ( ( = l( ( Z ( ( ( Z ( ( P( d l U U R : [ ] R R R R Conrol problem i lookin or admiion conrol o make perormance indicaor reachin he minimm vale on he ad Sppoin ha [5-6] (H ( θ θ l h i coninoly diereniable abo y y' z z' v v 'and he derivaive o h and i i linear rowh θ θ mee niorm Lipchiz condiion abo ( ' ' ' ( y z y z v v In oher word here exi Li Ki α j or i = y z y' z' v v' j = 34 makin θ ( y z y z θ ( y z y z L y y L z z L y y L z z L y z y z v Lv θ ( y z y z θ ( y z y z y y v v α3 α 4 K y y K y y K K z z z z = 6 ( [ ] ( yi zi yi zi i i R i l θ o d< = E E ( l l l θ ( w w' = θ ( w w' α3 α4 < Under he above ampion or any v( i ad here v v exi a niqe olion ( Z S ( ; R M ( ; R o he eqaion ( he perormance index deined i reaonable[7-8] Amed ( i he opimal conrol ( Z ( i he v correpondin opimal rajecory v ( mee ad becae o he convexiy o ad or any = v olion Z o ad here exi a niqe Lemma hypohei (H i eablihed or any [ ] E ( ˆ ( C E Z ( Zˆ ( d C Proo Noice ha MF-BDSDE: ( ˆ ( o mee he ollowin ˆ ( ( = [ Γ ( ( Z ( ( Γ ( ˆ ( Zˆ ( ˆ ( ] d [ Γ ( ( Z ( ( Γ ( ˆ ( Zˆ ( ˆ ( ] db( ( Z ( Zˆ ( dw( Applyin Io ormla o ( ˆ ( E( ( ˆ ( Z ( Zˆ ( d ˆ = E ( ( Γ ( ( Z ( ( Γ ( ˆ ( Zˆ ( ˆ ( d Accordin o (H here i E Γ ( ( Z ( ( Γ ( ˆ ( Zˆ ( ˆ ( d

3 3 Hon Zhan e al: Maximm Principle and he Applicaion o Mean-Field Backward Dobly Sochaic Syem E ( ˆ ( E Z ( Zˆ ( d k E ( ˆ( d k E ( d ki( i = i conan rely on (H Accordin Gronwall Ineqaliy and Brkholder-Davi-Gndy Ineqaliy rel are veriied For imple markin make ˆ α( = α( ˆ ( Zˆ ( ˆ ( ˆ ( Zˆ ( ˆ ( α ( = α( ( Z ( ( ( Z ( ( ( = ( F ( ( ( d G ( ( ( db ( ξ ψ ξ η ξ η η ( dw ( ( ( F ξ η = Ε θ y ξ θ z η θ y ξ θ z η ( G ξ η = Ε θ y ξ θ z η θ y ξ θ z η ( ( ( d ( ( db ψ = θ θ θ θ Ε Ε Marked y y Ε θ ξ θ ξ d = Ρ y y Ε θ ξ θ ξ d = Ρ Under he above ampion or any v( i ad here exi a niqe olion ( ξ ( η ( S ([ ];R ( ; he eqaion ( Lemma Marked ( ( = ξ ( z ( y Z Z = η ( M R o ( y z lim p Ε y ( = [ ] i he olion o he eqaion a ollow lim Ε z d = (3 δ = dy = y = Ε y y z z y y z z d Ε y y z z y y z z db z dw ( = λ ( = λ δ ( = ( δy θy Z Z dλ

4 Pre and Applied Mahemaic Jornal 5; 4(3: -8 4 and δ ( = δ θ ξ δ θ η δ θ ξ δ δ δ y y z z y y z δ θ η δ θ δ θ δ δ δ z Applyin Io ormla o y ( on [ ] Ε y ( Ε z ( d = Ε y ( Ε y z y z Accordin o (H here i y z y z d y z y z Ε Ε y z y z d k i conan when 时 opimal conrol Ε y Ε z d kε y d C C Accordin Gronwall Ineqaliy rel are veriied Becae o ( ( J J i he (4 Accordin o lemma here i lemma 3 Hypohei (H wa eablihed hen he ollowin variaion ineqaliy i eablihed [9-] : Ε Ε ξ η ξ η ly lz ly lz l l d ΕΕ h y ( ξ hy ( ξ (5 Ε l y ( ξ ( = l y ( ξ ( Ρ( d Proo ΕΕ h h = ΕΕ h ( y dλ ΕΕ ( y ( = λ ( hy ( ΕΕ hy ξ ξ h d { Ε Ε l ( l ( d } Ε Ε y ξ z η y ξ z η l l l l l l d λ o (5 i veriied Coniderin he adjoin eqaion: * * = Ε y Ε y ( ( G p q dw q db p h h F p q d (6 ( θ θ ( p ( F p q = Ε y p θ y q ly ( q ( l ( Ε θ * * * y y y

5 5 Hon Zhan e al: Maximm Principle and he Applicaion o Mean-Field Backward Dobly Sochaic Syem ( θ θ ( p ( * * * G p q = Ε z p θ z q lz Ε z θ z q lz Ε l = l Ρ d y y ( ( * * * y y ( ( * * * * * Ε θ p θ = p Ρ d Deine he Hamilonian ncion H :[ ] R R R R R R R R Ra ollow ( = θ ( θ ( H y z y z p q y z y z p y z y z q ( l y z y z (7 By he variaional ineqaliy (7 we preen MF - BDSDE ochaic conrol problem o ochaic maximm principle ˆ Z ˆ ˆ i he opimal rajecory o he conrol heorem (ochaic maximm principle Amed problem{((} v U ae [ ] a * * H Ε H ( Ε (8 where H ( H ( Z ( ( ( Z ( ( p ( q( = (9 Proo Applyin Io ormla o ξ ( p ( we can e Ε ξp = Ε Ε l y ξ lz η ly ξ lz η d Ε θ p θ q Ε d Accordin (5 we can e Accordin Hamilonian ncion we can e Ε ( ( p ( Ε θ θ q Ε d Ε θ p θ q l Ε d ( ( p ( Ε θ θ q l Ε d ( * * * * Ε H Z Z ( * * ( ( Ε ( ( ( ( ( ( ( ( p q H Z Z For any v U F i he any elemen o σ Alebra ( F ein p q d ( ( = ( ( [ [ [ [ ] F F We can know v( ad becae ( mee ( ( ein ad ( ( ( rewrien a = he above ineqaliie can be

6 Pre and Applied Mahemaic Jornal 5; 4(3: -8 6 Dierenial on a variable a = we can e So (8 i veriied * * Ε F Ε H Ε H d * * Ε F Ε H Ε H 3 Mean-Field Backward Dobly Sochaic LQ Problem hi ecion we apply he maximm vale principle o Mean-ield backward dobly ochaic LQ problem ( Z Z q Q Q ( = = A ( B ( Z C ( A ( B ( Z C ( ( Z Z = D ( E ( Z F ( D ( E ( Z F ( ( l Z Z = M ( N ( Z W ( M ( N ( Z W ( A [ ] R i : i bonded w w' A i w w' i i i i aiie he hypohei M N are nonneaive R i poiive he ae eqaion i M mearable (imilarly oher coeicien { E E E } E = ξ A B Z C d D E Z F db { E E E } Z dw Perormance indicaor i J ( ( A B Z C d { E D E ( Z F ( } db E ( E E = M N Z W d Q ( M N Z W d Q E E E i i In order o mark i imple p or A Hamilonian ncion i ( y z v y z v p q = H Accordin heorem we can e A ( A y B z C v A y B z C v p D y E z F v D y E z F v q M y N z W v M y N z W v ( C p F q W * * * = E C p F q W E (

7 7 Hon Zhan e al: Maximm Principle and he Applicaion o Mean-Field Backward Dobly Sochaic Syem ( ( ( = * E C p C p P d ( ( ( = E * W W * P d * ( ( ( = * * * * * E C p C p P d * = E ( E G ( p ( q ( dw ( q ( db ( p Q Q F p q d ( = A ( p( D ( q( M ( E * A ( p * ( D ( q * ( M ( F p q E ( = B ( p( E ( q( N ( E * B ( p * ( E ( q * ( N ( G p q E ( heorem Ame ha aiy (9 he (p i he olion o eqaion ( he above LQ problem have niqe olion Proo J ( J E E = M N Z Z d E E W M S d E E N Z Z W d EE Q Q ( M N Z Z Z d E E E E ( E E N Z Z Z W d Applyin Io ormla o p w on [ ] EE Q Q ( W M d EE Q Q ( ( = E E C p ( F q d ( E E ( ( ( E E M ( N Z Z Z d M N Z Z Z d E E We can e ( E E ( J J C p F q d C p F q d ( ( E E W W d E E C p F q d ( = E E C p F q d E E W ( * * E E * C ( p F q d heorem i veriied * ( E W ( ( d

8 Pre and Applied Mahemaic Jornal 5; 4(3: Smmary heorem (ochaic maximm principle Amed problem{((} where heorem Ame ha aiy (9 he (p i he olion o eqaion ( he above LQ problem have niqe olion Acknowledemen hi paper i nded by he projec o Naional Naral Science Fnd Loiic diribion o ariicial order pickin random proce model analyi and reearch (Projec nmber: 73733; and nded by inellien loiic yem Beijin Key Laboraory (NoBZ; and nded by cieniic-reearch bae--- Science & echnoloy Innovaion Plaorm---Modern loiic inormaion and conrol echnoloy reearch (Projec nmber: PXM5_44_; and nded by 4-5 chool year Beijin Wzi Univeriy Collee den' cieniic reearch and enreprenerial acion plan projec (No68; and nded by Beijin Wzi Univeriy nhe cholar proram(633/7; and nded by Beijin Wzi Univeriy Manaemen cience and enineerin Proeional rop o conrcion projec (No PXM5_44_39 Univeriy Clivaion Fnd Projec o 4-Reearch on Coneion Model and alorihm o pickin yem in diribion cener (54573 Reerence [ ] a v U ae [] A Szkala A Knee-ype heorem or eqaion x= ( x in locally convex pace Jornal or analyi and i applicaion 8 (999-6 ( ( Z ( ( * * H Ε H ( Ε i he opimal rajecory o he conrol H ( = H ( Z ( ( ( Z ( ( p ( q ( (3 (4 [] M an and Q Zhan Opimal variaional principle or backward ochaic conrol yem aociaed wih Levy procee Sci China Mah 55 ( [3] SevaS an and X Li Neceary condiion or opimal conrol o ochaic yem wih random jmp SIAM J Conrol Opim 3 ( [4] J Valero On he kneer propery or ome parapolic problem opoloy and i applicanon 55 ( [5] Z W Maximm principle or opimal conrol problem o lly copled orward-backward ochaic yem J Syem Sci Mah Sci ( [6] Z W Forward-backward ochaic dierenial eqaion wih Brownian Moion and Proce Poion Aca Mah Appl Sinica Enlih Serie 5 ( [7] Z W A maximm principle or parially oberved opimal conrol o orward-backward ochaic conrol yem Sci China Ser F 53 ( 5-4 [8] Z W and Z Flly copled orward-backward ochaic dierenial eqaion and relaed parial dierenial eqaion yem Chinee Ann Mah Ser A 5 ( [9] H Xiao and G Wan A neceary condiion or opimal conrol o iniial copled orward-backward ochaic dierenial eqaion wih parial inormaion J Appl Mah Comp 37 ( [] J Xion An Inrodcion o Sochaic Filerin heory London UK: Oxord Univeriy Pre 8

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