Al. I. Cuza Universiy of Iaşi, România 10ème Colloque Franco-Roumain de Mahémaiques Appliquées Augus 27, 2010, Poiiers, France
Shor hisory & moivaion Re eced Sochasic Di erenial Equaions were rs sudied by Skorohod (1961) and, afer ha, for general domains, by Lions, Szniman (1984), Saisho (1987). The re eced di usion processes give us a probabilisic inerpreaion of he soluion of elipic and parabolic PDE wih Neumann boundary condiions.
Shor hisory & moivaion Re eced Sochasic Di erenial Equaions were rs sudied by Skorohod (1961) and, afer ha, for general domains, by Lions, Szniman (1984), Saisho (1987). The re eced di usion processes give us a probabilisic inerpreaion of he soluion of elipic and parabolic PDE wih Neumann boundary condiions. The Euler approximaion for forward RSDE was considered by Chiashvili, Lazrieva (1981) and he Euler-Peano approximaion was used by Saisho (1987) o solve RSDEs wih Lipschiz condiions.
Shor hisory & moivaion Re eced Sochasic Di erenial Equaions were rs sudied by Skorohod (1961) and, afer ha, for general domains, by Lions, Szniman (1984), Saisho (1987). The re eced di usion processes give us a probabilisic inerpreaion of he soluion of elipic and parabolic PDE wih Neumann boundary condiions. The Euler approximaion for forward RSDE was considered by Chiashvili, Lazrieva (1981) and he Euler-Peano approximaion was used by Saisho (1987) o solve RSDEs wih Lipschiz condiions. In order o approximae he soluion of RSDEs he penalizaion mehod was also useful (see Menaldi, 1983).
Shor hisory & moivaion Re eced Sochasic Di erenial Equaions were rs sudied by Skorohod (1961) and, afer ha, for general domains, by Lions, Szniman (1984), Saisho (1987). The re eced di usion processes give us a probabilisic inerpreaion of he soluion of elipic and parabolic PDE wih Neumann boundary condiions. The Euler approximaion for forward RSDE was considered by Chiashvili, Lazrieva (1981) and he Euler-Peano approximaion was used by Saisho (1987) o solve RSDEs wih Lipschiz condiions. In order o approximae he soluion of RSDEs he penalizaion mehod was also useful (see Menaldi, 1983). Asiminoaei, R¼aşcanu (1997) and Ding, Zhang (2008) combined he penalizaion mehod wih he spliing-sep idea and propose new schemes for SDE wih re ecion a he boundary of he domain.
Shor hisory & moivaion Re eced Sochasic Di erenial Equaions were rs sudied by Skorohod (1961) and, afer ha, for general domains, by Lions, Szniman (1984), Saisho (1987). The re eced di usion processes give us a probabilisic inerpreaion of he soluion of elipic and parabolic PDE wih Neumann boundary condiions. The Euler approximaion for forward RSDE was considered by Chiashvili, Lazrieva (1981) and he Euler-Peano approximaion was used by Saisho (1987) o solve RSDEs wih Lipschiz condiions. In order o approximae he soluion of RSDEs he penalizaion mehod was also useful (see Menaldi, 1983). Asiminoaei, R¼aşcanu (1997) and Ding, Zhang (2008) combined he penalizaion mehod wih he spliing-sep idea and propose new schemes for SDE wih re ecion a he boundary of he domain. Moreover, he penalizaion mehod was used for he exisence of a soluion in he case of backward SDE wih a maximal monoone operaor of subdi erenial ype (Pardoux, R¼aşcanu, 1998 and Maiciuc, R¼aşcanu, 2007).
Shor hisory & moivaion Re eced Sochasic Di erenial Equaions were rs sudied by Skorohod (1961) and, afer ha, for general domains, by Lions, Szniman (1984), Saisho (1987). The re eced di usion processes give us a probabilisic inerpreaion of he soluion of elipic and parabolic PDE wih Neumann boundary condiions. The Euler approximaion for forward RSDE was considered by Chiashvili, Lazrieva (1981) and he Euler-Peano approximaion was used by Saisho (1987) o solve RSDEs wih Lipschiz condiions. In order o approximae he soluion of RSDEs he penalizaion mehod was also useful (see Menaldi, 1983). Asiminoaei, R¼aşcanu (1997) and Ding, Zhang (2008) combined he penalizaion mehod wih he spliing-sep idea and propose new schemes for SDE wih re ecion a he boundary of he domain. Moreover, he penalizaion mehod was used for he exisence of a soluion in he case of backward SDE wih a maximal monoone operaor of subdi erenial ype (Pardoux, R¼aşcanu, 1998 and Maiciuc, R¼aşcanu, 2007). Euler-ype approximaions for backward sochasic di erenial equaions were inroduced by Bouchard, Touzi (2004) and Zhang (2004).
Ouline Backward Sochasic Variaional Inequaliies (he Markovian framework)
Ouline Backward Sochasic Variaional Inequaliies (he Markovian framework) Approximaion schemes for BSVI
Ouline Backward Sochasic Variaional Inequaliies (he Markovian framework) Approximaion schemes for BSVI Generalized BSVI
Ouline Backward Sochasic Variaional Inequaliies (he Markovian framework) Approximaion schemes for BSVI Generalized BSVI Approximaion schemes for Generalized BSVI
Ouline Backward Sochasic Variaional Inequaliies (he Markovian framework) Approximaion schemes for BSVI Generalized BSVI Approximaion schemes for Generalized BSVI References
Backward Sochasic Variaional Inequaliies Firs, we remind he problem of he exisence and uniqueness of he soluion for he backward sochasic di erenial equaion involving an subdi erenial operaor: 8 dx = b(x )d + σ(x )dw, 2 [0, T ], >< dy + F (, X, Y, Z ) d 2 ϕ (Y ) d + Z dw, 2 [0, T ], (1) >: X 0 = x, Y T = g (X T ). Here (W ) 0 is a sandard Brownian moion de ned on a complee righ coninuous sochasic basis (Ω, F, P, F ). We will make he following assumpions on he coe cien funcions: The funcions are coninuous and b : R d! R d, σ : R d! R d d, F : [0, T ] R d R k R kd! R k, g : R d! R k (2)
here exis α 2 R and L, β, γ 0 such ha he coe ciens saisfy: (i) jb (x ) b ( x ) j L jx x j, (ii) jjσ (x ) σ ( x ) jj L jx x j, (iii) y ỹ, f (, x, y, z) f (, x, ỹ, z) αjy ỹ j 2, (iv ) jf (, x, y, z) f (, x, y, z)j β jjz zjj, (3) for all 2 [0, T ], x, x 2 R d, y, ỹ 2 R k, z, z 2 R kd. There exis some consans M > 0 and p, q 2 N such ha: (i) g (x ) M 1 + jx j q, (ii) f (, x, y, 0) M 1 + jx j p + jy j, (4) for all 2 [0, T ], x 2 R d, y 2 R k. The funcion ϕ : R k! (, + ] is a proper convex l.s.c. funcion, and here exis M > 0 and r 2 N such ha jϕ(g (x ))j M (1 + jx j r ), 8x 2 R d (5)
Theorem (see [Pardoux, R¼aşcanu, 98]) Under he assumpions (2)-(5), here exiss a unique riple of F -progressively measurable sochasic processes (Y, Z, U ) 2[0,T ], soluion of he BSVI: Y + U s ds = g (X T ) + f (s, X s, Y s, Z s )ds Z s dw s, for all 2 [0, T ], P-a.s., (6) wih (Y, U ) 2 ϕ, P (d ω) d. Moreover we have he following properies of he soluion of Eq.(6): 1 Y 2 L 2 ad (Ω; C ([0, T ] ; Rk )), Z 2 L 2 ad (Ω; L2 ([0, T ] ; R kd )) and U 2 L 2 ad (Ω; L2 ([0, T ] ; R k )).
Theorem (see [Pardoux, R¼aşcanu, 98]) Under he assumpions (2)-(5), here exiss a unique riple of F -progressively measurable sochasic processes (Y, Z, U ) 2[0,T ], soluion of he BSVI: Y + U s ds = g (X T ) + f (s, X s, Y s, Z s )ds Z s dw s, for all 2 [0, T ], P-a.s., (6) wih (Y, U ) 2 ϕ, P (d ω) d. Moreover we have he following properies of he soluion of Eq.(6): 1 Y 2 L 2 ad (Ω; C ([0, T ] ; Rk )), Z 2 L 2 ad (Ω; L2 ([0, T ] ; R kd )) and U 2 L 2 ad (Ω; L2 ([0, T ] ; R k )). 2 There exiss some consan C > 0, such ha, for all, 2 [0, T ], x, x 2 R d : (i) (ii) E sup jy j 2 C (1 + jx j 2 ) 2[0,T ] E sup jy x 2[0,T ] + f (s, Xs x, Ys x, Zs x ) 0 h Y x j 2 C E g (XT x ) g (X T x ) 2 f (s, Xs x, Ys x, Zs x ) i 2 ds.
The exisence resul for (6) is obained via Yosida approximaions. De ne for ε > 0 he convex C 1 -funcion ϕ ε by 1 ϕ ε (y ) := inf 2ε jy v j2 + ϕ (v ) : v 2 R k We denoe by and we have ha rϕ ε (y ) = y J ε (y ) := (I + ε ϕ) 1 (y ) J εy, rϕ ε (y ) 2 ϕ (J ε y ), ε ϕ ε (y ) = 1 2ε jy J εy j 2 + ϕ (J ε y ), 8y 2 R k. (7) For an arbirary (, x ) 2 [0, T ] R d, ε 2 (0, 1), by a classical resul here exiss (Y ε, Z ε ) 2[,T ] he unique soluion of he following approximaing BSDE: Ys ε + rϕ ε (Yr ε )dr = g (X T ) + f (r, X r, Yr ε, Zr ε )dr s s Zr ε dw r, 8s, P-a.s., s soluion ha converges o he soluion of Eq.(6).
Approximaion schemes for BSVI We inroduce an approximaion scheme for he soluion of Eq.(6): Consider a grid of [0, T ] : π = f i = ih, i ng, wih h := T /n, n 2 N,
Approximaion schemes for BSVI We inroduce an approximaion scheme for he soluion of Eq.(6): Consider a grid of [0, T ] : π = f i = ih, i ng, wih h := T /n, n 2 N, De ne X π he clasical Euler scheme for X,
Approximaion schemes for BSVI We inroduce an approximaion scheme for he soluion of Eq.(6): Consider a grid of [0, T ] : π = f i = ih, i ng, wih h := T /n, n 2 N, De ne X π he clasical Euler scheme for X, Firs, using he Yosida approximaion for ϕ, we have, for all 0, P-a.s.: Y ε + rϕ ε (Yr ε ) dr = g (XT π ) + F (r, Xr π, Yr ε, Zr ε ) dr Zr ε dw r, (8) Furher, we de ne an Euler-Yosida ype approximaion for Y ε (suggesed by [Bouchard, Touzi, 2004]).
Approximaion schemes for BSVI We inroduce an approximaion scheme for he soluion of Eq.(6): Consider a grid of [0, T ] : π = f i = ih, i ng, wih h := T /n, n 2 N, De ne X π he clasical Euler scheme for X, Firs, using he Yosida approximaion for ϕ, we have, for all 0, P-a.s.: Y ε + rϕ ε (Yr ε ) dr = g (XT π ) + F (r, Xr π, Yr ε, Zr ε ) dr Zr ε dw r, (8) Furher, we de ne an Euler-Yosida ype approximaion for Y ε (suggesed by [Bouchard, Touzi, 2004]). Le Y T := g (XT π ) and, for i = n 1, 0, Y ε i Y ε i+1 h f ( i, X π i, Y ε i, Z ε i ) rϕ ε (Y ε i ) Z ε i (W i+1 W i ) (9)
Approximaion schemes for BSVI We inroduce an approximaion scheme for he soluion of Eq.(6): Consider a grid of [0, T ] : π = f i = ih, i ng, wih h := T /n, n 2 N, De ne X π he clasical Euler scheme for X, Firs, using he Yosida approximaion for ϕ, we have, for all 0, P-a.s.: Y ε + rϕ ε (Yr ε ) dr = g (XT π ) + F (r, Xr π, Yr ε, Zr ε ) dr Zr ε dw r, (8) Furher, we de ne an Euler-Yosida ype approximaion for Y ε (suggesed by [Bouchard, Touzi, 2004]). Le Y T := g (XT π ) and, for i = n 1, 0, Y ε i Y ε i+1 h f ( i, X π i, Y ε i, Z ε i ) rϕ ε (Y ε i ) Z ε i (W i+1 W i ) (9) We ake he condiional expecaion E F i : Y ε i E F i (Y ε i+1 ) h f ( i, X π i, Y ε i, Z ε i ) rϕ ε (Y ε i )
Approximaion schemes for BSVI We inroduce an approximaion scheme for he soluion of Eq.(6): Consider a grid of [0, T ] : π = f i = ih, i ng, wih h := T /n, n 2 N, De ne X π he clasical Euler scheme for X, Firs, using he Yosida approximaion for ϕ, we have, for all 0, P-a.s.: Y ε + rϕ ε (Yr ε ) dr = g (XT π ) + F (r, Xr π, Yr ε, Zr ε ) dr Zr ε dw r, (8) Furher, we de ne an Euler-Yosida ype approximaion for Y ε (suggesed by [Bouchard, Touzi, 2004]). Le Y T := g (XT π ) and, for i = n 1, 0, Y ε i Y ε i+1 h f ( i, X π i, Y ε i, Z ε i ) rϕ ε (Y ε i ) Z ε i (W i+1 W i ) (9) We ake he condiional expecaion E F i : Y ε i E F i (Y ε i+1 ) h f ( i, X π i, Y ε i, Z ε i ) rϕ ε (Y ε i ) We muliply (9) wih W i+1 W i and we ake E F i : Z ε i 1 h EF i (Y ε i+1 (W i+1 W i ))
For he above Euler-Yosida approximaion scheme, we consider he sep of he grid h = ε 3 and we de ne, for i = n 1, 0 : 8 >< >: Y π i := E F i (Y π i+1 ) + h f ( i, X π i, Y π i, Z π i ) Y π T := g (X π T ), Z π i U π i := 1 h EF i (Y π i+1 (W i+1 W i )), := rϕ h 1/3 (E F i (Y π i+1 )). rϕ h 1/3 (Y π i ), (10) Consider now a coninuous version of (10). From he maringale represenaion heorem here exiss a square inegrable process Z such ha and, herefore, we de ne, for 2 ( i, i+1 ] Z i+1 Y π i+1 = E F i (Y π i+1 ) + Z s ds, i Y π := Y π i ( i ) f ( i, X π i, Y π i, Z π i ) rϕ h 1/3 (Y π i ) Z i+1 + Z s dw s. i From he isomery propery we noice ha Z π i = 1 Z i+1 h EF i Z s dw s, for i = 0, n i 1. (11)
An error esimae of he approximaion is given by he following resul. Theorem Under he assumpions (2)-(5), here exiss C > 0 which depends only on he Lipschiz consans of he coe ciens, such ha: sup E jy 2[0,T ] Y π j 2 + E jz 0 Z π j 2 d Ch. Proof (skech). For i = 0, n 1 le denoe: δy := Y Y π, δz := Z Z π. Applying he Energy equaliy, using he relaion (11) and he properies of he Yosida approximaion, we obain, by sandard calculus: E jδy i j 2 + 1 Z i+1 E jδz s j 2 ds (1 + Ch) h 2 + E Z i+1 δyi+1 2 + E Zs Z π 2 2 i i ds, i (12) where Z π i := 1 Z i+1 h EF i Z s ds i is he bes approximaion in L 2 of Z by a process which is consan of each inerval [ i, i+1 ).
Proof (con.) Ieraing (12) and summing upon i, we deduce: ( E jδy i j 2 + 1 E jδz s j 2 n 1 Z i+1 ds C h + 2 E ) Z s Z π 2 0 i=0 i ds. i Since we have max sup i=0,n 1 2[ i, i+1 ) E jy Y i j 2 n 1 + he conclusion of he heorem follows. i=0 Z i+1 E Z Z π 2 i d Ch, i
Generalized BSVI Consider now he following generalized backward sochasic variaional inequaliy (in he Markovian case): 8 dy >< + F (, X, Y, Z ) d + G (, X, Y ) da 2 2 ϕ (Y ) d + Z dw, 0 T, (13) >: Y T = g (X T ). Throughou his secion we suppose ha F, G sais es he same assumpion as F from he rs secion. I is proved (see [Maiciuc, R¼aşcanu, 2010]) ha he above equaion admis a unique soluion, i.e. where Y + U s ds = g (X T ) + F (s, X s, Y s, Z s ) ds + G (s, X s, Y s ) da s Z s dw s, for all 2 [0, T ] a.s., U 2 ϕ (Y ), a.e. on Ω [0, T ]. More deailed, le D be a open bounded subse of R d of he form D = fx 2 R d : ` (x ) < 0g, Bd (D) = fx 2 R d : ` (x ) = 0g, where ` 2 C 3 b R d, jr` (x )j = 1, for all x 2 Bd (D).
I follows, (using he paper of [Lions, Szniman, 1984]), ha for each (, x ) 2 R + D here exiss a unique pair of progressively measurable coninuous processes (Xs,x, A,x s ) s0, wih values in D R +, soluion of he re eced SDE: 8 >< >: Z s_ Z s_ Xs,x = x + b(r, Xr,x )dr + σ(r, X,x s 7! A,x A,x s = s is increasing Z s_ 1 fx,x r 2Bd (D)g da,x r. Moreover, i can be proved ha E sup X,x s2[0,t ] s X 0,x 0 s! p C r )dw r Z s_ x x 0 p + 0 p 2, r`(xr,x )da,x r, (14) and for all µ > 0 E i he µa,x T <.
Theorem Under he assumpions (2)-(5), he generalized BSVI (13) admis a unique soluion (Y, Z, U ) of F -progressively measurable processes. Moreover, for any 0 s T, we have, for some posiive consan C : (a) (b) Z E s jy r j 2 + jjz r jj 2 Z dr + jy r j 2 da r CM 1 s E sup jy r j 2 CM 1, sr (c) E ϕ (Y ) CM 2, Z (d ) E ju r j 2 dr CM 2, s (15) where M 1 = E jξj 2 + F (s, 0, 0) 2 ds + G (s, 0) 2 das, 0 M 2 = E jξj 2 + ϕ (ξ).
The main idea for he proof of he exisence for (13) consiss in aking he approximaing equaion Ys ε + s + G s rϕ ε (Y ε r ) dr = g r, X,x r, Yr ε dar X,x T + s s F r, Xr,x, Yr ε, Zr ε dr Z ε r dw r, 8s, P-a.s. Essenial for he esimaes of he process (Y ε ) ε>0 is he sochasic subdi erenial inequaliy given by: (16) Lemma (see [Pardoux, R¼aşcanu, 1998]) Le U, Y be k-dimensional c.p.m.s.p. (coninuous progressively measurable sochasic processes) and V be a real c.m.s.p. If Y is a semimaringale, V is a posiive bounded variaion sochasic process, ϕ : R k! ] U r 2 ϕ (Y r ),hen, + ] is a proper convex l.s.c. funcion, V r hu r, dy r i + V ϕ (Y ) + ϕ (Y r ) dv r V T ϕ (Y T ).
Consequenly, we can derive, for some consans λ, µ su cienly large, ha (a) (b) (c) E e λs+µas rϕ ε (Ys ε ) 2 ds CM 2, 0 E 0 e λs+µas ϕ J ε (Y ε s ) ds CM 2, Ee λs+µas Y ε s J ε (Y ε s ) 2 εcm 2, (d ) Ee λs+µas ϕ J ε (Y ε s ) CM 2. and, ha (Y ε, Z ε ) ε>0 is a Cauchy sequence, i.e.: E " sup e λ+µa Y ε 0T # Y δ Z 2 T + E 0 e λr +µar Z ε r Zr δ 2 dr C (ε + δ). The unique soluion (Y, Z, U ) of Eq.(13) is obained as he limi of he approximaing sequence (Ys ε, Zs ε, rϕ ε (Ys ε )).
Applicaions o PVI As usual, if we consider he deerminisic funcion u(, x ) = Y,x, (, x ) 2 [0, T ] D, we will obain a reprezenaion for he soluion of he following PVI: 8 u(, x ) L u (, x ) + ϕ u(, x ) 3 F, x, u(, x ), (ruσ)(, x ), >< > 0, x 2 D, u(, x ) 3 G, x, u(, x ) (17), > 0, x 2 Bd (D), n >: u(0, x ) = g (x ), x 2 D, where he operaor L is given by L v (x ) = 1 2 d (σσ ) ij (, x ) 2 v (x ) + i,j=1 x i x j d i=1 v (x ) b i (, x ), for v 2 C 2 (R d ). x i The funcions ha appear in (17) saisfy he condiions imposed in he previous slides (for k = 1).
We have he following resul. Theorem Under he considered hypoesis on b, σ, F, G and g, he Parabolic Variaional Inequaliy (17) admis a leas one viscosiy soluion u : [0, ) D! R, such ha u(0, x ) = h (x ), 8 x 2 D and u(, x ) 2 Dom (ϕ), 8(, x ) 2 (0, ) D. Moreover, if r! G (, x, r ) is a non-increasing funcion, for 0, x 2 Bd (D), and here exiss a coninuous funcion m : [0, )! [0, ), m (0) = 0, such ha F (, x, r, p) F (, y, r, p) m jx y j (1 + jpj), hen he uniqueness holds. 8 0, x, y 2 D, p 2 R d,
Approximaion schemes for Generalized BSVI For he generalized sysem considered above, we propose a mixed Euler-Yosida ype approximaion scheme. For he simpliciy of he presenaion, we consider he case ϕ 0 : Consider he grid of [0, T ] : π = f i = ih, i ng, wih h := T /n, n 2 N,
Approximaion schemes for Generalized BSVI For he generalized sysem considered above, we propose a mixed Euler-Yosida ype approximaion scheme. For he simpliciy of he presenaion, we consider he case ϕ 0 : Consider he grid of [0, T ] : π = f i = ih, i ng, wih h := T /n, n 2 N, De ne X π he Euler scheme for he re eced process X :
Approximaion schemes for Generalized BSVI For he generalized sysem considered above, we propose a mixed Euler-Yosida ype approximaion scheme. For he simpliciy of he presenaion, we consider he case ϕ 0 : Consider he grid of [0, T ] : π = f i = ih, i ng, wih h := T /n, n 2 N, De ne X π he Euler scheme for he re eced process X : X π 0 = x, Aπ 0 = 0,
Approximaion schemes for Generalized BSVI For he generalized sysem considered above, we propose a mixed Euler-Yosida ype approximaion scheme. For he simpliciy of he presenaion, we consider he case ϕ 0 : Consider he grid of [0, T ] : π = f i = ih, i ng, wih h := T /n, n 2 N, De ne X π he Euler scheme for he re eced process X : X π 0 = x, Aπ 0 = 0, ˆX π i+1 = X π i + b( i, X π i )( i+1 i ) + σ( i, X π i )(W i+1 W i ), Considering he projecion o he domain we de ne
Approximaion schemes for Generalized BSVI For he generalized sysem considered above, we propose a mixed Euler-Yosida ype approximaion scheme. For he simpliciy of he presenaion, we consider he case ϕ 0 : Consider he grid of [0, T ] : π = f i = ih, i ng, wih h := T /n, n 2 N, De ne X π he Euler scheme for he re eced process X : X π 0 = x, Aπ 0 = 0, ˆX π i+1 = X π i + b( i, X π i )( i+1 i ) + σ( i, X π i )(W i+1 W i ), Considering he projecion o he domain we de ne 8 < ˆX X π π i+1, ˆX π i+1 2 D, i+1 = and : Pr D ( ˆX π i+1 ), ˆX π i+1 /2 D,
Approximaion schemes for Generalized BSVI For he generalized sysem considered above, we propose a mixed Euler-Yosida ype approximaion scheme. For he simpliciy of he presenaion, we consider he case ϕ 0 : Consider he grid of [0, T ] : π = f i = ih, i ng, wih h := T /n, n 2 N, De ne X π he Euler scheme for he re eced process X : X π 0 = x, Aπ 0 = 0, ˆX π i+1 = X π i + b( i, X π i )( i+1 i ) + σ( i, X π i )(W i+1 W i ), Considering he projecion o he domain we de ne 8 < ˆX X π π i+1, ˆX π i+1 2 D, i+1 = and : Pr D ( ˆX π i+1 ), ˆX π i+1 /2 D, 8 < A π i+1 = : A π i, ˆX π i+1 2 D, A π i + jj Pr D ( ˆX π i+1 ) ˆX π i+1 jj, ˆX π i+1 /2 D;
Le YT π := g (X T π ) and, for i = n 1, 0 : Y i Y i+1 G (X π i+1, Y i+1 ) A π i Z i W i
Le YT π := g (X T π ) and, for i = n 1, 0 : Y i Y i+1 G (X π i+1, Y i+1 ) A π i Z i W i We ake he condiional expecaion E F i : Y i E F i (Y i+1 ) E F i [G (X π i+1, Y i+1 ) A π i ]
Le YT π := g (X T π ) and, for i = n 1, 0 : Y i Y i+1 G (X π i+1, Y i+1 ) A π i Z i W i We ake he condiional expecaion E F i : Y i E F i (Y i+1 ) E F i [G (X π i+1, Y i+1 ) A π i ] This sugges us o de ne he following approximaion scheme: 8 < : Y π i Z π i := E F i [Y π i+1 := 1 h EF i [Y π i+1 W i G (X π i+1, Y i+1 ) A π i ], Y π T := g (X π T ), G (X π i+1, Y i+1 ) A π i W i ], The proof of he convergence for he de ned approximaion scheme is sill in progress.
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