Approximation of backward stochastic variational inequalities
|
|
- Randolph Wiggins
- 5 years ago
- Views:
Transcription
1 Al. I. Cuza Universiy of Iaşi, România 10ème Colloque Franco-Roumain de Mahémaiques Appliquées Augus 27, 2010, Poiiers, France
2 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.
3 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.
4 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).
5 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.
6 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).
7 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).
8 Ouline Backward Sochasic Variaional Inequaliies (he Markovian framework)
9 Ouline Backward Sochasic Variaional Inequaliies (he Markovian framework) Approximaion schemes for BSVI
10 Ouline Backward Sochasic Variaional Inequaliies (he Markovian framework) Approximaion schemes for BSVI Generalized BSVI
11 Ouline Backward Sochasic Variaional Inequaliies (he Markovian framework) Approximaion schemes for BSVI Generalized BSVI Approximaion schemes for Generalized BSVI
12 Ouline Backward Sochasic Variaional Inequaliies (he Markovian framework) Approximaion schemes for BSVI Generalized BSVI Approximaion schemes for Generalized BSVI References
13 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)
14 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)
15 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 )).
16 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.
17 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).
18 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,
19 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,
20 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]).
21 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)
22 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 )
23 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 ))
24 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)
25 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 Z i+1 E jδz s j 2 ds (1 + Ch) h 2 + E Z i+1 δyi 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 ).
26 Proof (con.) Ieraing (12) and summing upon i, we deduce: ( E jδy i j 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
27 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).
28 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 <.
29 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).
30 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 ).
31 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 )).
32 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).
33 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,
34 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,
35 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 :
36 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,
37 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
38 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,
39 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;
40 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
41 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 ]
42 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.
43 References Asiminoaei I., R¼aşcanu A., Approximaion and simulaion of sochasic variaional inequaliies - spliing up mehod, Numerical Funcional Analysis and Opimizaion, 18, pp , Bouchard B., Touzi N. - Discree-ime approximaion anf Mone-Carlo simulaion of BSDEs, Sochasic Processes and heir Applicaions, 111, pp , Chiashvili R. J., Lazrieva N. L., Srong soluions of sochasic di erenial equaions wih boundary condiions, Sochasics, 5, pp , Ding D., Zhang Y. Y., A spliing-sep algorihm for re eced sochasic di erenial equaions in R 1 +, Compu. Mah. Appl., 55, pp , Lepingle D. - Euler scheme for re eced sochasic di erenial equaions, Mahemaics and Compuers in Simulaion, Vol. 38, pp , Lions P. L., Szniman S. - Sochasic di erenial equaions wih re ecing boundary condiions, Commun. on Pure and Appl. Mah., Vol. 37, pp , Maiciuc L., R¼aşcanu A. - Backward sochasic generalized variaional inequaliy, Applied analysis and di erenial equaions, pp , World Sci. Publ., Hackensack, NJ, Maiciuc L., R¼aşcanu A., Sochasic approach for a mulivalued Dirichle-Neumann problem, Sochasic Processes and heir Applicaions, 120, pp , 2010.
44 Menaldi J. L., Sochasic variaional inequaliy for re eced di usion, Indiana Universiy Mahemaics Journal, 32, pp , Pardoux E., R¼aşcanu A. - Backward sochasic di erenial equaions wih subdi erenial operaor and relaed variaional inequaliies, Sochasic Process. Appl. 76, no. 2, pp , 1998 Saisho Y. - Sochasic di erenial equaions for muli-dimensional domain wih re ecing boundary, Probab. Theory and Rel. Fields, Vol. 74, pp , Slominski L. - On approximaion of soluions of mulidimensional SDE s wih re ecing boundary condiions, Sochasic Processes and heir Applicaions, Vol. 50, pp , Zhang J. - A numerical scheme for BSDEs, Annals of Applied Probabiliy, 14, pp , 2004.
45 Thank you for your aenion!
An Introduction to Malliavin calculus and its applications
An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214
More informationTime discretization of quadratic and superquadratic Markovian BSDEs with unbounded terminal conditions
Time discreizaion of quadraic and superquadraic Markovian BSDEs wih unbounded erminal condiions Adrien Richou Universié Bordeaux 1, INRIA équipe ALEA Oxford framework Le (Ω, F, P) be a probabiliy space,
More informationBackward doubly stochastic di erential equations with quadratic growth and applications to quasilinear SPDEs
Backward doubly sochasic di erenial equaions wih quadraic growh and applicaions o quasilinear SPDEs Badreddine MANSOURI (wih K. Bahlali & B. Mezerdi) Universiy of Biskra Algeria La Londe 14 sepember 2007
More informationAn Introduction to Backward Stochastic Differential Equations (BSDEs) PIMS Summer School 2016 in Mathematical Finance.
1 An Inroducion o Backward Sochasic Differenial Equaions (BSDEs) PIMS Summer School 2016 in Mahemaical Finance June 25, 2016 Chrisoph Frei cfrei@ualbera.ca This inroducion is based on Touzi [14], Bouchard
More informationUniqueness of solutions to quadratic BSDEs. BSDEs with convex generators and unbounded terminal conditions
Recalls and basic resuls on BSDEs Uniqueness resul Links wih PDEs On he uniqueness of soluions o quadraic BSDEs wih convex generaors and unbounded erminal condiions IRMAR, Universié Rennes 1 Châeau de
More informationGeneralized Snell envelope and BSDE With Two general Reflecting Barriers
1/22 Generalized Snell envelope and BSDE Wih Two general Reflecing Barriers EL HASSAN ESSAKY Cadi ayyad Universiy Poly-disciplinary Faculy Safi Work in progress wih : M. Hassani and Y. Ouknine Iasi, July
More informationBackward stochastic dynamics on a filtered probability space
Backward sochasic dynamics on a filered probabiliy space Gechun Liang Oxford-Man Insiue, Universiy of Oxford based on join work wih Terry Lyons and Zhongmin Qian Page 1 of 15 gliang@oxford-man.ox.ac.uk
More informationOn a Fractional Stochastic Landau-Ginzburg Equation
Applied Mahemaical Sciences, Vol. 4, 1, no. 7, 317-35 On a Fracional Sochasic Landau-Ginzburg Equaion Nguyen Tien Dung Deparmen of Mahemaics, FPT Universiy 15B Pham Hung Sree, Hanoi, Vienam dungn@fp.edu.vn
More informationExistence and uniqueness of solution for multidimensional BSDE with local conditions on the coefficient
1/34 Exisence and uniqueness of soluion for mulidimensional BSDE wih local condiions on he coefficien EL HASSAN ESSAKY Cadi Ayyad Universiy Mulidisciplinary Faculy Safi, Morocco ITN Roscof, March 18-23,
More informationDual Representation as Stochastic Differential Games of Backward Stochastic Differential Equations and Dynamic Evaluations
arxiv:mah/0602323v1 [mah.pr] 15 Feb 2006 Dual Represenaion as Sochasic Differenial Games of Backward Sochasic Differenial Equaions and Dynamic Evaluaions Shanjian Tang Absrac In his Noe, assuming ha he
More informationThe Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales
Advances in Dynamical Sysems and Applicaions. ISSN 0973-5321 Volume 1 Number 1 (2006, pp. 103 112 c Research India Publicaions hp://www.ripublicaion.com/adsa.hm The Asympoic Behavior of Nonoscillaory Soluions
More informationSimulation of BSDEs and. Wiener Chaos Expansions
Simulaion of BSDEs and Wiener Chaos Expansions Philippe Briand Céline Labar LAMA UMR 5127, Universié de Savoie, France hp://www.lama.univ-savoie.fr/ Workshop on BSDEs Rennes, May 22-24, 213 Inroducion
More informationSimulation of BSDEs and. Wiener Chaos Expansions
Simulaion of BSDEs and Wiener Chaos Expansions Philippe Briand Céline Labar LAMA UMR 5127, Universié de Savoie, France hp://www.lama.univ-savoie.fr/ Sochasic Analysis Seminar Oxford, June 1, 213 Inroducion
More informationMonotonic Solutions of a Class of Quadratic Singular Integral Equations of Volterra type
In. J. Conemp. Mah. Sci., Vol. 2, 27, no. 2, 89-2 Monoonic Soluions of a Class of Quadraic Singular Inegral Equaions of Volerra ype Mahmoud M. El Borai Deparmen of Mahemaics, Faculy of Science, Alexandria
More informationAMartingaleApproachforFractionalBrownian Motions and Related Path Dependent PDEs
AMaringaleApproachforFracionalBrownian Moions and Relaed Pah Dependen PDEs Jianfeng ZHANG Universiy of Souhern California Join work wih Frederi VIENS Mahemaical Finance, Probabiliy, and PDE Conference
More informationBackward stochastic dierential equations with subdierential operator and related variational inequalities
Sochasic Processes and heir Applicaions 76 1998) 191 215 Bacward sochasic dierenial equaions wih subdierenial operaor and relaed variaional inequaliies Eienne Pardoux a;, Aurel Rascanu b;1 a Universie
More informationand Applications Alexander Steinicke University of Graz Vienna Seminar in Mathematical Finance and Probability,
Backward Sochasic Differenial Equaions and Applicaions Alexander Seinicke Universiy of Graz Vienna Seminar in Mahemaical Finance and Probabiliy, 6-20-2017 1 / 31 1 Wha is a BSDE? SDEs - he differenial
More informationMODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE
Topics MODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES 2-6 3. FUNCTION OF A RANDOM VARIABLE 3.2 PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE 3.3 EXPECTATION AND MOMENTS
More informationConvergence of the Neumann series in higher norms
Convergence of he Neumann series in higher norms Charles L. Epsein Deparmen of Mahemaics, Universiy of Pennsylvania Version 1.0 Augus 1, 003 Absrac Naural condiions on an operaor A are given so ha he Neumann
More informationUtility maximization in incomplete markets
Uiliy maximizaion in incomplee markes Marcel Ladkau 27.1.29 Conens 1 Inroducion and general seings 2 1.1 Marke model....................................... 2 1.2 Trading sraegy.....................................
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS TO THE BACKWARD STOCHASTIC LORENZ SYSTEM
Communicaions on Sochasic Analysis Vol. 1, No. 3 (27) 473-483 EXISTENCE AND UNIQUENESS OF SOLUTIONS TO THE BACKWARD STOCHASTIC LORENZ SYSTEM P. SUNDAR AND HONG YIN Absrac. The backward sochasic Lorenz
More informationChapter 3 Boundary Value Problem
Chaper 3 Boundary Value Problem A boundary value problem (BVP) is a problem, ypically an ODE or a PDE, which has values assigned on he physical boundary of he domain in which he problem is specified. Le
More informationLocal Strict Comparison Theorem and Converse Comparison Theorems for Reflected Backward Stochastic Differential Equations
arxiv:mah/07002v [mah.pr] 3 Dec 2006 Local Sric Comparison Theorem and Converse Comparison Theorems for Refleced Backward Sochasic Differenial Equaions Juan Li and Shanjian Tang Absrac A local sric comparison
More informationREFLECTED SOLUTIONS OF BACKWARD SDE S, AND RELATED OBSTACLE PROBLEMS FOR PDE S
The Annals of Probabiliy 1997, Vol. 25, No. 2, 72 737 REFLECTED SOLUTIONS OF BACKWARD SDE S, AND RELATED OBSTACLE PROBLEMS FOR PDE S By N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez
More information1. Introduction. Journal of Numerical Mathematics and Stochastics, 10(1) : 45-57, A.SGHIR 1, D.SEGHIR 2, and S. HADIRI 2
Journal of Numerical Mahemaics and Sochasics, 1(1) : 5-57, 218 hp://www.jnmas.org/jnmas1-.pdf JNM@S Euclidean Press, LLC Online: ISSN 2151-232 Numerical Mehods for Cerain Classes of Markovian Backward
More informationCONTRIBUTION TO IMPULSIVE EQUATIONS
European Scienific Journal Sepember 214 /SPECIAL/ ediion Vol.3 ISSN: 1857 7881 (Prin) e - ISSN 1857-7431 CONTRIBUTION TO IMPULSIVE EQUATIONS Berrabah Faima Zohra, MA Universiy of sidi bel abbes/ Algeria
More informationAlgorithmic Trading: Optimal Control PIMS Summer School
Algorihmic Trading: Opimal Conrol PIMS Summer School Sebasian Jaimungal, U. Torono Álvaro Carea,U. Oxford many hanks o José Penalva,(U. Carlos III) Luhui Gan (U. Torono) Ryan Donnelly (Swiss Finance Insiue,
More informationTO our knowledge, most exciting results on the existence
IAENG Inernaional Journal of Applied Mahemaics, 42:, IJAM_42 2 Exisence and Uniqueness of a Periodic Soluion for hird-order Delay Differenial Equaion wih wo Deviaing Argumens A. M. A. Abou-El-Ela, A. I.
More informationEMS SCM joint meeting. On stochastic partial differential equations of parabolic type
EMS SCM join meeing Barcelona, May 28-30, 2015 On sochasic parial differenial equaions of parabolic ype Isván Gyöngy School of Mahemaics and Maxwell Insiue Edinburgh Universiy 1 I. Filering problem II.
More informationBSDES UNDER FILTRATION-CONSISTENT NONLINEAR EXPECTATIONS AND THE CORRESPONDING DECOMPOSITION THEOREM FOR E-SUPERMARTINGALES IN L p
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 43, Number 2, 213 BSDES UNDER FILTRATION-CONSISTENT NONLINEAR EXPECTATIONS AND THE CORRESPONDING DECOMPOSITION THEOREM FOR E-SUPERMARTINGALES IN L p ZHAOJUN
More informationExistence Theory of Second Order Random Differential Equations
Global Journal of Mahemaical Sciences: Theory and Pracical. ISSN 974-32 Volume 4, Number 3 (22), pp. 33-3 Inernaional Research Publicaion House hp://www.irphouse.com Exisence Theory of Second Order Random
More informationHomogenization of random Hamilton Jacobi Bellman Equations
Probabiliy, Geomery and Inegrable Sysems MSRI Publicaions Volume 55, 28 Homogenizaion of random Hamilon Jacobi Bellman Equaions S. R. SRINIVASA VARADHAN ABSTRACT. We consider nonlinear parabolic equaions
More informationAn introduction to the theory of SDDP algorithm
An inroducion o he heory of SDDP algorihm V. Leclère (ENPC) Augus 1, 2014 V. Leclère Inroducion o SDDP Augus 1, 2014 1 / 21 Inroducion Large scale sochasic problem are hard o solve. Two ways of aacking
More informationSingular control of SPDEs and backward stochastic partial diffe. reflection
Singular conrol of SPDEs and backward sochasic parial differenial equaions wih reflecion Universiy of Mancheser Join work wih Bern Øksendal and Agnès Sulem Singular conrol of SPDEs and backward sochasic
More informationOn Oscillation of a Generalized Logistic Equation with Several Delays
Journal of Mahemaical Analysis and Applicaions 253, 389 45 (21) doi:1.16/jmaa.2.714, available online a hp://www.idealibrary.com on On Oscillaion of a Generalized Logisic Equaion wih Several Delays Leonid
More informationVariational Iteration Method for Solving System of Fractional Order Ordinary Differential Equations
IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 1, Issue 6 Ver. II (Nov - Dec. 214), PP 48-54 Variaional Ieraion Mehod for Solving Sysem of Fracional Order Ordinary Differenial
More informationarxiv: v1 [math.pr] 21 May 2010
ON SCHRÖDINGER S EQUATION, 3-DIMENSIONAL BESSEL BRIDGES, AND PASSAGE TIME PROBLEMS arxiv:15.498v1 [mah.pr 21 May 21 GERARDO HERNÁNDEZ-DEL-VALLE Absrac. In his work we relae he densiy of he firs-passage
More informationOscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson
PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 4 7, 00, Wilmingon, NC, USA pp 0 Oscillaion of an Euler Cauchy Dynamic Equaion S Huff, G Olumolode,
More informationarxiv: v1 [math.ca] 15 Nov 2016
arxiv:6.599v [mah.ca] 5 Nov 26 Counerexamples on Jumarie s hree basic fracional calculus formulae for non-differeniable coninuous funcions Cheng-shi Liu Deparmen of Mahemaics Norheas Peroleum Universiy
More informationExample on p. 157
Example 2.5.3. Le where BV [, 1] = Example 2.5.3. on p. 157 { g : [, 1] C g() =, g() = g( + ) [, 1), var (g) = sup g( j+1 ) g( j ) he supremum is aken over all he pariions of [, 1] (1) : = < 1 < < n =
More informationarxiv: v1 [math.pr] 14 Jul 2008
Refleced Backward Sochasic Differenial Equaions Driven by Lévy Process arxiv:0807.2076v1 [mah.pr] 14 Jul 2008 Yong Ren 1, Xiliang Fan 2 1. School of Mahemaics and Physics, Universiy of Tasmania, GPO Box
More informationIn nite horizon optimal control of forward-backward stochastic di erential equations with delay.
In nie horizon opimal conrol of forward-backward sochasic di erenial equaions wih delay. Nacira AGAM and Bern ØKSENDAL yz 25 January 213 Absrac Our sysem is governed by a forward -backward sochasic di
More informationCouplage du principe des grandes déviations et de l homogénisation dans le cas des EDP paraboliques: (le cas constant)
Couplage du principe des grandes déviaions e de l homogénisaion dans le cas des EDP paraboliques: (le cas consan) Alioune COULIBALY U.F.R Sciences e Technologie Universié Assane SECK de Ziguinchor Probabilié
More informationQuadratic and Superquadratic BSDEs and Related PDEs
Quadraic and Superquadraic BSDEs and Relaed PDEs Ying Hu IRMAR, Universié Rennes 1, FRANCE hp://perso.univ-rennes1.fr/ying.hu/ ITN Marie Curie Workshop "Sochasic Conrol and Finance" Roscoff, March 21 Ying
More informationUndetermined coefficients for local fractional differential equations
Available online a www.isr-publicaions.com/jmcs J. Mah. Compuer Sci. 16 (2016), 140 146 Research Aricle Undeermined coefficiens for local fracional differenial equaions Roshdi Khalil a,, Mohammed Al Horani
More informationOn Gronwall s Type Integral Inequalities with Singular Kernels
Filoma 31:4 (217), 141 149 DOI 1.2298/FIL17441A Published by Faculy of Sciences and Mahemaics, Universiy of Niš, Serbia Available a: hp://www.pmf.ni.ac.rs/filoma On Gronwall s Type Inegral Inequaliies
More informationAnn. Funct. Anal. 2 (2011), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL:
Ann. Func. Anal. 2 2011, no. 2, 34 41 A nnals of F uncional A nalysis ISSN: 2008-8752 elecronic URL: www.emis.de/journals/afa/ CLASSIFICAION OF POSIIVE SOLUIONS OF NONLINEAR SYSEMS OF VOLERRA INEGRAL EQUAIONS
More informationDifferential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
More informationA class of multidimensional quadratic BSDEs
A class of mulidimensional quadraic SDEs Zhongmin Qian, Yimin Yang Shujin Wu March 4, 07 arxiv:703.0453v mah.p] Mar 07 Absrac In his paper we sudy a mulidimensional quadraic SDE wih a paricular class of
More information6. Stochastic calculus with jump processes
A) Trading sraegies (1/3) Marke wih d asses S = (S 1,, S d ) A rading sraegy can be modelled wih a vecor φ describing he quaniies invesed in each asse a each insan : φ = (φ 1,, φ d ) The value a of a porfolio
More informationL p -L q -Time decay estimate for solution of the Cauchy problem for hyperbolic partial differential equations of linear thermoelasticity
ANNALES POLONICI MATHEMATICI LIV.2 99) L p -L q -Time decay esimae for soluion of he Cauchy problem for hyperbolic parial differenial equaions of linear hermoelasiciy by Jerzy Gawinecki Warszawa) Absrac.
More informationKalman Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems
J. Mah. Anal. Appl. 34 8) 18 196 www.elsevier.com/locae/jmaa Kalman Bucy filering equaions of forward and backward sochasic sysems and applicaions o recursive opimal conrol problems Guangchen Wang a,b,,zhenwu
More informationSome New Uniqueness Results of Solutions to Nonlinear Fractional Integro-Differential Equations
Annals of Pure and Applied Mahemaics Vol. 6, No. 2, 28, 345-352 ISSN: 2279-87X (P), 2279-888(online) Published on 22 February 28 www.researchmahsci.org DOI: hp://dx.doi.org/.22457/apam.v6n2a Annals of
More informationExistence of positive solution for a third-order three-point BVP with sign-changing Green s function
Elecronic Journal of Qualiaive Theory of Differenial Equaions 13, No. 3, 1-11; hp://www.mah.u-szeged.hu/ejqde/ Exisence of posiive soluion for a hird-order hree-poin BVP wih sign-changing Green s funcion
More informationPositive continuous solution of a quadratic integral equation of fractional orders
Mah. Sci. Le., No., 9-7 (3) 9 Mahemaical Sciences Leers An Inernaional Journal @ 3 NSP Naural Sciences Publishing Cor. Posiive coninuous soluion of a quadraic inegral equaion of fracional orders A. M.
More informationEngineering Letter, 16:4, EL_16_4_03
3 Exisence In his secion we reduce he problem (5)-(8) o an equivalen problem of solving a linear inegral equaion of Volerra ype for C(s). For his purpose we firs consider following free boundary problem:
More informationCorrespondence should be addressed to Nguyen Buong,
Hindawi Publishing Corporaion Fixed Poin Theory and Applicaions Volume 011, Aricle ID 76859, 10 pages doi:101155/011/76859 Research Aricle An Implici Ieraion Mehod for Variaional Inequaliies over he Se
More informationEXISTENCE AND UNIQUENESS THEOREMS ON CERTAIN DIFFERENCE-DIFFERENTIAL EQUATIONS
Elecronic Journal of Differenial Equaions, Vol. 29(29), No. 49, pp. 2. ISSN: 72-669. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu EXISTENCE AND UNIQUENESS THEOREMS ON CERTAIN
More informationA Necessary and Sufficient Condition for the Solutions of a Functional Differential Equation to Be Oscillatory or Tend to Zero
JOURNAL OF MAEMAICAL ANALYSIS AND APPLICAIONS 24, 7887 1997 ARICLE NO. AY965143 A Necessary and Sufficien Condiion for he Soluions of a Funcional Differenial Equaion o Be Oscillaory or end o Zero Piambar
More informationarxiv: v1 [math.pr] 7 Jan 2018
Refleced forward-backward sochasic differenial equaions driven by G-Brownian moion wih coninuous arxiv:181.2271v1 [mah.pr] 7 Jan 218 monoone coefficiens Bingjun Wang 1,2 Hongjun Gao 1 Mei Li 3 1. Insiue
More informationarxiv: v1 [math.pr] 28 Nov 2016
Backward Sochasic Differenial Equaions wih Nonmarkovian Singular Terminal Values Ali Devin Sezer, Thomas Kruse, Alexandre Popier Ocober 15, 2018 arxiv:1611.09022v1 mah.pr 28 Nov 2016 Absrac We solve a
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationOn R d -valued peacocks
On R d -valued peacocks Francis HIRSCH 1), Bernard ROYNETTE 2) July 26, 211 1) Laboraoire d Analyse e Probabiliés, Universié d Évry - Val d Essonne, Boulevard F. Mierrand, F-9125 Évry Cedex e-mail: francis.hirsch@univ-evry.fr
More informationHamilton- J acobi Equation: Explicit Formulas In this lecture we try to apply the method of characteristics to the Hamilton-Jacobi equation: u t
M ah 5 2 7 Fall 2 0 0 9 L ecure 1 0 O c. 7, 2 0 0 9 Hamilon- J acobi Equaion: Explici Formulas In his lecure we ry o apply he mehod of characerisics o he Hamilon-Jacobi equaion: u + H D u, x = 0 in R n
More informationA proof of Ito's formula using a di Title formula. Author(s) Fujita, Takahiko; Kawanishi, Yasuhi. Studia scientiarum mathematicarum H Citation
A proof of Io's formula using a di Tile formula Auhor(s) Fujia, Takahiko; Kawanishi, Yasuhi Sudia scieniarum mahemaicarum H Ciaion 15-134 Issue 8-3 Dae Type Journal Aricle Tex Version auhor URL hp://hdl.handle.ne/186/15878
More informationLecture 20: Riccati Equations and Least Squares Feedback Control
34-5 LINEAR SYSTEMS Lecure : Riccai Equaions and Leas Squares Feedback Conrol 5.6.4 Sae Feedback via Riccai Equaions A recursive approach in generaing he marix-valued funcion W ( ) equaion for i for he
More informationLECTURE 1: GENERALIZED RAY KNIGHT THEOREM FOR FINITE MARKOV CHAINS
LECTURE : GENERALIZED RAY KNIGHT THEOREM FOR FINITE MARKOV CHAINS We will work wih a coninuous ime reversible Markov chain X on a finie conneced sae space, wih generaor Lf(x = y q x,yf(y. (Recall ha q
More informationOptimal Investment Strategy Insurance Company
Opimal Invesmen Sraegy for a Non-Life Insurance Company Łukasz Delong Warsaw School of Economics Insiue of Economerics Division of Probabilisic Mehods Probabiliy space Ω I P F I I I he filraion saisfies
More informationINVARIANCE OF CLOSED CONVEX CONES FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
INVARIANCE OF CLOSED CONVEX CONES FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS STEFAN TAPPE Absrac. The goal of his paper is o clarify when a closed convex cone is invarian for a sochasic parial differenial
More informationOn the probabilistic stability of the monomial functional equation
Available online a www.jnsa.com J. Nonlinear Sci. Appl. 6 (013), 51 59 Research Aricle On he probabilisic sabiliy of he monomial funcional equaion Claudia Zaharia Wes Universiy of Timişoara, Deparmen of
More informationSobolev-type Inequality for Spaces L p(x) (R N )
In. J. Conemp. Mah. Sciences, Vol. 2, 27, no. 9, 423-429 Sobolev-ype Inequaliy for Spaces L p(x ( R. Mashiyev and B. Çekiç Universiy of Dicle, Faculy of Sciences and Ars Deparmen of Mahemaics, 228-Diyarbakir,
More informationHamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation:
M ah 5 7 Fall 9 L ecure O c. 4, 9 ) Hamilon- J acobi Equaion: Weak S oluion We coninue he sudy of he Hamilon-Jacobi equaion: We have shown ha u + H D u) = R n, ) ; u = g R n { = }. ). In general we canno
More informationarxiv: v1 [math.pr] 18 Feb 2015
Non-Markovian opimal sopping problems and consrained BSDEs wih jump arxiv:152.5422v1 [mah.pr 18 Feb 215 Marco Fuhrman Poliecnico di Milano, Diparimeno di Maemaica via Bonardi 9, 2133 Milano, Ialy marco.fuhrman@polimi.i
More informationNonlinear Fuzzy Stability of a Functional Equation Related to a Characterization of Inner Product Spaces via Fixed Point Technique
Filoma 29:5 (2015), 1067 1080 DOI 10.2298/FI1505067W Published by Faculy of Sciences and Mahemaics, Universiy of Niš, Serbia Available a: hp://www.pmf.ni.ac.rs/filoma Nonlinear Fuzzy Sabiliy of a Funcional
More informationDynamical systems method for solving linear ill-posed problems
ANNALES POLONICI MATHEMATICI * (2*) Dynamical sysems mehod for solving linear ill-posed problems by A. G. Ramm (Manhaan, KS) Absrac. Various versions of he Dynamical Sysems Mehod (DSM) are proposed for
More informationCash Flow Valuation Mode Lin Discrete Time
IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728,p-ISSN: 2319-765X, 6, Issue 6 (May. - Jun. 2013), PP 35-41 Cash Flow Valuaion Mode Lin Discree Time Olayiwola. M. A. and Oni, N. O. Deparmen of Mahemaics
More informationProperties Of Solutions To A Generalized Liénard Equation With Forcing Term
Applied Mahemaics E-Noes, 8(28), 4-44 c ISSN 67-25 Available free a mirror sies of hp://www.mah.nhu.edu.w/ amen/ Properies Of Soluions To A Generalized Liénard Equaion Wih Forcing Term Allan Kroopnick
More informationarxiv: v1 [math.fa] 9 Dec 2018
AN INVERSE FUNCTION THEOREM CONVERSE arxiv:1812.03561v1 [mah.fa] 9 Dec 2018 JIMMIE LAWSON Absrac. We esablish he following converse of he well-known inverse funcion heorem. Le g : U V and f : V U be inverse
More informationarxiv: v1 [math.pr] 6 Oct 2008
MEASURIN THE NON-STOPPIN TIMENESS OF ENDS OF PREVISIBLE SETS arxiv:8.59v [mah.pr] 6 Oc 8 JU-YI YEN ),) AND MARC YOR 3),4) Absrac. In his paper, we propose several measuremens of he nonsopping imeness of
More informationHeat kernel and Harnack inequality on Riemannian manifolds
Hea kernel and Harnack inequaliy on Riemannian manifolds Alexander Grigor yan UHK 11/02/2014 onens 1 Laplace operaor and hea kernel 1 2 Uniform Faber-Krahn inequaliy 3 3 Gaussian upper bounds 4 4 ean-value
More informationEXERCISES FOR SECTION 1.5
1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler
More informationDynamics of a stochastic predator-prey model with Beddington DeAngelis functional response
SCIENTIA Series A: Mahemaical Sciences, Vol. 22 22, 75 84 Universidad Técnica Federico Sana María Valparaíso, Chile ISSN 76-8446 c Universidad Técnica Federico Sana María 22 Dynamics of a sochasic predaor-prey
More informationThe Existence, Uniqueness and Stability of Almost Periodic Solutions for Riccati Differential Equation
ISSN 1749-3889 (prin), 1749-3897 (online) Inernaional Journal of Nonlinear Science Vol.5(2008) No.1,pp.58-64 The Exisence, Uniqueness and Sailiy of Almos Periodic Soluions for Riccai Differenial Equaion
More informationOption pricing and implied volatilities in a 2-hypergeometric stochastic volatility model
Opion pricing and implied volailiies in a 2-hypergeomeric sochasic volailiy model Nicolas Privaul Qihao She Division of Mahemaical Sciences School of Physical and Mahemaical Sciences Nanyang Technological
More informationExistence of multiple positive periodic solutions for functional differential equations
J. Mah. Anal. Appl. 325 (27) 1378 1389 www.elsevier.com/locae/jmaa Exisence of muliple posiive periodic soluions for funcional differenial equaions Zhijun Zeng a,b,,libi a, Meng Fan a a School of Mahemaics
More informationSingular perturbation control problems: a BSDE approach
Singular perurbaion conrol problems: a BSDE approach Join work wih Francois Delarue Universié de Nice and Giuseppina Guaeri Poliecnico di Milano Le Mans 8h of Ocober 215 Conference in honour of Vlad Bally
More informationStable approximations of optimal filters
Sable approximaions of opimal filers Joaquin Miguez Deparmen of Signal Theory & Communicaions, Universidad Carlos III de Madrid. E-mail: joaquin.miguez@uc3m.es Join work wih Dan Crisan (Imperial College
More informationMost Probable Phase Portraits of Stochastic Differential Equations and Its Numerical Simulation
Mos Probable Phase Porrais of Sochasic Differenial Equaions and Is Numerical Simulaion Bing Yang, Zhu Zeng and Ling Wang 3 School of Mahemaics and Saisics, Huazhong Universiy of Science and Technology,
More informationarxiv: v1 [math.pr] 19 Feb 2011
A NOTE ON FELLER SEMIGROUPS AND RESOLVENTS VADIM KOSTRYKIN, JÜRGEN POTTHOFF, AND ROBERT SCHRADER ABSTRACT. Various equivalen condiions for a semigroup or a resolven generaed by a Markov process o be of
More informationReflected Discontinuous Backward Doubly Stochastic Differential Equations With Poisson Jumps
Journal of Numerical Mahemaics and Sochasics, () : 73-93, 8 hp://www.jnmas.org/jnmas-5.pdf JNM@S uclidean Press, LLC Online: ISSN 5-3 Refleced Disconinuous Backward Doubly Sochasic Differenial quaions
More informationApproximating positive solutions of nonlinear first order ordinary quadratic differential equations
Dhage & Dhage, Cogen Mahemaics (25, 2: 2367 hp://dx.doi.org/.8/233835.25.2367 APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE Approximaing posiive soluions of nonlinear firs order ordinary quadraic
More informationResearch Article Existence and Uniqueness of Periodic Solution for Nonlinear Second-Order Ordinary Differential Equations
Hindawi Publishing Corporaion Boundary Value Problems Volume 11, Aricle ID 19156, 11 pages doi:1.1155/11/19156 Research Aricle Exisence and Uniqueness of Periodic Soluion for Nonlinear Second-Order Ordinary
More informationRecursive Least-Squares Fixed-Interval Smoother Using Covariance Information based on Innovation Approach in Linear Continuous Stochastic Systems
8 Froniers in Signal Processing, Vol. 1, No. 1, July 217 hps://dx.doi.org/1.2266/fsp.217.112 Recursive Leas-Squares Fixed-Inerval Smooher Using Covariance Informaion based on Innovaion Approach in Linear
More information!dp= fj f dv for any f e C* (S).
252 [Vol. 42, 59. Fundamenal Equaions o f Branching Markov Processes By Nobuyuki IKEDA, Masao NAGASAWA, and Shinzo WATANABE Osaka Universiy, Tokyo Insiue of Technology, and Kyoo Universiy (Comm. by Kinjiro
More informationAttractors for a deconvolution model of turbulence
Aracors for a deconvoluion model of urbulence Roger Lewandowski and Yves Preaux April 0, 2008 Absrac We consider a deconvoluion model for 3D periodic flows. We show he exisence of a global aracor for he
More informationBackward Stochastic Differential Equations with Nonmarkovian Singular Terminal Values
Backward Sochasic Differenial Equaions wih Nonmarkovian Singular Terminal Values Ali Sezer, Thomas Kruse, Alexandre Popier, Ali Sezer To cie his version: Ali Sezer, Thomas Kruse, Alexandre Popier, Ali
More informationSupplement for Stochastic Convex Optimization: Faster Local Growth Implies Faster Global Convergence
Supplemen for Sochasic Convex Opimizaion: Faser Local Growh Implies Faser Global Convergence Yi Xu Qihang Lin ianbao Yang Proof of heorem heorem Suppose Assumpion holds and F (w) obeys he LGC (6) Given
More informationDifferential Harnack Estimates for Parabolic Equations
Differenial Harnack Esimaes for Parabolic Equaions Xiaodong Cao and Zhou Zhang Absrac Le M,g be a soluion o he Ricci flow on a closed Riemannian manifold In his paper, we prove differenial Harnack inequaliies
More informationEIGENVALUE PROBLEMS FOR SINGULAR MULTI-POINT DYNAMIC EQUATIONS ON TIME SCALES
Elecronic Journal of Differenial Equaions, Vol. 27 (27, No. 37, pp. 3. ISSN: 72-669. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu EIGENVALUE PROBLEMS FOR SINGULAR MULTI-POINT DYNAMIC EQUATIONS ON
More informationPOSITIVE SOLUTIONS OF NEUTRAL DELAY DIFFERENTIAL EQUATION
Novi Sad J. Mah. Vol. 32, No. 2, 2002, 95-108 95 POSITIVE SOLUTIONS OF NEUTRAL DELAY DIFFERENTIAL EQUATION Hajnalka Péics 1, János Karsai 2 Absrac. We consider he scalar nonauonomous neural delay differenial
More information