EMS SCM joint meeting. On stochastic partial differential equations of parabolic type

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1 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

2 I. Filering problem II. Equaions of nonlinear filering III. Some problems in filering heory Innovaion problem Robusness of he filer Acceleraed numerical schemes 2

3 I. Filering heory 1. Esimaing fuure values of signals in he framework of saionary processes 3

4 2. Sae-space filering model dx = bx d + θ dw, X 0 = ξ N(m, R) dy = BX d + dv, Y 0 = η N( m, R), (W, V ) 0 is muli-dimensional Wiener process, b, B and θ are given marices. 4

5 Task: A esimae ϕ(x ) from observaions (Y s ) s [0,] =: Y. wih smalles mean square error; i.e., le ˆϕ denoe he opimal esimae Clearly, where E ˆϕ ϕ(x ) 2 = min f E f(y ) ϕ(x ) 2. ˆϕ = P (ϕ) := E(ϕ(X ) Y s, s ) = = R d ϕ(x)π (x) dx, R d ϕ(x)p (dx) P (dx) := P (X dx Y s, s ) = p (x) dx. 5

6 Noe ha (X, Y ) is Gaussian process. Hence given Y, P (dx) is also Gaussian. I is sufficien o deermine is mean, m and is covariance C, or equivalenly, m = E(X Y s, s ), γ = E{(X m )(X m ) }. dm = bm d + γ B d V, m 0 = EX 0 γ = aγ +γ a γ B Bγ +θθ, γ 0 = E{(X 0 m 0 )(X 0 m 0 ) }, V := Y 0 Bm s ds. Remark. ( V ) 0 is called innovaion process. I is a Wiener process. 6

7 Inerpreaion: V + V = Y + Y Hm carries new informaion, relaive o (Y s ) s [0,]. Clearly, for each 0 σ( V s : s ) σ(y s : s ). Innovaion problem: σ( V s : s ) = σ(y s : s )? Does he innovaion process carry he same informaion as he observaion process? 7

8 3. Nonlinear filering Sae: dx = b(z ) d + θ(z ) dw + ρ(z ) dv, X 0 = ξ Observaion: dy = B(Z ) d + dv, Y 0 = 0, where Z := (X, Y ) R d+d 1, (W, V ) is a mulidimensional Wiener; b and B are Lipschiz coninuous vecor fields, and θ and ρ are Lipschiz coninuous marix fields on R d+d 1; ξ is a random vecor, independen of (W, V ). As we know, for funcions of X, ϕ(x ), he opimal esimaor, given he observaions (Y ) s [0,], is P (ϕ) := E(ϕ(Z ) Y s, s ). Innovaion process: V = Y 0 P s (B) ds. 8

9 Quesions: 1. Innovaion problem: σ( V s, s ) = σ(y s, s )? 2. How o compue P (ϕ)? 3. Robusness of he filer: does he compuaion of P (ϕ) depend coninuously on he observaions? 4. How o calculae P (ϕ) numerically? 9

10 1. Innovaion problem Assumpion 1. There is δ > 0 such ha ρρ δi for all λ R d, z R d+d 1, where (ρρ ) ij := ρ ir ρ jr. Assumpion 2. ξ has a probabiliy densiy π 0 H 1 p for some p 2, Assumpion 3. (1 + x 2 ) α p 0 L 2, for some α > d/2. Theorem 1.(N.V. Krylov, I.G.) Le Assumpions 1-3 hold. Then σ( V s, s ) = σ(y s, s ). Generalizes a resul of N.V. Krylov

11 2. Equaions of nonlinear filering Some noaion: b (x) := b(x, Y ), ρ (x) = ρ(x, Y ), θ (x) := θ(x, Y ), B (x) = B(x, Y ), a (x) := (ρ ρ (x)+θ θ (x))/2, x R d L = a ij (x)d id j + b (x) i D i, Filering equaion: M r = θ ir (x)d i + B i (x) dp (ϕ) =P (Lϕ) d + {P (M r ϕ) P (B r )P (ϕ)} d V r (1) P 0 (ϕ) =E(ϕ(X 0 )), (2) where V is he innovaion process. If π (x) := P (dx)/dx exiss, hen i saisfies dπ (x) =L π (x) d + {M r π (x) (π, B r )π (x)} d V r, π 0 (x) =P (X 0 dx)/dx =: p 0. (Kushner-Shiryayev equaion) 11

12 Idea of he proof of Thm1: Consider he sysem dπ (x) =L π (x) d + {M r p (x) (π, B r )π (x)} d V r, π 0 = p 0 dy =(π, B ) d + d V r, Y 0 = 0. Sep 1. Approximaion π (0) = p 0, Y (0) = 0, n = 1, 2,... dπ (n) = L (n) π (n) dy (n) Sep 2. (π (n) d + {(M (n) π n (π (n 1), B n )π n } d V r = (π (n 1), B (n) ) d + d W., Y (n) ), which is σ( V s : s )-measurable. Sep 3. Show ha in probabiliy sup π (n) π L2 0, sup Y (n) Y 0. [0,T ] [0,T ] 12

13 Exisence and analyic propery of he densiy π. The Kushner-Shiryayev equaion can be ransform ino a linear SPDE (Zakai equaion): du (x) =L u (x) d + M r u (x) dy r u 0 (x) =p 0 (x) Under Assumpions 1 and 2 a (generalised) soluion (u ) [0,T ] L p ([0, T ], H 1 p ), (u, 1) > 0 and π (x) = u (x) (u, 1). 13

14 3. Robusness of he filer Le Y (n) be coninuous processes of bounded variaion, Y (n) 0 = 0. Quesion: sup T Y (n) Consider du (n) Y 0 π (n) π? (x) =L (n) u (n) (x) d + M r (n) u (n) (x) dy r(n) u 0 (x) =p 0 (x) Define A (n)ij A ij := := 0 Y (n)i s 0 Y i s dy j s S (n)ij := dy s (n)j 0 Y (n)j s dy (n)i s,, 0 Y j s dy i s, i, j = 1,..., d 1. 0 (Y i s Y s (n)i )dy s (n)j. 14

15 Theorem 2. (I.G. 1988) Le m 0. Le (i) sup [0,T ] Y (n) Y 0, sup [0,T ] S (n) S 0, S (n) T = o(ln n) (ii) b(x, y), ρ(x, y) sufficienly smooh in x; B(x, y), θ(x, y) are sufficienly smooh in (x, y), (iii) p 0 H k 2 wih sufficienly high k. Then sup [0,T ] u (n) ũ H m 2 = 0 (in probabiliy), where ũ is he soluion of L := L M r dũ = L ũ d + M r ũ dy r, ũ 0 = p 0, M r + r N r, N r = θ ir y r(x, Y )D i +B r y r(x, Y ). 15

16 Consider wih L (n) du (n) (x) =L (n) u 0 (x) =p 0 (x). := L (n) u (n) 1 2 M r(n) M r(n) + r N r(n), (x) d + M r (n) u (n) (x) dy r(n) Then H m -valued unique soluions u and u (n). sup u (n) u H m 0, sup π [0,T ] 2 (n) π H m 0, [0,T ] 2 where π (n) := u (n) /(u (n), 1). Theorem 3. (P. Singa, I.G. 2013) For some κ > 0 assume sup T Y (n) Y = O(n κ ), sup S (n) S = O(n κ ) (a.s.). T Then for any γ < κ sup T u (n) u H m 2 = O(n γ ), π (n) π H m 2 = O(n γ ). 16

17 4. Acceleraed numerical schemes Richardson s idea: Assume for q(h) q and for h 0 we have q(h) = q + q 1 h + O(h). Then q(h) := 2q(h/2) q(h) = q + O(h 2 ). 17

18 More generally, assume h 2 q(h) = q + q 1 h + q 2 hen h k q k k! + O(hk+1 ), q(h) := λ 0 q(h)+λ 1 q(h/2)+...+λ k q(h/2 k ) = q+o(h k+1 ), wih consans λ 0, λ 1,..., λ k, defined by (λ 0, λ 1,..., λ k ) = (1, 0,..., 0)V 1, where V 1 is he inverse of V = (V ij ), V ij = 2 (i 1)(j 1), j = 1, 2,..., k

19 Acceleraed finie difference schemes Noaion: For h > 0, vecors e i, x R d δ h,ei ϕ(x) := 1 h (ϕ(x+he i) ϕ(x)), δ h i ϕ(x) := 1 2 (δ h,e i +δ h,ei ) Consider D i δ h i, L L h, M r M h,r du h (x) =Lh uh h,r (x) d + M u h (x) dy r u h 0 (x) =p 0(x) [0, T ], x G h := hz d. Infinie sysem of SDEs 19

20 Truncaed finie difference schemes For R > 0 le ζ R C 0 (Rd ), s.., ζ R (x) = 1 if x R, ζ R (x) = 0 if x ρ > R. Se L h R := ϕ RL h, M h,r R := ϕ RM h,r, p R 0 = p 0ζ R. Consider du h,r (x) =L h R, uh,r (x) d + M h,r u h,r 0 (x) =pr 0 (x) R, uh,r for [0, T ] and x G h,ρ := G h { x ρ}. (x)) dy r Finie sysem of linear SDEs. a unique soluion u h,r. Se v h,r := k j=0 λ j u h/2j,r. 20

21 Theorem 4.(M. Gerencsér, N.V. Krylov, I.G.) Le k 0. Assume (i) b, ρ, θ, B are sufficienly smooh (ii) p 0 H k 2 Then he Zakai equaion has a classical soluion u, and for R > 0, κ (0, 1), q > 0 E sup [0,T ] sup vr, h (x) u (x) q N 1 h q(k+1) +N 2 e νr2 x G h { x κr} wih posiive consans N 1, N 2, ν, independen of h. 21

22 Theorem 5. Assume (i), (ii) and (1 + x 2 ) α p 0 L 2 for some α > d. Then for we have π R,h (x) := vh,r (x) (v h,r, 1) E sup [0,T ] sup πr, h (x) π (x) q N 1 h q(k+1) +N 2 e νr2 x G h { x κr} wih posiive consans N 1, N 2, ν, independen of h. 22

23 Summary: Via problems from filering heory we showed a iny bi of he heory and is applicaions. of he heory of parabolic SPDEs. 23

24 Moles Gràcies!