Nonlinear Filtering with Lévy noise

Transcription

1 Nonlinear Filtering with Lévy noise Hausenblas Erika (joint work with Paul Razafimandimby) Montanuniversität Leoben, Österreich April 10, 2014 Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

2 Nonlinear Filtering - an introduction Gliederung 1 Nonlinear Filtering - an introduction 2 Nonlinear Filtering with Lévy noise A Lévy Process Nonlinear Filtering 3 Correlated noise Copulas and Lévy Copulas 4 Nonlinear Filtering with correlated Lévy noise 5 Pseudo Differential Operators Lévy s symbol Analytic semigroups Lévy s symbol and pseudo differential operators 6 Future works Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

3 Nonlinear Filtering - an introduction Nonlinear Filtering - an Introduction Statement of the Problem Let I be an index set, i.e. I = IN or I = R + 0. Given: a state process X = {X(t) : t I} with noise, an observable process Y = {Y(t) : t I} depending on X and some additional (independent or dependent) noise Task: Let f : R R be a nice function. At each time t I give an estimate of f(x(t)). Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

4 Nonlinear Filtering - an introduction Nonlinear Filtering - an Introduction Statement of the Problem Let I be an index set, i.e. I = IN or I = R + 0. Given: a state process X = {X(t) : t I} with noise, an observable process Y = {Y(t) : t I} depending on X and some additional (independent or dependent) noise Task: Let f : R R be a nice function. At each time t I give an estimate of f(x(t)). Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

5 Nonlinear Filtering - an introduction Nonlinear Filtering - the Brownian case Given: Two independent Brownian motions W = {W(t) : 0 t < } and V = {V(t) : 0 t < } ; A signal process X = {X(t) : 0 t < }; dx(t) = b(x(t))dt +σ(x(t))dw(t) + σ(x(t))dv(t). An observable process Y = {Y(t) : 0 t < } dy(t) = h(x(t))dt + dv(t). Task: Let f : R R be a nice function. Given the path of Y up to time t, give an estimate of f(x(t)). Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

6 Nonlinear Filtering - an introduction Nonlinear Filtering - the Brownian case Solution: For f : R R, the best estimator of f(x(t)) is given by the conditional probability E[f(X(t)) F Y t ] where F Y t is the σ field generated by {Y s : 0 s t}. Definition The conditional density process π = {π t : t 0} is the measure valued process defined by R [ π t (x)f(x)dx = E f(x(t)) Ft Y ], f continuous. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

7 Nonlinear Filtering - an introduction Nonlinear Filtering - the Brownian case Solution: For f : R R, the best estimator of f(x(t)) is given by the conditional probability E[f(X(t)) F Y t ] where F Y t is the σ field generated by {Y s : 0 s t}. Definition The conditional density process π = {π t : t 0} is the measure valued process defined by R [ π t (x)f(x)dx = E f(x(t)) Ft Y ], f continuous. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

8 Nonlinear Filtering - an introduction Nonlinear Filtering - Kallianpur Striebel Formula Let Q be a probability measure, equivalent to P such that Y is a Brownian motion over (Ω,F Y t,q). Let Z t = dp, t 0. dq Ft Let ρ = {ρ t : t 0} be the measure valued process with ρ t (f) = E Q[ ] Z(t)f(X(t)) Ft Y. Then π t (f) = EQ[ Z(t)f(X(t)) Ft Y ] E Q[ ] Z(t) Ft Y = ρ t(f) ρ t (1). Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

9 Nonlinear Filtering - an introduction Nonlinear Filtering - Kallianpur Striebel Formula Let Q be a probability measure, equivalent to P such that Y is a Brownian motion over (Ω,F Y t,q). Let Z t = dp, t 0. dq Ft Let ρ = {ρ t : t 0} be the measure valued process with ρ t (f) = E Q[ ] Z(t)f(X(t)) Ft Y. Then π t (f) = EQ[ Z(t)f(X(t)) Ft Y ] E Q[ ] Z(t) Ft Y = ρ t(f) ρ t (1). Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

10 Given: Nonlinear Filtering - an introduction Two independent Brownian motions W = {W(t) : 0 t < } and V = {V(t) : 0 t < } ; A signal process X = {X(t) : 0 t < }; dx(t) = b(x(t))dt +σ(x(t))dw(t) + σ(x(t))dv(t). An observable process Y = {Y(t) : 0 t < } dy(t) = h(x(t))dt + dv(t). Task: Let f : R R be a nice function. Given the path of Y up to time t, give an estimate of f(x(t)). Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

11 Nonlinear Filtering - an introduction Nonlinear Filtering - the Brownian case Definition Let A 0 be the infinitesimal generator of the Markovian semigroup of X, i.e. Af(x) = 1 ( σσ ) 2 x x f(x) +b(x) x f(x), f C(2) (R). Theorem (Zakai and Kushner) Now, under appropriate assumptions one can show that ρ is a solution to the so called Zakai equation, i.e. we have Q a.s. for all t 0 t t ρ t (f) = π 0 (f)+ ρ s (Af)ds + ρ s (fh+b)dy s, f Cb 2 (R), 0 0 where B = σ(x) x. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

12 Nonlinear Filtering with Lévy noise Gliederung 1 Nonlinear Filtering - an introduction 2 Nonlinear Filtering with Lévy noise A Lévy Process Nonlinear Filtering 3 Correlated noise Copulas and Lévy Copulas 4 Nonlinear Filtering with correlated Lévy noise 5 Pseudo Differential Operators Lévy s symbol Analytic semigroups Lévy s symbol and pseudo differential operators 6 Future works Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

13 Nonlinear Filtering with Lévy noise A Lévy Process A Lévy Process Definition An R valued stochastic process L = {L(t) : 0 t < } is a Lévy process over (Ω;F;P) iff L(0) = 0; L has independent and stationary increments; L is stochastically continuous, i.e. for any A B(E) the function t E1 A (L(t)) is continuous on R + ; L has a.s. càdlàg a paths; a càdlàg = continue à droite, limitée à gauche. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

14 Nonlinear Filtering with Lévy noise Nonlinear Filtering Nonlinear Filtering - the Lévy case Given: Two independent Brownian motions V and W and two independent Lévy processes L 1 and L 2 ; A signal process X = {X(t) : 0 t < }; dx(t) = b(x(t))dt +σ(x(t))dw(t) + dl 1 (t). An observable process Y = {Y(t) : 0 t < } dy(t) = h(x(t))dt + dv(t)+dl 2 (t). Task: Let f : R R be a nice function. Given the path of Y up to time t, give an estimate of f(x(t)). Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

15 Nonlinear Filtering with Lévy noise Nonlinear Filtering Nonlinear Filtering - the Lévy case Definition Let A 0 be the infinitesimal generator of the Markovian semigroup of X, i.e. A 0 f(x) = 1 2 σ2 (x) 2 x 2 f(x) +b(x) x f(x)+ (f(x +y) f(y) f (x)y) ν(dy), f C (2) (R). Theorem R Now, under appropriate assumptions one can show that ρ is a solution to the so called Zakai equation, i.e. we have Q a.s. for all t 0 t t ρ t (f) = π 0 (f)+ ρ s (Af)ds + ρ s (fh)dy s. 0 0 Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

16 Nonlinear Filtering with Lévy noise Nonlinear Filtering Nonlinear Filtering - the Lévy case Ahn H. and Feldman R. Optimal filtering of a Gaussian signal in the presence of Lévy noise. SIAM J. Appl. Math. 60 no. 1, (2000). Meyer-Brandis T. and Proske F. Explicit solution of a non-linear filtering problem for Lévy processes with application to finance. Appl. Math. Optim. 50 no. 2, (2004). Sornette D. and Ide K. The Kalman-Lévy filter. Phys. D 151 no. 2-4, (2001). C. Ceci and K. Colaneri. Nonlinear Filtering for Jump Diffusion Observations. Advances in Applied Probability. 44: , R. Frey, T. Schimdt and L. Xu. On Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations, Uni Leipzig, (2011). B. Grigelionis and R. Mikulevicius. Nonlinear filtering equations for stochastic processes with jumps. The Oxford handbook of nonlinear filtering, S. Popa and S. S. Sritharan. Nonlinear filtering of Itô-Lévy stochastic differential equations with continuous observations. Commun. Stoch. Anal. 3(3): , Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

17 Correlated noise Gliederung 1 Nonlinear Filtering - an introduction 2 Nonlinear Filtering with Lévy noise A Lévy Process Nonlinear Filtering 3 Correlated noise Copulas and Lévy Copulas 4 Nonlinear Filtering with correlated Lévy noise 5 Pseudo Differential Operators Lévy s symbol Analytic semigroups Lévy s symbol and pseudo differential operators 6 Future works Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

18 Correlated noise Copulas and Lévy Copulas Copulas Copulas are tools for modelling dependence of several random variables. Definition A d dimensional Copula C : [0,1] d R is a function which is a cummulative distribution function with uniform marginals. As commulative density functions are always increasing; The marginal in component i is obtained by setting u j = 1 for all j i and has to be uniformly distributed; Clearly, for a i b i the probability P(U 1 [a 1 ;b 1 ],...,[a d ;b d ]) must be nonnegative, which leads to the so called rectangle inequality 2 i 1 =1 2 ( 1) i 1+ +i d C(u 1,i1,...,u d,id ) 0. i d =1 Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

19 Correlated noise Copulas and Lévy Copulas Copulas Copulas are tools for modelling dependence of several random variables. Definition A d dimensional Copula C : [0,1] d R is a function which is a cummulative distribution function with uniform marginals. As commulative density functions are always increasing; The marginal in component i is obtained by setting u j = 1 for all j i and has to be uniformly distributed; Clearly, for a i b i the probability P(U 1 [a 1 ;b 1 ],...,[a d ;b d ]) must be nonnegative, which leads to the so called rectangle inequality 2 i 1 =1 2 ( 1) i 1+ +i d C(u 1,i1,...,u d,id ) 0. i d =1 Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

20 Correlated noise Copulas and Lévy Copulas Copulas Copulas are tools for modelling dependence of several random variables. Definition A d dimensional Copula C : [0,1] d R is a function which is a cummulative distribution function with uniform marginals. As commulative density functions are always increasing; The marginal in component i is obtained by setting u j = 1 for all j i and has to be uniformly distributed; Clearly, for a i b i the probability P(U 1 [a 1 ;b 1 ],...,[a d ;b d ]) must be nonnegative, which leads to the so called rectangle inequality 2 i 1 =1 2 ( 1) i 1+ +i d C(u 1,i1,...,u d,id ) 0. i d =1 Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

21 Correlated noise Copulas and Lévy Copulas Copulas Definition A d dimensional Copula C : [0,1] d [0,1] is a function which is a cummulative distribution function with uniform marginals. Theorem (Sklars Theorem (1959)) Consider a d dimensional cumulative distribution function F with marginals F 1,...,F d. Then there exists a copula C, such that F(x 1,...,x d ) = C (F 1 (x 1 ),...,F d (x d )), x = (x 1,...,x d ) R d. If all F i are continuous, then C is uniquely defined, otherwise C is only uniquely determined on RanF 1 RanF d. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

22 Correlated noise Copulas and Lévy Copulas Copulas - some examples the independent copula is given by C(u 1,u 2,...,u d ) = u 1 u 2 u d, 1 the independent copula 1 comonotonic copula the commonotonic copula is given by C(u 1,u 2 ) = min(u 1,u 2 ), Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

23 Correlated noise Copulas and Lévy Copulas Copulas - some examples the Clayton copula (1978) (θ ((0, )) ) C(u 1,u 2,...,u d ) = max [u1 θ +u2 θ + u θ d (d 1)] θ,0 1 ; Some properties: the parameter θ > 0 determines the dependence of the jump sizes. Larger values of θ indicate a stronger dependence, smaller values of θ indicate independence. That means, in the limit θ 0 the Clayton copula converges to the independence copula, and for θ the Clayton copula converges to the comonotonic copula. the Clayton copula turns out to have a lower tail dependence. This significate, roughly spoken, if X 1 and X 2 are defined by its marginal distribution and a Clayton copula, there is a strong tendency for X 2 to be small, if X 1 is small and vice versa. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

24 Correlated noise Copulas and Lévy Copulas Copulas - some examples the Clayton copula (1978) (θ( (0, )) ) C(u 1,u 2,...,u d ) = max [u1 θ +u2 θ + u θ d (d 1)] θ,0 1 ; 1 Clayton copula θ=1 1 Clayton copula θ=5 1 Clayton copula θ= Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

25 Correlated noise Copulas and Lévy Copulas Copulas - some examples the Gumbel copula (θ [1, )) C(u 1,u 2,...,u d ) ( = exp [ ) ( lnu 1 ) θ +( lnu 2 ) θ + +( lnu d ) θ]1 θ. Some properties: Again, the parameter θ > 1 determines the dependence of the jump sizes. In particular, if θ = 1 one obtains for the Gumbel copula the independence copula, and for θ converges the Gumbel copula to the comonotonic copula. The Gumbel copula turns out to have upper tail dependence. This significate, roughly spoken, if X 1 and X 2 are defined by its marginal distribution and a Gumbel copula, there is a strong tendency for X 2 to be extreme, if X 1 is extreme and vice versa. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

26 Correlated noise Copulas and Lévy Copulas Copulas - some examples the Gumbel copula (θ [1, )) C(u 1,u 2,...,u d ) ( = exp [ ) ( lnu 1 ) θ +( lnu 2 ) θ + +( lnu d ) θ]1 θ. 1 Gumbel copula, θ=2 1 Gumbel θ=5 1 Gumbel θ= Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

27 Correlated noise Copulas and Lévy Copulas Lévy Copulas Let ν 1, ν 2 be two Lévy measures in R +1, and U 1, and U 2 the corresponding tail integrals given by U i(x) = ν i(x, ), i = 1,2. Definition A 2 dimensional Copula H : [0, ] 2 R is a function which is a cumulative distribution function with uniform marginals. Theorem (Sklars Theorem for Lévy copula) (see the book of Cont and Tankov (2004)) Consider a 2 dimensional Lévy measure ν with marginal intensity functions ν 1 and ν 2. Then there exists a copula H, such that U(x 1,x 2) = H (U 1(x 1),U 2(x 2)), x = (x 1,x 2) R 2. Here U(x 1,x 2) = ν([x 1, ) [x 2, )). If all U i are continuous, then H is uniquely defined, otherwise H is only uniquely determined on RanU 1 RanU 2. 1 One can extend the definition to Lévy measures in R, however, for simplicity, we consider here in the talk only Lévy measures on R +. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

28 Correlated noise Copulas and Lévy Copulas Lévy Copulas If L 1 and L 2 are independent, the copula H is given by H (z 1,z 2 ) = z 1 1 z2 = +z 2 1 z1 =, z 1,z 2 R +, If L 1 and L 2 are completely dependent, the copula H is given by H (z 1,z 2 ) = min(z 1,z 2 ), z 1,z 2 R +. the Clayton copula (θ (0, )) H(z 1,z 2 ) = ( ) 1 z1 θ +z2 θ θ, z 1,z 2 0. For further Examples, we refer to the book of Cont and Tankov, articles of Tankov and Kallsen. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

29 Correlated noise Copulas and Lévy Copulas Setting of nonlinear Filtering with Lévy noise The noise: L = {L(t) = (L 1 (t),l 2 (t)) R 2 : t 0} be a two dimensional pure jump Lévy process with marginal intensities ν 1 and ν 2 ; L 0 be a general Lévy process and W(t) be a Wiener process; L, L 0 and W 2 are pairwise independent processes. The Processes: Let X be the signal process. In particular X solves the following SDE dx(t) = b(x(t))dt +dl 0 (t)+dl 1 (t), t > 0, X(0) = x 0. Let Y be the observable. In particular Y solves the following SDE dy(t) = h(x(t))dt +dl 2 (t)+dw(t), t > 0, Y(0) = y 0. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

30 Correlated noise Copulas and Lévy Copulas Setting of nonlinear Filtering with Lévy noise The noise: L = {L(t) = (L 1 (t),l 2 (t)) R 2 : t 0} be a two dimensional pure jump Lévy process with marginal intensities ν 1 and ν 2 ; L 0 be a general Lévy process and W(t) be a Wiener process; L, L 0 and W 2 are pairwise independent processes. The Processes: Let X be the signal process. In particular X solves the following SDE dx(t) = b(x(t))dt +dl 0 (t)+dl 1 (t), t > 0, X(0) = x 0. Let Y be the observable. In particular Y solves the following SDE dy(t) = h(x(t))dt +dl 2 (t)+dw(t), t > 0, Y(0) = y 0. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

31 Nonlinear Filtering with correlated Lévy noise Gliederung 1 Nonlinear Filtering - an introduction 2 Nonlinear Filtering with Lévy noise A Lévy Process Nonlinear Filtering 3 Correlated noise Copulas and Lévy Copulas 4 Nonlinear Filtering with correlated Lévy noise 5 Pseudo Differential Operators Lévy s symbol Analytic semigroups Lévy s symbol and pseudo differential operators 6 Future works Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

32 Nonlinear Filtering with correlated Lévy noise Nonlinear Filtering - the Brownian case Let W be nondegenerate. Let Q be a probability measure, equivalent to P such that Y is a Brownian motion over (A,F Y t,q). Let Z(t) = dp, t 0. dq Ft Let ρ = {ρ t : t 0} be the measure valued process with ρ t (f) = E Q [Z(t)f(X(t))]. Then π t (f) = EQ [Z(t)f(X(t))] E Q [Z(t)] = ρ t(f) ρ t (1). Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

33 Nonlinear Filtering with correlated Lévy noise Nonlinear Filtering - the Brownian case Let W be nondegenerate. Let Q be a probability measure, equivalent to P such that Y is a Brownian motion over (A,F Y t,q). Let Z(t) = dp, t 0. dq Ft Let ρ = {ρ t : t 0} be the measure valued process with ρ t (f) = E Q [Z(t)f(X(t))]. Then π t (f) = EQ [Z(t)f(X(t))] E Q [Z(t)] = ρ t(f) ρ t (1). Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

34 Nonlinear Filtering with correlated Lévy noise Nonlinear Filtering - the Lévy case Setting Remark ν 1 and ν 2 are Lévy measures, U 1, U 2 are the tail integrals of ν 1 and ν 2, respectively; H be a twice differentiable copula; In case the Lévy measure of L is infinite, the copula cannot be arbitrary and has to satisfy certain scaling properties. In particular, there exists a constant c > 0 such that lim γ H(γu,γv) H(γ,γ) = ch(u,v). Let for A B(R) ν 1,z2 (A) = A 2 H(u 1,u 2 ) u 1 u 2 u1 =U 1 (z 1 ) u 2 =U 2 (z 2 ) ν 1 (dz 1 ). Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

35 Nonlinear Filtering with correlated Lévy noise Nonlinear Filtering - the Lévy case Setting Remark ν 1 and ν 2 are Lévy measures, U 1, U 2 are the tail integrals of ν 1 and ν 2, respectively; H be a twice differentiable copula; In case the Lévy measure of L is infinite, the copula cannot be arbitrary and has to satisfy certain scaling properties. In particular, there exists a constant c > 0 such that lim γ H(γu,γv) H(γ,γ) = ch(u,v). Let for A B(R) ν 1,z2 (A) = A 2 H(u 1,u 2 ) u 1 u 2 u1 =U 1 (z 1 ) u 2 =U 2 (z 2 ) ν 1 (dz 1 ). Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

36 Nonlinear Filtering with correlated Lévy noise Nonlinear Filtering - the Lévy case Theorem Under the previous assumption, the unconditional density estimator ρ = {ρ t : t 0} solves the equation t ρ t,f = ρ 0,f + 0 ρ s,f dy c s t + ρ s,a 0 f ds + 0 t 0 R ρ s,b z2 f η 2 (dz 2,dt), f C (2) b (R), where η 2 is the to L 2 associated Poisson random measure, the operator B z2 is given for z 2 R\{0} by B z2 f(x) = [f(x +z 1 ) f(x) z 1 f (x)]ν 1,z2 (dz 1 ), f C (2) b (R). R\{0} Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

37 Nonlinear Filtering with correlated Lévy noise Nonlinear Filtering - the Lévy case In practice one is often interested in entities like P(X(t) > a), a R. This corresponds to a function f = 1 (,a]. One has to solve in another sense. dρ t = A 0 ρ tds +ρ t dy c t + R B z 2 ρ t η 2 (dz 2,dt) In the setting of semigroups, the variation of constant formula gives ρ t = T A 0 (t)ρ 0 ds + t 0 T A a (t s)ρ 0 s dys c + t 0 R T A (t s)b 0 z 2 ρ s η 2 (dz 2,dt) a Here T A 0 denotes the semigroup generated by A 0. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

38 Nonlinear Filtering with correlated Lévy noise Nonlinear Filtering - the Lévy case In practice one is often interested in entities like P(X(t) > a), a R. This corresponds to a function f = 1 (,a]. One has to solve in another sense. dρ t = A 0 ρ tds +ρ t dy c t + R B z 2 ρ t η 2 (dz 2,dt) In the setting of semigroups, the variation of constant formula gives ρ t = T A 0 (t)ρ 0 ds + t 0 T A a (t s)ρ 0 s dys c + t 0 R T A (t s)b 0 z 2 ρ s η 2 (dz 2,dt) a Here T A 0 denotes the semigroup generated by A 0. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

39 Nonlinear Filtering with correlated Lévy noise Nonlinear Filtering - the Lévy case In practice one is often interested in entities like P(X(t) > a), a R. This corresponds to a function f = 1 (,a]. One has to solve in another sense. dρ t = A 0 ρ tds +ρ t dy c t + R B z 2 ρ t η 2 (dz 2,dt) In the setting of semigroups, the variation of constant formula gives ρ t = T A 0 (t)ρ 0 ds + t 0 T A a (t s)ρ 0 s dys c + t 0 R T A (t s)b 0 z 2 ρ s η 2 (dz 2,dt) a Here T A 0 denotes the semigroup generated by A 0. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

40 Nonlinear Filtering with correlated Lévy noise Nonlinear Filtering - the Lévy case In practice one is often interested in entities like P(X(t) > a), a R. This corresponds to a function f = 1 (,a]. One has to solve in another sense. dρ t = A 0 ρ tds +ρ t dy c t + R B z 2 ρ t η 2 (dz 2,dt) In the setting of semigroups, the variation of constant formula gives ρ t = T A 0 (t)ρ 0 ds + t 0 T A a (t s)ρ 0 s dys c + t 0 R T A (t s)b 0 z 2 ρ s η 2 (dz 2,dt) a Here T A 0 denotes the semigroup generated by A 0. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

41 Pseudo Differential Operators Gliederung 1 Nonlinear Filtering - an introduction 2 Nonlinear Filtering with Lévy noise A Lévy Process Nonlinear Filtering 3 Correlated noise Copulas and Lévy Copulas 4 Nonlinear Filtering with correlated Lévy noise 5 Pseudo Differential Operators Lévy s symbol Analytic semigroups Lévy s symbol and pseudo differential operators 6 Future works Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

42 Pseudo Differential Operators Lévy s symbol Lévy s symbol and pseudo differential operators Let B = {B(t) : t 0} be a Brownian motion. Then the Markovian semigroup (P t ) t 0 defined by has as infinitesimal generator P t f(x) := E[f(B t +x)] 1 A 0 f := lim h 0 h (P h P 0 ) f = 2 x2f, f C(2) 2 (R). Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

43 Pseudo Differential Operators Lévy s symbol Lévy s symbol and pseudo differential operators Let L = {L(t) : t 0} be a Lévy process. Then the Markovian semigroup (P t ) t 0 defined by P t f(x) := E[f(L(t)+x)] has as infinitesimal generater 1 A 0 f := lim h 0 h (P h P 0 ) f = with ψ(ξ) = 1 t ln(eeiξl(t) ) = R\{0} R e iξx ψ(ξ)ˆf(ξ)dξ, f H 2,2 (R), t > 0. (1 cos(yξ)) ν(dξ), ξ R, t > 0. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

44 Pseudo Differential Operators Lévy s symbol The SPDE which have to be solved Fix ρ R. Let us assume that there exists some ρ 0 > 1 such that u 0 H ρ 0 2 (Rd ); A has symbol ψ with lower index α 0 ; for δ f < α 0 there exists a C f > 0 with f(x) f(y) ρ+δ H f C f x y ρ, x,y H Hρ (Rd ) for δ Σ < α 0 2 there exists a C Σ > 0 with Σ(x) Σ(y) 2 H ρ+δ C Σ Σ x y 2 H ρ, x,y H ρ 2 (Rd ); 2 2 for δ G < α 0 p there exists a C G > 0 with z 1 G(r,x,z) G(r,x,z) p ν(dz) C G x y p H ρ+β+ 2 (R d ) H ρ 2 The equation: { du(t) = Au(t)dt +Σ(u(t))dB(t)+ G(t,u(t),z)η(dz,dt), R u(0) = u 0. we are interested in a solution u such that for ρ = ρ 0 = 1 2. u D((0,T],H ρ 2 (Rd )) D([0,T];H ρ 0 2 (Rd )),, y,x H ρ 2 (Rd ). Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

45 Pseudo Differential Operators Lévy s symbol Lévy s symbol and pseudo differential operators Let H s (R) := { ( ) } f : R R : (1+x 2 ) s 2 2 ˆf 2 (x) dx <. R Let (P t ) t 0 the Markovian semigroup of the Brownian motion. Then we have In particular, P t f H 2 C t f H 0, f H0 (R). P t f H 0 C t f H 0, f H 0 (R). Question: Does some similar inequality holds for the infinitesimal generator of a Lévy process? Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

46 Pseudo Differential Operators Lévy s symbol Lévy s symbol and pseudo differential operators Let H s (R) := { ( ) } f : R R : (1+x 2 ) s 2 2 ˆf 2 (x) dx <. R Let (P t ) t 0 the Markovian semigroup of the Brownian motion. Then we have In particular, P t f H 2 C t f H 0, f H0 (R). P t f H 0 C t f H 0, f H 0 (R). Question: Does some similar inequality holds for the infinitesimal generator of a Lévy process? Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

47 Pseudo Differential Operators Analytic semigroups Analytic semigroups Definition Let X be a Banach space and let A be the generator of a degenerate analytic C 0 -semigroup on X. We say that A is of type (ω,θ,k), where ω R, θ (0, π 2 ) and K > 0, if ω +Σπ 2 +θ ρ(a) and λ ω (λ+a) 1 L(X) K for all λ ω +Σπ 2 +θ. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

48 Pseudo Differential Operators Analytic semigroups Analytic semigroups Proposition Let X be a Banach space. Let A 0 be the generator of a degenerate analytic C 0 -semigroup T on X and let B be a possibly unbounded operator acting on X. Suppose A 0 is of type (ω,θ,k) for some ω R,θ (0, π 2 ) and K > 0. Suppose there exist an ǫ [0,1) and a constant C(A 0,B) such that for all λ ω +Σπ 2 +θ one has: Then for all t > 0 we have: (λ A 0 ) 1 B L(X,X) C(A 0,B) λ ω ε 1. T 0 (t)b L(X,X) 2Γ(ε)[sinθ] 1 C(A 0,B)e ωt t ε. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

49 Pseudo Differential Operators Analytic semigroups Analytic semigroups Idea of the proof: Fix θ (0,θ). Use the following identity (see Pazy) T 0 (s) = (2πi) 1 e λs (λ A 0 ) 1 dλ; where Γ θ is the path composed from the two rays re i(π 2 +θ ) and re i(π 2 +θ ), 0 r <, and is oriented such that Imλ increases along Γ θ. Evaluate carefully the integral. Γ θ Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

50 Pseudo Differential Operators Lévy s symbol and pseudo differential operators Lévy s symbol and pseudo differential operators Definition We call a symbol of type (ω,θ), ω R, θ (0, π 2 ), iff Rg(ψ) C\ω +Σ θ+ π 2. Remark For λ C\Rg(ψ) we have (λ A 0 ) 1 1 dist(k,λ). Moreover, the set K = Rg(ψ) equals the spectrum of A 0. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

51 Pseudo Differential Operators Lévy s symbol and pseudo differential operators Lévy s symbol and pseudo differential operators Definition We call a symbol of type (ω,θ), ω R, θ (0, π 2 ), iff Rg(ψ) C\ω +Σ θ+ π 2. Remark For λ C\Rg(ψ) we have (λ A 0 ) 1 1 dist(k,λ). Moreover, the set K = Rg(ψ) equals the spectrum of A 0. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

52 Pseudo Differential Operators Lévy s symbol and pseudo differential operators Lévy s symbol and pseudo differential operators Definition Let L be a Lévy process with symbol ψ. Then the upper index β + and lower index β are defined by { } β + ψ(ξ) (ψ) := inf lim sup λ>0 ξ ξ λ = 0 and { } β ψ(ξ) (ψ) := inf lim inf λ>0 ξ ξ λ = 0. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

53 Pseudo Differential Operators Lévy s symbol and pseudo differential operators Lévy s symbol and pseudo differential operators Corollary Then Given an infinitesimal operators of a Lévy process A 0 with symbol ψ such that ψ is of type (ω,θ); has lower index α An pseudodifferential operator B with symbol ϕ such that ϕ has upper index β +. T A0 (t)bx H s (R) C β + sinθ t α x H s (R), x L 2 (R). (1) where T A0 = (T A0 (t)) t 0 is the Markovian semigroup associated to the Lévy process with infinitesimal generator A 0. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

54 Pseudo Differential Operators Lévy s symbol and pseudo differential operators Lévy s symbol and pseudo differential operators Fix α (0,2). L be a symmetric α stable process without drift. Then ψ(ξ) = ξ α, L be a tempered α stable process, α < 1, then ν(a) = A R + \0 z α 1 e ρ z dz. and ψ(ξ) Γ( α) (ρ iξ) α ρ α. L be a tempered α stable process, then ν(a) = A\0 z α 1 e ρ z dz. and ψ(ξ) Γ( α)c(ρ) ξ α. L be the Meixner process, then for ( m R, δ,a > 0, b ( π,π). log cosh ψ m,δ,a,b (ξ) = imξ +2δ Here the upper and lower index is 1. ( aξ ib 2 ) logcos ( b 2) ), ξ R, L be the normal inverse Gaussian ( process; Then a2 ψ NIG (ξ) = imξ +δ (b +iξ) ) a 2 b 2, ξ R, where m R, δ > 0, 0 < b < a. The upper and lower index is 1. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

55 Pseudo Differential Operators Lévy s symbol and pseudo differential operators Nonlinear Filtering - the Lévy case Setting ν 1 and ν 2 are tempered β stable distribution, H be the Clayton copula with index β, i.e. L 0 be an α 0 stable Lévy process, H(z 1,z 2 ) = ( z1 θ +z2 θ ) 1 θ, z 1,z 2 0. µ X 0 has a density and EX 2 0 < ; Let us assume that there exist numbers u [1,2) and p (β,2] such that p < u α, (βθ +u)p > β. Then for s 1 < p u there exists a unique solution in Hmin(s1, 1 2 ) of equation dρ t = A 0 ρ t dt +ρ t dy c t + R B z 2 ρ t η 2 (dz 2,dt), ρ 0 = µ. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

56 Pseudo Differential Operators Lévy s symbol and pseudo differential operators Nonlinear Filtering - the Lévy case Setting ν 1 and ν 2 are tempered β stable distribution, H be the Clayton copula with index β, i.e. L 0 be an α 0 stable Lévy process, H(z 1,z 2 ) = ( z1 θ +z2 θ ) 1 θ, z 1,z 2 0. µ X 0 has a density and EX 2 0 < ; Let us assume that there exist numbers u [1,2) and p (β,2] such that p < u α, (βθ +u)p > β. Then for s 1 < p u there exists a unique solution in Hmin(s1, 1 2 ) of equation dρ t = A 0 ρ t dt +ρ t dy c t + R B z 2 ρ t η 2 (dz 2,dt), ρ 0 = µ. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

57 Pseudo Differential Operators Lévy s symbol and pseudo differential operators Nonlinear Filtering - the Lévy case Theorem Let us assume that X 0 has distribution function F, which has a L 2 integrable density with respect to the Lebesgue measure; the symbol ψ 0 associated to L 0 has lower index α 0 > 1, g H δ 2(R) C (2) b (R) with δ > 1 α 0 2 ; the symbol f Bz associated to the operator B z, has upper index β +, there exists some function k : R + 0 R+ 0, k(0) = 0, continuous at 0, such that ψ limsup z2 (ξ) ξ k(z ξ 2), z β+ 2 R. If there exists a number p (1,2] such that β + < 1 and α p z k(z2) p ν 2(dz 2) <, then there exists a unique family of probabilities kernels π = {π t : t 0} such that π t(f) = E[f(X t) Y t], f B b (R) < c. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

58 Pseudo Differential Operators Lévy s symbol and pseudo differential operators A result of Getoor remark The key step for the existence of π is a lemma of the existence of probability kernels by Getoor (1975). Let ( Ω, A) be a measurable space. Now, the lemma says, that if a operator T : B b ((R,B(R)) B b (( Ω,A)) is linear a.e. and positive a.e.; for any sequence of functions {f n : n IN} and f B b (R), 0 f n f implies Tf n f then there exists a bounded kernel µ(, ) from ( Ω,A) to (R,B(R)) such that Tf(ω) = R f(u)µ(ω,du), for all f B b(r) and ω Ω. Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

59 Future works Gliederung 1 Nonlinear Filtering - an introduction 2 Nonlinear Filtering with Lévy noise A Lévy Process Nonlinear Filtering 3 Correlated noise Copulas and Lévy Copulas 4 Nonlinear Filtering with correlated Lévy noise 5 Pseudo Differential Operators Lévy s symbol Analytic semigroups Lévy s symbol and pseudo differential operators 6 Future works Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

60 Future works Remaining work for the future Extension to L p spaces, can it be formulated with initial condition a measure? Get rid of W(t) in the observable process innovation approach; What happens if Y is only known at a grid; Approximations of ρ, respectively, π;... Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46

61 Future works Thank you for the attention Hausenblas (Leoben) Nonlinear Filtering with Lévy noise April 10, / 46