Fractional Pearson diffusions

Transcription

1 Fractional Pearson diffusions N.N. Leonenko, School of Mathematics, Cardiff University, UK joint work with Alla Sikorskii (Michigan State University, US), Ivan Papić and Nenad Šuvak (J.J. Strossmayer University of Osijek, Croatia) Fractional Pearson diffusions 1 / 38

2 Fractional Pearson diffusions (FPD) Outline Fractional Pearson diffusions - Spectral representation of the transition densities - Stationary (limiting) distributions - Strong solutions of time-fractional Kolmogorov backward equation for fractional Fisher-Snedecor diffusion - Correlation structure Continuous time random walks (CTRWs) and fractional Pearson diffusions - Bernoulli-Laplace urn scheme model: fractional Ornstein-Uhlenbeck process - Wright-Fisher urn scheme model: fractional Cox-Ingersoll-Ross and Jacobi diffusion Fractional Pearson diffusions 2 / 38

3 Fractional Pearson diffusions (FPD) Fractional diffusion definition X 1 = (X 1(t), t 0) Markovian diffusion with transition densities p 1(x, t; y) D = (D t, t 0) standard stable subordinator independent of the diffusion X 1, with the Laplace transform E[e sdt ] = exp ( ts α ), s 0, 0 < α < 1 E t = inf {x > 0: D x > t} - inverse of the α-stable subordinator D (E t, t 0) non-markovian and non-decreasing, for every t random variable E t has a density f t( ) with the Laplace transform E[e set ] = 0 e sx f t(x)dx = E α( st α ), where E α( st α ) is the Mittag-Leffler function E α( st α ( st α ) j ) = Γ(1 + αj) fractional diffusion non-markovian process defined via time-change of the diffusion X 1(t) by the inverse E t of the α-stable subordinator, i.e. j=0 X α(t) = X 1 (E t), t 0 Fractional Pearson diffusions 3 / 38 (1)

4 Fractional Pearson diffusions (FPD) Fractional diffusions applications hydrology modeling sticking and trapping of contaminant particles in a porous medium (Meerschaert et al., 2003) or a river flow (Chakraborty et al., 2009) finance modeling delays between trades (Scalas, 2006) statistical physics fractional time derivative appears in the equation for a continuous time random walk limit and reflects random waiting times between particle jumps (Meerschaert, 2004) Fractional Pearson diffusions 4 / 38

5 Pearson diffusions (PD) Fractional Pearson diffusion definition fractional Pearson diffusion non-markovian process (X α(t), t 0) = (X 1(E t), t 0), where (X 1(t), t 0) is the Pearson diffusion Pearson diffusion a unique strong solution (Øksendal, Theorem 5.2.1) of the SDE dx t = µ(x t)dt + σ(x t)dw t, t 0 (2) with polynomial infinitesimal parameters µ(x) = a 0 + a 1x, σ(x) = 2b(x) = 2(b 2x 2 + b 1x + b 0) p(x) the stationary density of the diffusion (2) belongs to the Pearson family of continuous distributions µ(x) and b(x) are related to the polynomials in the Pearson differential equation p (x) p(x) = (a1 2b2)x + (a0 b1) b 2x 2 + b 1x + b 0 Fractional Pearson diffusions 5 / 38

6 Pearson diffusions (PD) Pearson diffusions - classification six subfamilies of Pearson diffusions according to the degree of polynomial b(x) and, in the quadratic case, to the sign of b 2 and the sign of its discriminant : constant b(x) OU process (Gaussian stationary distribution) linear b(x) CIR process (gamma stationary distribution) quadratic b(x) with b 2 < 0 Jacobi diffusion (beta stationary distribution) quadratic b(x) with b 2 > 0 and > 0 Fisher-Snedecor (FS) diffusion quadratic b(x) with b 2 > 0 and = 0 reciprocal gamma (RG) diffusion quadratic b(x) with b 2 > 0 and < 0 Student diffusion important references: Kolmogorov (1931), Wong (1964), Forman & Sørensen (2008), Avram et al. (2013a, 2013b) Fractional Pearson diffusions 6 / 38

7 Pearson diffusions (PD) Pearson and fractional Pearson diffusions sample paths Sample paths of fractional and non-fractional RG and FS diffusions with parameters γ = 10, β = 20, θ = 0.01 and α = 0.7 based on points with inital state X 0 = 0.4 fractional reciprocal gamma diffusion fractional Fisher Snedecor diffusion reciprocal gamma diffusion Fisher Snedecor diffusion Fractional Pearson diffusions 7 / 38

8 Transition densities of PD and FPD Non-heavy-tailed Pearson diffusions OU, CIR and Jacobi diffusions transition densities p(x, t; y) = P(Xt x X0 = y) x closed-form expressions S. Karlin and H.M. Taylor (1981) A Second Course in Stochastic Processes, Academic Press, New York spectral representations of transition densities given in terms of the pure-point spectrum of the infinitesimal generator and the corresponding eigenfunctions (Hermite, Laguerre and Jacobi polynomials, respectively) spectral analysis overview of existing results given in N.N. Leonenko, M.M. Meerschaert and A. Sikorskii (2013) Fractional Pearson diffusions, Journal of Mathematical Analysis and Applications, 403(2): Fractional Pearson diffusions 8 / 38

9 Transition densities of PD and FPD Heavy-tailed Pearson diffusions reciprocal gamma, Fisher-Snedecor and Student diffusions transition densities representable in terms of the spectrum of the corresponding infinitesimal generator and related functions infinitesimal generator of heavy-tailed Pearson diffusion Gf (x) = µ(x)f (x) + σ2 (x) f (x) (3) 2 µ(x) - linear; σ 2 (x) - quadratic, with positive leading coefficient spectrum of the Sturm-Liouville operator ( G) discrete spectrum σ d [0, Λ - finite set of eigenvalues eigenfunctions are finite systems of orthogonal polynomials (Bessel, Fisher-Snedecor and Romanovski polynomials, respectively) absolutely continuous spectrum σ ac(g) in Λ, functions related to the σ ac(g) - confluent (RG) and Gauss (FS, Student) hypergeometric functions Student diffusion heavy-tailed Pearson diffusions with σ ac(g) of multiplicity two (still not completely resolved) Fractional Pearson diffusions 9 / 38

10 Transition densities of PD and FPD Fisher-Snedecor (FS) diffusion FS diffusion SDE dx 1(t) = θ ( X 1(t) β ) dt + β 2 where t 0 and θ > 0 (autocorrelation parameter) stationary density 4θ X1(t) (γx1(t) + β) dw (t), (4) γ(β 2) fs(x) = β β 2 B ( γ, ) β 2 2 (γx) γ 2 1 (γx + β) γ 2 + β 2 γ I 0, (x), γ > 0, β > 2 transition density spectral representation p 1(x, t; y) = p d (x, t; y) + p c(x, t; y) (5) derived in F. Avram, N.N. Leonenko and N. Šuvak. (2013) Spectral representation of transition density of Fisher-Snedecor diffusion, Stochastics, 85(2): Fractional Pearson diffusions 10 / 38

11 Transition densities of PD and FPD FS diffusion discrete part of transition density transition density discrete part eigenvalues of the SL operator ( G) λ n = β 4 p d (x, t; y) = fs(x) e λnt F n(y) F n(x) (6) n=0 θ n(β 2n), n {0, 1,..., β/4 }, β > 2 (7) β 2 eigenfunctions of the SL operator ( G) Fisher-Snedecor polynomials F n(x) = K n x 1 γ 2 (γx + β) γ 2 + β 2 d n { } 2 n x γ dx n 2 +n 1 (γx + β) n γ 2 β 2 (8) Fractional Pearson diffusions 11 / 38

12 Transition densities of PD and FPD FS diffusion continuous part of transition density transition density continuous part p c(x, t; y) = fs(x) 1 π Λ= θβ2 8(β 2) e λt a(λ) f 1(y, λ)f 1(x, λ) dλ (9) function f 1 solution of the SL equation Gf (x) = λf (x) for λ > Λ ( f 1(x, λ) = 2F 1 β 4 + ik(λ), β 4 ik(λ); γ 2 ; γ ) β x, (10) normalization constant a(λ) = k(λ) B 1 2 k(λ) = i β 2 16 λ(β 2) 2θ ( γ, ) ( β 2 2 Γ β + ik(λ)) Γ ( γ + β + ik(λ)) Γ ( ) γ 2 Γ (1 + 2ik(λ)) (11) Fractional Pearson diffusions 12 / 38

13 Transition densities of PD and FPD Fractional FS diffusion transition density fractional FS diffusion (X α(t), t 0), where X α(t) = X 1(E t), t 0 (X 1(t), t 0) FS diffusion given by the SDE (4) (E t, t 0), where E t = inf {x > 0: D x > t} inverse of the α-stable subordinator, 0 < α < 1 transition density defined as P(X α(t) B X α(0) = y) = for any Borel set B from B (0, ) B p α(x, t; y)dx (12) Fractional Pearson diffusions 13 / 38

14 Transition densities of PD and FPD Fractional FS diffusion transition density transitions density p = p α(x, t; y) of the FS diffusion satisfies the following equations: fractional forward (Fokker-Planck) equation α p t α = 2 ( µ(x)p) + x x 2 ( σ 2 (x) 2 ) p with the point-source initial condition p(x, 0; y) = δ(x y) fractional backward equation α p t α p = µ(y) y + σ2 (y) 2 p 2 y 2 α / t α Caputo fractional derivative of order 0 < α < 1 d α f (x) dx α = 1 Γ(1 α) 0 d dx f (x y)y α dy Fractional Pearson diffusions 14 / 38

15 Transition densities of PD and FPD Fractional FS diffusion transition density Theorem The transition density of fractional FS diffusion is given by β 4 p α(x, t; y) = fs(x) F n(y) F n(x) E α( λ nt α )+ (13) + fs(x) π θβ 2 8(β 2) n=0 E α( λt α ) a(λ) f 1(y, λ), f 1(x, λ) dλ, where F n are FS polynomials given by (8), f 1 is the solution of the non-fractional SL problem given by (10), a(λ) is given by (11) and E α( λt α ) is the Mittag-Leffler function given by (1). detailed proof could be found in N.N. Leonenko, I. Papić, A. Sikorskii and N. Šuvak. (2017) Heavy-tailed fractional Pearson diffusions, Stochastic Processes and their Applications, accepted for publication Fractional Pearson diffusions 15 / 38

16 Transition densities of PD and FPD Fractional FS diffusion transition density, sketch of the proof P(X α(t) B X α(0) = y) = P(X 1 (τ) B X 1 (0) = y) f t (τ) dτ 0 = p 1 (x, τ; y) f t (τ) dx dτ 0 = B (p d (x, τ; y) + p c (x, τ; y)) f t (τ) dτ dx = (14) B 0 β 4 = fs(x) F n(y)f n(x)e λnτ f t (τ)dτ + 1 e λτ f t (τ)a(λ)f 1 (y, λ)f 1 (x, λ)dλdτ B 0 n=0 π dx 0 Λ β 4 = fs(x) F n(y) F n(x) E α( λ nt α ) + 1 E α( λt α ) a(λ) f 1 (y, λ) f 1 (x, λ) dλ B n=0 π dx (15) Λ change of the order of integration in (14) follows from the non-negativity of p 1 and f t (Fubini-Tonelli theorem) change of the order of integration in (15) follows by the Fubini theorem since Λ e λτ f t (τ) a(λ) f 1 (y, λ) f 1 (x, λ) dτ dλ < 0 (for bounds regarding the Gauss hypergeometric functions we refer to Erdelyi, Equation 17, page 77) Fractional Pearson diffusions 16 / 38

17 Transition densities of PD and FPD Fractional FS diffusion - limiting density Theorem Let X α(t) be the fractional FS diffusion and let p α(x, t) be the density of X α(t). Assume that X α(0) has a twice continuously differentiable density f that vanishes at infinity. Then p α(x, t) fs(x) as t, where fs(x) is the stationary density of the non-fractional FS diffusion. detailed proof - Leonenko et al. (2017) Fractional Pearson diffusions 17 / 38

18 Transition densities of PD and FPD Fractional FS diffusion - strong solutions of time-fractional Kolmogorov backward equation Theorem For any g from the domain of the generator G, a strong solution of the fractional Cauchy problem α q(y, t) t α = Gq(y, t), q(y, 0) = g(y), (16) where α t α is the Caputo fractional derivative of order 0 < α < 1, is given by q(t; y) = 0 p α(x, t; y)g(x) dx, (17) where the transition density p α(x, t; y) of the fractional FS diffusion is given by (13). detailed proof - Leonenko et al. (2017) Fractional Pearson diffusions 18 / 38

19 Transition densities of PD and FPD Correlation structure of the fractional Pearson diffusion Theorem Let us assume that X α(0) has the probability density m, where m( ) is the stationary density of the corresponding Pearson diffusion. Then for t s > 0. Corr [X α(t), X α(s)] = E α( θt α ) + θαtα Γ(1 + α) s/t 0 E α( θt α (1 z) α ) z 1 α dz (18) N.N. Leonenko, M.M. Meerschaert and A. Sikorskii (2013) Correlation structure of fractional Pearson diffusions, Computers and Mathematics with Applications, 66(5): fractional diffusion: diffusion Corr[X α(t), X α(s)] = ( ) 1 1 t α Γ(1 α) θ + s α (1 + o(1)), t Γ(1 + α) Corr[X 1(t), X 1(s)] = e θ(t s) Fractional Pearson diffusions 19 / 38

20 CTRWs and fractional Pearson diffusions Bernoulli-Laplace urn scheme - historical roots Bernoulli-Laplace urn scheme P. Laplace (1812) Théorie Analytique des Probabilités, Ve. Courcier, Paris two urns, A and B, each contains n of total 2n balls n balls are black and n are white from each urn we randomly choose one ball which is then placed in the opposite urn number of white balls in urn A after r draws? (Z r (n), r N 0), for each n N - Markov chain with state space {0, 1, 2,..., n} transition probabilities: p x,x+1 = ( 1 x ) 2, px,x = 2 x n n ( 1 x ) ( x ) 2, p x,x 1 =, 0 otherwise (19) n n Fractional Pearson diffusions 20 / 38

21 CTRWs and fractional Pearson diffusions Bernoulli-Laplace urn scheme - historical roots Laplace was interested in finding heat kernel z x,r - probability that urn A contains x white balls after r draws. ( ) 2 x + 1 z x, r+1 = z x+1, r + 2 x ( 1 x ) ( z x, r + 1 x 1 ) z x 1, r (20) n n n n in order to approximate the solution of the equation (20), Laplace introduced the following transformations: x = 1 2 (n + µ n), r = nr Fractional Pearson diffusions 21 / 38

22 CTRWs and fractional Pearson diffusions Bernoulli-Laplace urn scheme - historical roots z x+1, r z x, r + zx, r x z x 1, r z x, r zx, r x z x, r+1 z x, r + zx, r r z x, r 2 x z x, r 2 x 2 µ = 2 n, z x, r = U(µ, r ) z x+1, r = U(µ + µ, r ) U + µ U µ U 2 ( µ)2 µ 2 z x 1, r = U(µ µ, r ) U µ U µ U 2 ( µ)2 µ 2 z x, r+1 = U(µ, r + 1 n ) U + 1 n U r Fractional Pearson diffusions 22 / 38

23 CTRWs and fractional Pearson diffusions Bernoulli-Laplace urn scheme - historical roots U = r µ ( 2µU) U (2U) = 2U + 2µ µ 2 µ + 2 U (21) µ 2 special case of Kolmogorov forward equation (Fokker-Planck equation) for p(x, t) t = x (µ(x)p(x, t)) ( ) σ 2 (x)p(x, t) 2 x 2 µ(x) = 2x, σ 2 (x) = 2 which are the infinitesimal parameters of specifically parametrized Ornstein-Uhlenbeck process. Fractional Pearson diffusions 23 / 38

24 CTRWs and fractional Pearson diffusions Theorem Let (Y (n), n N) be a sequence of discrete-time Markov chains on S with transition operators (U n, n N). Consider a Feller process X on S generator A. Fix a core D for the generator A, and assume that (h n, n N) is the sequence of positive reals tending to zero as n. Let A n = hn 1 (U n I ), Xt n = Y n t/h n. Then the following statements are equivalent: a) If f D, there exist some f n Dom(A n) with f n f and A nf n Af as n b) if X (n) 0 X 0 in S, then X n X in the Skorokhod space D(S) with the J 1 topology. Teorem 19.28, O. Kallenberg (2002) Foundations of Modern Probability, 2nd edition, Springer Series in Statistics, Probability and its applications, Springer Fractional Pearson diffusions 24 / 38

25 CTRWs and fractional Pearson diffusions Bernoulli-Laplace urn scheme - OU diffusion space variable transformation H (n) r = 1 a n time variable transformation ( 2Z r (n) n b n X (n) t ), a, b R, a 0 (22) := H (n) θ nt, θ > 0 (23) 2 Let X = (X t, t 0) be the OU diffusion, i.e. the solution of the SDE ( dx t = θ X t + b ) θ dt + dw (t), t 0, (24) a a2 where (W (t), t 0) is the standard Brownian motion. infinitesimal generator: Af (x) = θ core of the generator: C 3 c (R) ( x + b ) f (x) + 1 a 2 θ a f (x) 2 Fractional Pearson diffusions 25 / 38

26 CTRWs and fractional Pearson diffusions Bernoulli-Laplace urn scheme - OU diffusion Theorem Let (H (n) r (X (n) t, r N 0), for each n N, be the Markov chain defined by (22). Let, t 0) for each n N, be the time-changed process (23). Then X n X, n in Skorohod space D(R), where X = (X t, t 0) is the Ornstein-Uhlenbeck process defined by (24). Fractional Pearson diffusions 26 / 38

27 CTRWs and fractional Pearson diffusions Wright-Fisher urn scheme - Jacobi and CIR diffusion Wright-Fisher urn scheme S. Karlin and H.M. Taylor (1981) A Second Course in Stochastic Processes, Academic Press, New York describes gene mutations (in some genetic pool) over time, strongly influencing selection in the corresponding population population has n individuals, where in the current generation, i individuals are of type A and n i are of type a Once born, individual of A-type can mutate in a-type with probability α and individual of a-type can mutate in A-type with probability β survival ability of each type is modeled by parameter parametar s: the ratio of A-types over a-types is equal to 1 + s fraction of mature A-types in population before reproduction is p i = (1 + s) [i (1 α) + (n i) β] (1 + s) [i (1 α) + (n i) β] + [iα + (n i) (1 β)] (25) Fractional Pearson diffusions 27 / 38

28 CTRWs and fractional Pearson diffusions Wright-Fisher urn scheme - Jacobi and CIR diffusion Wright-Fisher urn scheme Assumption of the model: the composition of the next generation is determined through n binomial trials, where the probability of producing an A-type in each trial is p i, where p i is given via (25) the number of A-types in population over time is described by Markov chain (G r (n), r N 0) with state space {0, 1, 2,..., n} and transition probabilities p ij = ( n j ) p j i (1 p i) n j (26) parameters of the model: α, β i s Fractional Pearson diffusions 28 / 38

29 CTRWs and fractional Pearson diffusions Wright-Fisher urn scheme - Jacobi diffusion parameters of the model: space variable transformation time variable transformation α = a n, β = b, s = 0, a, b > 0 n Y (n) t H (n) r = 1 n G (n) r (27) := H (n) θnt, θ > 0 (28) Let Y = (Y t, t 0) be the Jacobi diffusion, i.e. solution of the SDE ( dy t = θ(a + b) Y t b a + b where (W (t), t 0) is standard Brownian motion infinitesimal generator: ) dt + θy t(1 Y t)dw (t), t 0, (29) Af (x) = θ( y(a + b) + b)f (y) θy(1 y)f (y) core of the generator: C 3 c ([0, 1]) Fractional Pearson diffusions 29 / 38

30 CTRWs and fractional Pearson diffusions Wright-Fisher urn scheme - Jacobi diffusion Theorem Let (H (n) r (Y (n) t, r N 0), for each n N, be the Markov chain defined by (27). Let, t 0) for each n N, be the time-changed process (28). Then Y n Y, n in Skorohod space D([0, 1]), where Y = (Y t, t 0) is the Jacobi diffusion defined by (29). Fractional Pearson diffusions 30 / 38

31 CTRWs and fractional Pearson diffusions Wright-Fisher urn scheme - CIR diffusion parameters of the model: α = a n, β = b, 0 < d < 1, a, b > 0, s = 0 d n space variable transformation time variable transformation Z (n) t H (n) r = G r (n) (30) n d := H (n) θ a nd t, θ > 0 (31) Let Z = (Z t, t 0) be the CIR diffusion, i.e. the solution of the SDE ( dz t = θ Z t b ) θ dt + ZtdWt, t 0, θ > 0, a > 0, b > 0, (32) a a where (W (t), t 0) is standard Brownian motion infinitesimal generator: ( Af (x) = θ z b ) f (z) + 1 θ a 2 a zf (z) core of the generator: C 3 c ([0, )) Fractional Pearson diffusions 31 / 38

32 CTRWs and fractional Pearson diffusions Wright-Fisher urn scheme - CIR diffusion Theorem Let (H (n) r (Z (n) t, r N 0), for each n N, be the Markov chain defined by (30). Let, t 0) for each n N, be the time-changed process (31). Then Z n Z, n in Skorohod space D(R + ), where Z = (Z t, t 0) is the CIR diffusion defined by (32). Fractional Pearson diffusions 32 / 38

33 CTRWs and fractional Pearson diffusions Fractional OU, Jacobi and CIR diffusions as the correlated CTRWs limits T 0 = 0, T r = G G r, where G r 0 are iid waiting times between particle jumps that are independent of the Markov chain (H (n) r, r N 0) G 1 is in the domain of attraction of the β-stable distribution with index 0 < β < 1 the waiting time of the Markov chain (H r (n), r N 0) until its r-th move is described by G r is the number of jumps up to time t 0 N t = max{r 0: T r t} (33) ( ) H (n) (N t), t 0 is the correlated continuous time random walks (CTRW) and describes the state of the Markov chain at time t 0 in Skorohod space D(R + ) with J 1 topology n 1 β T ( nt ) D t, n (34) Fractional Pearson diffusions 33 / 38

34 CTRWs and fractional Pearson diffusions Fractional OU, Jacobi and CIR diffusions as the correlated CTRWs limits Theorem Let (H r (n), r N 0) be the Markov chain defined by (22) in the case of OU diffusion, by (27) in the case of Jacobi diffusion and by (30) in the case of CIR diffusion. Let (X (n) (t), t 0), (Y (n) (t), t 0) and (Z (n) (t), t 0) be the corresponding rescaled Markov chains given by (23), (28) and (31) respectively. Let (N(t), t 0) be the renewal process defined in (33), and (E t, t 0) be the inverse of the standard β-stable subordinator (D(t), t 0) with 0 < β < 1. Then X (n) (n 1 N(n 1 β t)) X (E t), n, Y (n) (n 1 N(n 1 β t)) Y (E t), n, Z (n) (n 1 N(n 1 β t)) Z(E t), n, in the Skorokhod space D(S) with J 1 topology, where (X (t), t 0) is Ornstein-Uhlenbeck diffusion defined by (24), (Y (t), t 0) is Jacobi diffusion defined by (29) and (Z(t), t 0) is CIR diffusion defined by (32). Fractional Pearson diffusions 34 / 38

35 CTRWs and fractional Pearson diffusions Fractional OU, Jacobi and CIR diffusions as the correlated CTRWs limits ( ) X (n) (n 1 N(n β 1 t)) = H (n) 2 1 θn(n β 1 t), ( ) Y (n) (n 1 N(n β 1 t)) = H (n) θn(n β 1 t), ( ) Z (n) (n 1 N(n β 1 t)) = H (n) a 1 θn d 1 N(n β 1 t), where (H r (n), r N 0) is corresponding Markov chain given by (22) for the OU, (27) for Jacobi and by (30) for CIR case Fractional Pearson diffusions 35 / 38

36 CTRWs and fractional Pearson diffusions References I Avram, F., Leonenko, N.N., Šuvak, N. (2013a) Spectral representation of transition density of Fisher-Snedecor diffusion, Stochastics, 85(2): Avram, F., Leonenko, N.N., Šuvak, N. (2013b) On spectral analysis of heavy-tailed Kolmogorov-Pearson diffusions, Markov Processes and Related Fields, 19(2): Borodin, S., Salminen, P. (1996) Handbook of Brownian Motion: Facts and Formulae, Springer Chakraborty, P., Meerschaert, M.M., Lim, C.Y. (2009) Parameter estimation for fractional transport: A particle-tracking, Water Resources Research 45(10) Erdelyi, A. (1981) Higher Transcendental Functions, Volume II, Krieger Pub Co. Forman, J.L., Sørensen, M. (2008). The Pearson diffusions: A class of statistically tractable diffusion processes. Scand. J. Statist. 35: Karlin, S., Taylor, H.M. (1981) A Second Course in Stochastic Processes, Academic Press, New York Kolmogorov, A.N. (1931) On analytical methods in probability theory, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann., 104: Kallenberg, O. (2002) Foundations of Modern Probability, Springer Series in Statistics, Probability and its applications, Springer Laplace, P. (1812) Théorie Analytique des Probabilités, Ve. Courcier, Paris Fractional Pearson diffusions 36 / 38

37 CTRWs and fractional Pearson diffusions References II Leonenko, N.N., Meerschaert, M.M., Sikorskii, A. (2013) Fractional Pearson diffusions, Journal of Mathematical Analysis and Applications, 403(2): Leonenko, N.N., Meerschaert, M.M., Sikorskii, A. (2013) Correlation structure of fractional Pearson diffusions, Computers and Mathematics with Applications, 66(5): Leonenko, N.N., Papić, I., Sikorskii, A., Šuvak, N. (2017) Heavy-tailed fractional Pearson diffusions, Stochastic Processes and their Applications, accepted for publication Leonenko, N.N., Papić, I., Sikorskii, A., Šuvak, N. (2017) Correlated continuous time random walks and fractional Pearson diffusions, Bernoulli, accepted for publication McKean, H.P. (1956) Elementary solutions for certain parabolic partial differential equations, Transactions of the American Mathematical Society, 82(2): Meerschaert, M.M., Scheffler, H.P. (2004) Limit theorems for continuous-time random walks with infinite mean waiting times, Journal of Applied Probability 41(3): Meerschaert, M.M., Sikorskii, A. (2011) Stochastic Models for Fractional Calculus, De Gruyter Øksendal, B. (2000) Stochastic Differential Equations, Springer Scalas, E. (2006) Five years of continuous-time random walks in econophysics, Complex Netw. Econ. Interactions 567(1): 3 16 Schumer, R., Benson, D.A., Meerschaert, M.M., Baeumer, B. (2003) Fractal mobile/immobile solute transport, Water Resources Research 39(1): Fractional Pearson diffusions 37 / 38

38 CTRWs and fractional Pearson diffusions References III Wong, E. (1964). The construction of a class of stationary Markov processes. Sixteen Symposium in Applied mathematics - Stochastic processes in mathematical Physics and Engineering, American Mathematical Society, Ed. R. Bellman 16: Fractional Pearson diffusions 38 / 38