Stochastic Functional Differential Equations

Transcription

1 Stochastic Functional Differential Equations Evelyn Buckwar Heriot-Watt University, until 31st of August 2011, then off to JKU Linz! MFO Oberwolfach, 26th August 2011

2 Stochastic functional differential equations X (t) = X (0) + t 0 F (s, X s)ds + t 0 G(s, X s)dw (s) memory functional or segment process X t (u) = {X (t + u) : u [ τ, 0]} for X t C([ τ, 0]; R n ); initial data: X (t) = ψ(t) for t [ τ, 0]; coefficients: (globally Lipschitz) F : [0, T ] C([ τ, 0]; R n ) R n, G = (G 1,..., G m ) : [0, T ] C([ τ, 0]; R n ) R n m ; Wiener process: W = {W (t, ω), t [0, T ], ω Ω} is an m-dim. Wiener process on (Ω, F, {F t } t [0,T ], P).

3 The memory functional often takes special forms, e.g., by setting F and/or G to 0 H 4 (s, X (s), H 1 (s, X (s)), H 2 (s, X (s), X (s τ)), H 3 (s, X (s), X (s τ(s))), τ K(s, u, X (s + u))du), (no delay) (constant/discrete delay) (variable delay) (distributed delay). We assume that there exists a path-wise unique strong solution X ( ) of the above equation. (Analysis: Itô, Nisio (1964); Mao; Mohammed)

4 Some references to SDDEs in Biosciences Human pupil light reflex (Longtin et al, 90) Human postural sway (Eurich et al, 96) Neurological diseases (Beuter et al, 93, Tass et al 05) Infectious diseases (Beretta et al, 98) Chemical kinetics (Burrage et al, 07) Population dynamics (Carletti, Beretta, Tapaswi & Mukhopadhyay); Neural field models (Hutt et al) Cowan-Wilson networks with delayed feedback (Jirsa et al) FitzHugh-Nagumo networks with delayed feedback (Schöll et al) Hematopoietic Stem Cell Regulation System (Mackey et al, 07)... and many more

5 Explicit Solutions and the method of steps

6 Solutions of stochastic differential equations dx (t)=ax (t)dt+bdw (t), X (t)=e at (1 + b t 0 e as dw (s)) dx (t)=ax (t)dt +bx (t)dw (t), X (t)=exp((a 1 2 b2 )t + bw (t))

7 Solutions of SDDEs via Method of steps dx (t)=x (t 1)dt +βdw (t), t 0 and X (t)=φ 1 (t)=1 + t, t [ 1, 0] t [0, 1] dx (t) = Φ 1 (t 1)dt + βdw (t) = t dt + βdw (t) X (t) = 1 + t2 2 + βw (t) =: Φ 2(t) t [1, 2] dx (t) = Φ 2 (t 1)dt + βdw (t) X (t) = 1 6 t3 1 2 t t + 1 t 3 + β( W (s 1)ds + W (t)) 1

8 Itô formulas and Fokker-Planck equations

9 Stochastic ordinary differential equations Consider X (t) = X (0) + t 0 F (X (s)) ds + t 0 G(X (s)) dw (s) Itô formula for φ(x) function, suff. differentiable, everything scalar: φ(x (t)) = φ(x (0)) t 0 t 0 φ x (X (s))f (X (s))ds +φ x (X (s))g(x (s))dw (s) φ xx (X (s)) G 2 (X (s)) ds

10 Itô-formula, distributed delay X (t) = X (0) + t 0 F (s, X (s), Y (s)) ds + t G(s, X (s), Y (s)) dw (s) 0 Y (t) = t t τ K(t, s t, X (s)) ds (A. Friedman (1975), M. Arriojas, PhD thesis (1997)): φ[t] φ[t ] = a 2 (t) := d dt t t τ t t D 1 φ[s] + D 2 φ[s]f [s] D2 2 φ[s](g[s]) 2 + D 3 φ[s]a 2 (s) ds + t t τ t t K(t, s t, X (s)) ds = D 2 φ[s] G[s] dw (s) D 1 K(t, s t, X (s) D 2 K(t, s t, X (s)) ds +K(t, 0, X (t)) K(t, τ, X (t τ)) f [t] := f (t, X (t), Y (t)), D i derivative w.r.t. the i th argument

11 Itô-formula, discrete delay Itô-formula from Hu, Mohammed & Yan (AoP 32(1A), 2004) for SDDE with m = 1, n = 1, r = 2, s = 2, τ 1 = σ 1 = 0 dx (t)=f (X (t), X (t τ 2 ))dt + G(X (t), X (t σ 2 ))dw (t), t > 0 X (t) = ψ(t), τ < t < 0, τ := τ 2 σ 2, and function φ(x (t), X (t δ)) δ > 0

12 dφ ( X (t), X (t δ) ) = φ x 2 (X (t), X (t δ)) 1 [0,δ) (t)dψ(t δ) + φ x 2 (X (t), X (t δ))1 [δ, ) (t) [ F ( X (t δ), X (t τ 2 δ) ) dt +G ( X (t δ), X (t σ 2 δ) ) dw (t δ) ] + 2 φ ( ) ( X (t), X (t δ) G X (t δ), X (t σ2 δ) ) 1 [δ, ) (t)d t δ X (t)dt x 1 x φ ( ) ( X (t), X (t δ) G X (t δ), X (t σ2 2 x2 2 δ) ) 2 1[δ, ) (t)dt + φ ( ) [ ( X (t), X (t δ) F X (t), X (t τ2 ) ) dt + G ( X (t), X (t σ 2 ) ) dw (t) ] x φ ( ) ( X (t), X (t δ) G X (t), X (t σ2 2 x1 2 ) ) 2 dt,

13 Fokker-Planck equation etc. : Problems arising due to: infinite dimensional nature of the segment process, the segment process is in general not a semi-martingale, non-markovian structure of the solution of an SFDE. an approximate Fokker-Planck equation using the method of steps is presented by A. Longtin et al, Physical Review E, 1999 weak infinitesimal generator of SFDEs and a Feynman-Kac formula are derived in SEA Mohammed and F Yan, Stochastic Anal. Appl. 2005, using Malliavin calculus

14 Oscillatory behaviour

15 Oscillatory Behaviour Deterministic theory (math.): Well-known: DDEs display a rich variety of dynamical behaviour, already in the scalar case: Oscillations, bifurcations, multi-stability, etc... Books by Gopalsamy (1992) and Ladde, Lakshmikantham & Zhang (1987) and lots of articles on the subject exist.

16 Stochastic theory Physics literature: many references (Garcia-Ojalvo and Roy, Mackey and Glass, Longtin, etc.) Mathematical literature: SODEs (e.g., Mao 1997, Baxendale, Namachchivaya,..) SDDEs???? What does a mathematician do? Take a simple example...

17 What is the effect of the delays? Example: y (t) = λy(t), t 0, λ R, y(0) = 1 versus x (t) = λx(t) + µx(t 1), t 0, λ, µ R, x(t) = 1 + t, t 0

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20 Results for a simple example, joint work with J. Appleby dx (t) = (ax (t) + bx (t τ)) dt + σx (t) dw (t) X (t) = ψ(t), τ t 0, With (Φ(t)) t τ given by Φ(t) = 1 for t [ τ, 0] and Φ(t) = exp((a σ 2 /2)t + σw (t)) for t 0 (Φ(t) solves dφ(t) = aφ(t)dt + σφ(t) dw (t)) set Y (t) = X (t)/φ(t) then Y (t) = Y (0) + t 0 b Y (s τ) Φ(s τ) Φ(s) 1 ds, t 0, (X (t)) t 0 ( satisfies X (t) = Φ(t) ψ(0) + ) t 0 b X (s τ) Φ(s) 1 ds. Now Y C 1 ((0, ); R), thus Y (t) = b Φ(t τ) Φ(t) 1 Y (t τ)

21 Now, for Y (t) = b Φ(t τ) Φ(t) 1 Y (t τ) use path-wise results for deterministic DDE: b > 0, ψ(t) 0, solutions of SDDE are always positive and non-oscillatory b < 0, any ψ, solutions of the SDDE are always oscillatory for σ 0. For σ = 0 there are parameter ranges where non-oscillatory solutions exist! Further, in contrast to the zeros of the Wiener process and additive noise SODEs and SDDEs, the zeros of the solution X are at least a τ apart. more general approach to SDDE RDDE: H Lisei, Conjugation of Flows for Stochastic and Random Functional Differential Equations. Stochastics and Dynamics 1(2), (2001)

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23 Stability issues

24 Lyapunov stability of equilibrium solutions of ODEs Consider ODE (1) x (t) = f (x(t)), x(0) = x 0, and assume (the constant function) x E is an equilibrium solution of (1) then the equilibrium x E is said to be Lyapunov stable, if for every ɛ > 0, there exists a δ = δ(ɛ), such that for some norm. if x(0) x E < δ, then x(t) x E < ɛ for all t 0; the equilibrium is said to be asymptotically Lyapunov stable, if it is Lyapunov stable and there exists a δ > 0 such that if x(0) x E < δ, then lim t x(t) x E = 0.

25 What do we consider as equilibrium solutions of SDDEs? dx (t) = F (X (t), X (t τ))dt + G(X (t), X (t τ))dw (t) Obvious choice: a constant function X E such that F (X E, X E ) = G(X E, X E ) = 0. Look at some example models...

26 Time-delayed feedback in neurosystems Schöll et al, 2009 FitzHugh-Nagumo system ɛ 1 x 1(t) = x 1 (t) x 3 1 (t)/3 y 1 (t) + C[x 2 (t τ) x 1 (t)] y 1(t) = x 1 (t) + a + D 1 ξ 1 (t) ɛ 2 x 2(t) = x 2 (t) x 3 2 (t)/3 y 2 (t) + C[x 1 (t τ) x 2 (t)] y 2(t) = x 2 (t) + a + D 2 ξ 2 (t) x 1, y 1, x 2, y 2 correspond to single neurons or populations, linearly coupled with coupling strength C x 1, x 2 are related to transmembrane voltage, y 1, y 2 are connected to electrical conductance of ion currents a excitability parameter, a > 1 excitable, a < 1 self-sustained periodic firing ɛ i time-scale parameters 1, fast activator variables x i, y 1 slow inhibitor variables ξ 1, ξ 2 Gaussian white noise, mean zero, unit variance, noise intensities D 1, D 2

27 Time-delayed feedback in neurosystems Schöll et al, 2009 FitzHugh-Nagumo system ɛ 1 x 1(t) = x 1 (t) x 3 1 (t)/3 y 1 (t) + C[x 2 (t τ) x 1 (t)] y 1(t) = x 1 (t) + a + D 1 ξ 1 (t) ɛ 2 x 2(t) = x 2 (t) x 3 2 (t)/3 y 2 (t) + C[x 1 (t τ) x 2 (t)] y 2(t) = x 2 (t) + a + D 2 ξ 2 (t) Modelling approach: deterministic system + extrinsic additive noise Equilibrium solution of deterministic system (fix ɛ 1 = ɛ 2 = 0.01 and a = 1.05): x E = (x E,1, y E,1, x E,2, y E,2 ) with x E,i = a and y E,i = a 3 /3 a... which is not an equilibrium solution of the stochastic system

28 Infectious Diseases Model Beretta et al, 1998 ds(t) dt di (t) dt dr(t) dt = ( βs(t) h 0 +σ 1 (S(t) S E )dw 1 (t) = (βs(t) h 0 f (s)i (t s)ds µ 1 S(t) + b)dt f (s)i (t s)ds (µ 2 + λ)i (t))dt +σ 2 (I (t) I E )dw 2 (t) = (λi (t) µ 3 R(t))dt + σ 3 (R(t) R E )dw 3 (t). Modelling approach: deterministic system + white noise perturbations proportional to distances S(t) S E, etc.

29 Example: Hematopoietic Stem Cell Regulation System Mackey et al, 2007 dq(t) dt = ( b 1Q(t) 1 + Q 4 (t) δq(t) + b 1µ 1 Q(t 1) 1 + Q 4 (t 1) )dt +σq(t)dw (t). Non-dimensional form with Q modelling quiescent stem cells with maturation delay Modelling approach: deterministic system + white noise perturbations around mean of parameter δ Equilibrium solution of deterministic system: ( ) 4 b1 (µ Q E = 1 1) 1 δ... which is not an equilibrium solution of the stochastic system

30 Chemical Kinetics (genetic regulatory network) Burrage et al, 1998 Chemical Langevin Equation with Delay dx (t) = M 1 M N υ j a j (X (t T j )dt + υ j a j (X (t)dt + b j (X (t))dw j (t). j=m j+1 j=1 j=1 a j propensity functions, υ j state change vector, b j is a column vector of a principal root of a certain matrix Has deterministic equilibrium solution X E 0 Modeling approach: As limit of a diffusion approximation of a discrete stochastic system See Diffusion approximation of birth-death processes: Comparison in terms of large deviations and exit points, K Pakdaman, M Thieullen, G Wainrib, Statistics and Probability Letters 80 (2010) , for a discussion

31 What do we consider as equilibrium solutions of SDDEs? Observation: Transition from deterministic to stochastic system may be equilibrium preserving or not! Possibility: Consider stability of deterministic equilibrium X E (Mackey et al, 2007) NOT the same concept as standard Lyapunov stability as X E not even a solution of the stochastic system! Practically no mathematical theory developed. However: A stochastic system may have a stochastic equilibrium...

32 Stationary solutions as stochastic equilibrium General additive noise equation: dx (t) = f (X (t)) dt + σ dw (t). (1) A special case: dx (t) = λx (t)dt + σdw (t), (2)

33 The Ornstein-Uhlenbeck stochastic stationary process The linear, scalar SDE (2) has the explicit solution X (t) = e λ(t t 0) X 0 + σe λt t t 0 e λs dw (s) (3) for all t t 0 and the initial value X (t 0) = X 0. This expression has no forward limit but the pathwise pullback limit (i.e., as t 0 with t fixed) exists and is given by t Ô(t) := σe λt e λs dw (s), (4) which is known as the scalar Ornstein-Uhlenbeck stochastic stationary process. It is Gaussian with zero mean and variance σ 2 /2λ. (This requires the use of a two-sided Wiener process, i.e., with W (t) defined for all t R rather than just for t R + ).

34 The Ornstein-Uhlenbeck stochastic stationary process The Ornstein-Uhlenbeck process (4) is a stochastic stationary solution of the linear SDE (2). Moreover, it attracts all other solutions of this SDE forwards in time in the pathwise sense. To see this simply subtract one solution of (2) from another to obtain ( ) X 1 (t) X 2 (t) = e λ(t t 0) X0 1 X0 2, and then replace X 2 (t) by the Ornstein-Uhlenbeck process Ô(t).

35 Stationary solutions of SDDEs Ito, K.; Nisio, M., On stationary solutions of a stochastic differential equation. J. Math. Kyoto Univ. 4, 1 75 (1964). Küchler, U.; Mensch, B., Langevins stochastic differential equation extended by a time-delayed term. Stochastics 40(1), (1992). Mohammed, Salah EA, book and articles Scheutzow, M, several articles Bakhtin, Yu; Mattingly, JC., Stationary solutions of stochastic differential equations with memory and stochastic partial differential equations. Commun. Contemp. Math. 7, No. 5, (2005)

36 Example: stochastic logistic DDE Scheutzow, 1984 The stochastic logistic DDE dx (t) = (k 1 k 2 X k 3 (t 1))X (t)dt + k 4 X (t) dw (t) has a unique, positive-valued stationary solution, if k 1 k 2 4 /2, k 2, k 3, k 4 are all positive. All its moments are finite.

37 Next choice: Norm Definition 1. The equilibrium solution X S of an SDDE is mean-square stable/a.s. stable if and only if, for each ɛ > 0, there exists a δ 0 such that E X (t) X S p < ɛ, t 0, / X (t) X S < ɛ, t 0, a.s. whenever E X (0) X S p < δ / X (0) X S < δ; 2. The equilibrium solution is asymptotically mean-square stable/a.s. stable if and only if it is mean-square stable/a.s. stable, and for all X (0) X S R, lim E X (t) X S p = 0 / lim X (t) X S = 0 t t a.s.

38 Mean-square vs. Almost sure stability d ( ) X1 (t) = X 2 (t) ( λ 0 0 λ ) ( ) X1 (t) dt + X 2 (t) ( σ 0 0 σ ) ( ) X1 (t) dw (t) X 2 (t) Stability conditions: MS-stab λ σ2 < 0, a.s. stab λ 1 2 σ2 < 0 Choose λ = 0.1, σ = 0.5, we see that λ σ2 = > 0, λ 1 2 σ2 = < 0, and therefore the equilibrium solution X E 0 is simultaneously mean-square unstable and a.s. asymptotically stable.

39 Figure: Simulations of System above with λ = 0.1, σ = 0.5. X 1 and X 2 represent the components of a single path of the simulation, whereas MS-X 1 and MS-X 2 represent the estimated mean-square norm of each solution component.

40 Existing techniques Lyapunov functional techniques, mostly MS-stab: VB Kolmanovskii, L Shaikhet Razumikhin techniques, LaSalle principle, MS-stab & a.s. stab: X Mao Random dynamical systems techniques (i.e., pathwise results) giving Lyapunov exponents: SEA Mohammed, M Scheutzow Random dynamical systems techniques for SPDEs with memory, also considering ms and a.s. stab of stationary solutions: T Caraballo, M Garrido-Atienza, B Schmalfuss Halanay inequalities, MS stab.: CTH Baker & EB

41 Open problems Linear and nonlinear stability analysis for SDDEs arising, e.g., in the models above Investigating oscillatory and other dynamics for SDDEs Averaging techniques Amplitude equations Large deviations