An adaptive numerical scheme for fractional differential equations with explosions

An adaptive numerical scheme for fractional differential equations with explosions Johanna Garzón Departamento de Matemáticas, Universidad Nacional de Colombia Seminario de procesos estocásticos Jointly with Soledad Torres and Ciprian Tudor Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 1 / 36

Contents 1 Introduction 2 Deterministic case 3 Fractional Brownian motion 4 Adaptive Euler scheme 5 Numerical example Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 2 / 36

Introduction We consider a fractional stochastic differential equation (FSDE) dx(t) = b(x(t))dt + σ(x(t))db H t, X(0) = x 0, where B H is a fractional Brownian motion with Hurst index H > 1/2, X(0) = x 0 > 0. The functions σ, b are smooth and b is positive. The equation explodes in a finite time, if there exist a time T e < such that X(t) is defined in [0, T e ) and lim X(t) =. t T e Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 4 / 36

Introduction We consider a fractional stochastic differential equation (FSDE) dx(t) = b(x(t))dt + σ(x(t))db H t, X(0) = x 0, where B H is a fractional Brownian motion with Hurst index H > 1/2, X(0) = x 0 > 0. The functions σ, b are smooth and b is positive. The equation explodes in a finite time, if there exist a time T e < such that X(t) is defined in [0, T e ) and lim X(t) =. t T e The goal of this talk is to give a numerical approximation to the explosion time of FSDE when this occurs. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 4 / 36

Deterministic case Let y(t) the solution of the ordinary differential equation y (t) = b(y (t)), y (0) = z > 0 where b is a positive, increasing and smooth function. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 6 / 36

Deterministic case Let y(t) the solution of the ordinary differential equation y (t) = b(y (t)), y (0) = z > 0 where b is a positive, increasing and smooth function. This equation explodes (blows up) in finite time if and only if ds <. The explosion time is z 1 b(s) T e = z 1 b(s) ds. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 6 / 36

Deterministic case y (t) = b(y (t)), y (0) = z > 0 where t 0 = 0 and t k+1 = t k + h y h (t k+1 ) = y h (t k ) + b(y h (t k ))h Euler Scheme : the time step h is constant. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 7 / 36

Adaptive method Let h > 0 be the parameter of the method, where t 0 = 0, t k+1 = t k + τ h k y h (t k+1 ) = y h (t k ) + b(y h (t k ))τ h k τ h k = and h, k = 0, 1,... b(y h (t k )) Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 8 / 36

Adaptive method Let h > 0 be the parameter of the method, where t 0 = 0, t k+1 = t k + τ h k y h (t k+1 ) = y h (t k ) + b(y h (t k ))τ h k and In this case, τ h k = h, k = 0, 1,... b(y h (t k )) y h (t k+1 ) = y h (t k ) + h. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 8 / 36

Adaptive method Let h > 0 be the parameter of the method, where t 0 = 0, t k+1 = t k + τ h k y h (t k+1 ) = y h (t k ) + b(y h (t k ))τ h k and In this case, and hence, τ h k = h, k = 0, 1,... b(y h (t k )) y h (t k+1 ) = y h (t k ) + h. y h (t k+1 ) = y h (t 0 ) + kh = z + kh Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 8 / 36

Adaptive method Let h > 0 be the parameter of the method, where t 0 = 0, t k+1 = t k + τ h k y h (t k+1 ) = y h (t k ) + b(y h (t k ))τ h k and In this case, and hence, therefore, τ h k = h, k = 0, 1,... b(y h (t k )) y h (t k+1 ) = y h (t k ) + h. y h (t k+1 ) = y h (t 0 ) + kh = z + kh τ h k = h b(y h (t k )) = h b(z + kh). Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 8 / 36

If then y h explodes in a finite time. τk h h = k=0 k=0 b(y h (t k )) < Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 9 / 36

If then y h explodes in a finite time. τk h h = k=0 k=0 b(y h (t k )) < The explosion time is T h = k=0 τ h k. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 9 / 36

If then y h explodes in a finite time. τk h h = k=0 k=0 b(y h (t k )) < The explosion time is T h = k=0 τ h k. We have T h = τk h = k=1 h b(z) + k=1 h b(z + kh) < h b(z) + 0 h b(z + sh) ds = h b(z) + T e (T e is the explosion time of X) Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 9 / 36

If then y h explodes in a finite time. τk h h = k=0 k=0 b(y h (t k )) < The explosion time is T h = k=0 τ h k. We have T h = τk h = k=1 h b(z) + k=1 h b(z + kh) < h b(z) + 0 h b(z + sh) ds = h b(z) + T e (T e is the explosion time of X) and T h T e = h b(z). Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 9 / 36

Brownian motion case dx(t) = b(x(t))dt + σ(x(t))dw t, X(0) = x 0, (1) where W is a Brownian motion. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 10 / 36

Brownian motion case dx(t) = b(x(t))dt + σ(x(t))dw t, X(0) = x 0, (1) where W is a Brownian motion. If b and σ satisfy c 1 σ 2 (x) c 2 b(x) b is nondecreasing and 0 1 dx <, b(x) then the solution of (1) explodes in finite time with probability one. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 10 / 36

Brownian motion case dx(t) = b(x(t))dt + σ(x(t))dw t, X(0) = x 0, (1) where W is a Brownian motion. If b and σ satisfy c 1 σ 2 (x) c 2 b(x) b is nondecreasing and 0 1 dx <, b(x) then the solution of (1) explodes in finite time with probability one. Dávila et al [1] proposed a numerical scheme to approximate the time of explosion. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 10 / 36

Fractional Brownian motion Definition A fractional Brownian motion (fbm) with Hurst parameter H (0, 1), B H = {Bt H : t R + } is a centered Gaussian stochastic process with covariance function given by Q B (s, t) = E(Bt H Bs H ) = 1 2 (t2h + s 2H t s 2H ). Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 12 / 36

Fractional Brownian motion Definition A fractional Brownian motion (fbm) with Hurst parameter H (0, 1), B H = {Bt H : t R + } is a centered Gaussian stochastic process with covariance function given by Q B (s, t) = E(Bt H Bs H ) = 1 2 (t2h + s 2H t s 2H ). If H = 1/2, B H is a Brownian motion. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 12 / 36

B 1/4 B 1/2 B 3/4 Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 13 / 36

Properties B H is self-similar (with index H), (B H at) t 0 d = (a H B H t ) t 0. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 14 / 36

Properties B H is self-similar (with index H), B H has stationary increments (B H at) t 0 d = (a H B H t ) t 0. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 14 / 36

Properties B H is self-similar (with index H), (Bat) H d t 0 = (a H Bt H ) t 0. B H has stationary increments B H is κ-hölder continuous for any exponent κ < H. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 14 / 36

Properties B H is self-similar (with index H), (Bat) H d t 0 = (a H Bt H ) t 0. B H has stationary increments B H is κ-hölder continuous for any exponent κ < H. B H is neither a Markov process nor a semimartingale for H 1. 2 Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 14 / 36

Fractional stochastic integral The pathwise integral fractional Brownian motion B H is defined by b a fdb H = b a (Da+f α )(s)(db 1 α BH b )(s)ds, Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 15 / 36

The pathwise integral w.r.t. fractional Brownian motion B H is defined by b a fdb H = b a (Da+f α )(s)(db 1 α BH b )(s)ds, Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 16 / 36

The pathwise integral w.r.t. fractional Brownian motion B H is defined by where (D α a+f )(s) = b a fdb H = b a (Da+f α )(s)(db 1 α BH b )(s)ds, [ 1 f (s) ] s Γ(1 α) (s a) + α f (s) f (u) α a (s u) du I α (a,b) (s), [ (Db 1 α B b )(s) = exp iπα Bb (s) Γ(α) (b s) 1 α +(1 α) are fractional derivatives, and b s B b (s) B b (u) (u s) 2 α du B b (s) = (B s B b )I (a,b) (s). ] I (a,b) (s), Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 16 / 36

[ b b fdb H F 0 a a f (s) b s (s a) ds + α a a ] f (s) f (u) duds, (s u) α+1 Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 17 / 36

[ b b fdb H F 0 a a and for all p [1, ) E(F p 0 ) = E f (s) b s (s a) ds + α a a ( ] f (s) f (u) duds, (s u) α+1 p sup Dt 1 α B t (s)) <. 0 s t T Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 17 / 36

Fractional stochastic differential equation or X(t) = x 0 + t 0 b(x(s))ds + t 0 σ(x(s))db H s, t [0, T ] dx(t) = b(x(t))dt + σ(x(t))db H t, X(0) = x 0, Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 18 / 36

Fractional stochastic differential equation or X(t) = x 0 + t 0 b(x(s))ds + t 0 σ(x(s))db H s, t [0, T ] dx(t) = b(x(t))dt + σ(x(t))db H t, X(0) = x 0, H 1 1 Local Lipschitz continuity : for any N > 0 there exists K N < 0 such that b(x) b(y) K N x y, x, y N. 2 Linear growth b(x) L( x + 1). Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 18 / 36

dx(t) = b(x(t))dt + σ(x(t))db H t, X(0) = x 0, H 2 σ(x) is differenciable in x, there exists some M > 0, κ (1 H, 1], such that 1 Lipschitz continuity σ(x) σ(y) M x y, x, y R, t [0, T ], 2 Local Hölder continuity x σ(x) x σ(y) M N x y κ, x, y N. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 19 / 36

Fractional stochastic differential equation dx(t) = b(x(t))dt + σ(x(t))db H t, X(0) = x 0. (2) Nualart-Răşcanu If the coefficients b and σ satisfy assumptions H 1 and H 2 then the equation (2) has a unique solution {X(t), t [0T ]}. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 20 / 36

León and Villa [2] X(t) = x 0 + t 0 b(x(s))ds + B H t, (3) where b : R R + is a positive, locally Lipschitz and non-decreasing function. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 21 / 36

León and Villa [2] X(t) = x 0 + t 0 b(x(s))ds + B H t, (3) where b : R R + is a positive, locally Lipschitz and non-decreasing function. The solution of the equation (3) explodes in finite time with probability 1 if and only if 1 ds <. b(s) Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 21 / 36

Narita [4] X(t) = x 0 + where b is a function t 0 b(x(s))ds + t positive, continuous, nondecreasing, b(x) b( x ) for x r for some constant r > 0, r 0 1 ds =. b(s) X(s)dB H s, (4) Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 22 / 36

Narita [4] X(t) = x 0 + where b is a function t 0 b(x(s))ds + t positive, continuous, nondecreasing, b(x) b( x ) for x r for some constant r > 0, r 0 1 ds =. b(s) X(s)dB H s, (4) Then the explosion time T of the solution X of the equation (4) satisfies P(T = ) = 1. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 22 / 36

Adaptive Euler scheme dx(t) = b(x(t))dt + σ(x(t))db H t, X(0) = x 0. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 24 / 36

Adaptive Euler scheme dx(t) = b(x(t))dt + σ(x(t))db H t, X(0) = x 0. We assume that b is bounded away from 0, 0 < L 1 b(x), x R. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 24 / 36

Adaptive Euler scheme dx(t) = b(x(t))dt + σ(x(t))db H t, X(0) = x 0. We assume that b is bounded away from 0, 0 < L 1 b(x), x R. Let 0 < h < 1 be the parameter of the method. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 24 / 36

Adaptive Euler scheme dx(t) = b(x(t))dt + σ(x(t))db H t, X(0) = x 0. We assume that b is bounded away from 0, 0 < L 1 b(x), x R. Let 0 < h < 1 be the parameter of the method. Y h (t 0 ) = x 0, Y h := Y h (t n+1 ) = Y h (t n ) + b(y h (t n ))τn h +σ(y h (t n ))[B(t n+1 ) B(t n )], where t 0 = 0 and τ h n = t n+1 t n = h b(y h (t k )). Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 24 / 36

Adaptive Euler scheme Let Y h (t 0 ) = x 0, Y h = Y h (t n+1 ) = Y h (t n ) + b(y h (t n ))τn h +σ(y h (t n ))[B(t n+1 ) B(t n )], τ h n = t n+1 t n = T h = τn h, n=0 h b(y h (t k )). we can observe that if T h (ω) < then Y h explodes and the explosion time is T h = τn h <. n=0 Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 25 / 36

Adaptive Euler scheme dx(t) = b(x(t))dt + σ(x(t))db H t, X(0) = x 0, Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 26 / 36

Adaptive Euler scheme dx(t) = b(x(t))dt + σ(x(t))db H t, X(0) = x 0, Y h (t 0 ) = x 0, Y h = Y h (t n+1 ) = Y h (t n ) + b(y h (t n ))τn h +σ(y h (t n ))[B(t n+1 ) B(t n )], Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 26 / 36

Adaptive Euler scheme dx(t) = b(x(t))dt + σ(x(t))db H t, X(0) = x 0, Y h (t 0 ) = x 0, Y h = Y h (t n+1 ) = Y h (t n ) + b(y h (t n ))τn h +σ(y h (t n ))[B(t n+1 ) B(t n )], Theorem (Convergence) If we assume that 0 < L 1 b(x) L 2 σ(x) M, then lim E h 0 ( ) sup X(t) Y h (t) 2 = 0. 0 t T Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 26 / 36

Theorem (General case) Assume that σ and b are local Lipschitz continuous and 0 < L 1 b(x). Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 27 / 36

Theorem (General case) Assume that σ and b are local Lipschitz continuous and 0 < L 1 b(x). We fix a time S > 0 and a constant M > 0 and we consider the times given by R M := inf{t > 0 : X(t) M}, R h M := inf{t > 0 : Y h (t) M} R h = min{r M, R h M, S}. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 27 / 36

Theorem (General case) Assume that σ and b are local Lipschitz continuous and 0 < L 1 b(x). We fix a time S > 0 and a constant M > 0 and we consider the times given by R M := inf{t > 0 : X(t) M}, R h M := inf{t > 0 : Y h (t) M} R h = min{r M, R h M, S}. Then lim E h 0 ( sup X(t) Y h (t) 2 0 t R h ) = 0. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 27 / 36

Theorem (Approximation to the explosion time ) dx(t) = b(x(t))dt + σ(x(t))db H t, X(0) = x 0, Assume that X explodes in a finite time T e. Let M > 0 and consider the times given by R h M := inf{t > 0 : Y h (t) M} Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 28 / 36

Theorem (Approximation to the explosion time ) dx(t) = b(x(t))dt + σ(x(t))db H t, X(0) = x 0, Assume that X explodes in a finite time T e. Let M > 0 and consider the times given by R h M := inf{t > 0 : Y h (t) M} For any ε > 0, then lim lim P ( RM h T e > ε ) = 0. M h 0 Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 28 / 36

Numerical example dx(t) = ( X(t) 1.1 + 0.1)dt + We assume that M = 10 5. X(t) 1.1 + 0.1dB H t, X(0) = x 0. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 30 / 36

Numerical example dx(t) = ( X(t) 1.1 + 0.1)dt + H = 0.7 X(t) 1.1 + 0.1dB H t, X(0) = x 0. 10 x 104 9 8 7 6 X(t) 5 4 3 2 1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 1: Three sample paths with explosions for H=0,7 Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 31 / 36

35 30 25 Frequency 20 15 10 5 0 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 Histogram time explosion for H=0.7 Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 32 / 36

H = 0.9 10 x 104 9 8 7 6 X(t) 5 4 3 2 1 0 0 1 2 3 4 5 6 Figure 3: Three sample paths with explosions for H=0,9 Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 33 / 36

Frequency 20 18 16 14 12 10 8 6 4 2 0 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 Histogram time explosion for H=0.9 Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 34 / 36

Thanks Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 35 / 36

References 1 J. Dávila, J. Fernández, J. D. Rossi, P. Groisman, M. Sued, Numerical analysis of stochastic differential equations with explosions. Stochastic Analysis and Applications, 23 (2005) 809-826. 2 J. A. León, J. Villa, An Osgood criterion for integral equations with applications to stochastic differential equations with an additive noise. Stat. Prob. Let., (2011) 4, 470-477. 3 Y. Mishura, G. Shevchenko, The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion, Stochastics 80 (2008) 489-511. 4 K. Narita, Nonexplosion Criteria for Solutions of SDE with Fractional Brownian Motion, Stochastic Analysis and Applications 25 (2007) 77-88. 5 D. Nualart, A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002) 55-81. Johanna Garzón (U. Nacional) An adaptive numerical scheme for fsde with explosions 36 / 36