Stochastic calculus for uncoupled continuous-time ... - CCP 2010
Stochastic calculus for uncoupled continuous-time random walks Guido Germano1 , Mauro Politi1,2 , Enrico Scalas3 , Ren´e L. Schilling4 1 Department of Chemistry and WZMW, Philipps-University Marburg, Germany 2 Department of Physics, University of Milan, Italy 3 DISTA, University of East Piedmont, Alessandria, Italy 4 Institut f¨ ur Mathematische Stochastik, TU Dresden, Germany
The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications not only in physics, but also in insurance, finance and economics. A definition is given for a class of stochastic integrals driven by a CTRW, which includes the It¯o and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingale transform theorem, if the CTRW is a martingale, the It¯o integral is a martingale too. It is shown how the definition of the stochastic integrals can be used to easily compute them by Monte Carlo simulation. The relations between a CTRW, its quadratic variation, its Stratonovich integral and its It¯o integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric L´evy α-stable distribution and its waiting times have a one-parameter MittagLeffler distribution. Remarkably, these distributions have fat tails and thus the CTRW has an unbounded quadratic variation. In the diffusive limit of vanishing scale parameters, the probability density of this kind of CTRW satisfies the space-time fractional diffusion equation (FDE) or more in general the fractional Fokker-Planck equation, which generalizes the standard diffusion equation solved by the probability density of the Wiener process, and thus provides a phenomenologic model of anomalous diffusion. We also provide an analytic expression for the probability density function of the quadratic variation, also called integrated volatility, of the stochastic process described by the FDE, and check it by Monte Carlo.
References [1] G. Germano, M. Politi, E. Scalas, R. L. Schilling, Phys. Rev. E 79, 066102 (2009). [2] G. Germano, M. Politi, E. Scalas, R. L. Schilling, Commun. Nonlin. Sci. Numer. Simul. 15, 1583 (2010).
3 α = 2.0, β = 1.0 γt = 0.1
Probability density function
Probability density function
3
pX(x) uX(x) pS(x) 2uX|dX/dS|(x) pI(x) 2pS(x)∗p[X](-2x) p[X](-x) u[X](-x)
2
1
0
α = 1.9, β = 1.0 γt = 0.1 pX(x) uX(x) pS(x) 2uX|dX/dS|(x) pI(x) 2pS(x)∗p[X](-2x) p[X](-x) u[X](-x)
2
1
0 -3
-2
-1
0
1
2
3
-3
-2
-1
x
pX(x) uX(x) pS(x) 2uX|dX/dS|(x) pI(x) 2pS(x)∗p[X](-2x) p[X](-x) u[X](-x)
2
1
0
2
3
α = 1.9, β = 1.0 γt = 0.01 pX(x) uX(x) pS(x) 2uX|dX/dS|(x) pI(x) 2pS(x)∗p[X](-2x) p[X](-x) u[X](-x)
2
1
0 -3
-2
-1
0
1
2
3
-3
-2
-1
x
0
1
2
3
x
3
3 α = 2.0, β = 1.0 γt = 0.001
Probability density function
Probability density function
1
3 α = 2.0, β = 1.0 γt = 0.01
Probability density function
Probability density function
3
pX(x) uX(x) pS(x) 2uX|dX/dS|(x) pI(x) 2pS(x)∗p[X](-2x) p[X](-x) u[X](-x)
2
1
0
α = 1.9, β = 1.0 γt = 0.001 pX(x) uX(x) pS(x) 2uX|dX/dS|(x) pI(x) 2pS(x)∗p[X](-2x) p[X](-x) u[X](-x)
2
1
0 -3
-2
-1
0
1
2
3
-3
-2
-1
x
0
1
2
3
x
3
3 α = 2.0, β = 1.0 γt = 0.0001
Probability density function
Probability density function
0 x
pX(x) uX(x) pS(x) 2uX|dX/dS|(x) pI(x) 2pS(x)∗p[X](-2x) p[X](-x) u[X](-x)
2
1
0
α = 1.9, β = 1.0 γt = 0.0001 pX(x) uX(x) pS(x) 2uX|dX/dS|(x) pI(x) 2pS(x)∗p[X](-2x) p[X](-x) u[X](-x)
2
1
0 -3
-2
-1
0
1
2
3
-3
x
-2
-1
0
1
2
3
x
Convergence of the empirical probability densities p from 1 × 106 Monte Carlo runs (points) to the analytic probability densities u (lines) in the diffusive limit for a CTRW X(t), its Stratonovich integral S(t), its It¯o integral I(t), and its quadratic variation [X](t), with t = 1 and different choices of the index parameters α, β and of the scale parameters γx , γt , where γxα /γtβ = D = 1.
2
The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications not only in physics, but also in insurance, finance and economics. A definition is given for a class of stochastic integrals driven by a CTRW, which includes the It¯o and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingale transform theorem, if the CTRW is a martingale, the It¯o integral is a martingale too. It is shown how the definition of the stochastic integrals can be used to easily compute them by Monte Carlo simulation. The relations between a CTRW, its quadratic variation, its Stratonovich integral and its It¯o integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric L´evy α-stable distribution and its waiting times have a one-parameter MittagLeffler distribution. Remarkably, these distributions have fat tails and thus the CTRW has an unbounded quadratic variation. In the diffusive limit of vanishing scale parameters, the probability density of this kind of CTRW satisfies the space-time fractional diffusion equation (FDE) or more in general the fractional Fokker-Planck equation, which generalizes the standard diffusion equation solved by the probability density of the Wiener process, and thus provides a phenomenologic model of anomalous diffusion. We also provide an analytic expression for the probability density function of the quadratic variation, also called integrated volatility, of the stochastic process described by the FDE, and check it by Monte Carlo.
References [1] G. Germano, M. Politi, E. Scalas, R. L. Schilling, Phys. Rev. E 79, 066102 (2009). [2] G. Germano, M. Politi, E. Scalas, R. L. Schilling, Commun. Nonlin. Sci. Numer. Simul. 15, 1583 (2010).
3 α = 2.0, β = 1.0 γt = 0.1
Probability density function
Probability density function
3
pX(x) uX(x) pS(x) 2uX|dX/dS|(x) pI(x) 2pS(x)∗p[X](-2x) p[X](-x) u[X](-x)
2
1
0
α = 1.9, β = 1.0 γt = 0.1 pX(x) uX(x) pS(x) 2uX|dX/dS|(x) pI(x) 2pS(x)∗p[X](-2x) p[X](-x) u[X](-x)
2
1
0 -3
-2
-1
0
1
2
3
-3
-2
-1
x
pX(x) uX(x) pS(x) 2uX|dX/dS|(x) pI(x) 2pS(x)∗p[X](-2x) p[X](-x) u[X](-x)
2
1
0
2
3
α = 1.9, β = 1.0 γt = 0.01 pX(x) uX(x) pS(x) 2uX|dX/dS|(x) pI(x) 2pS(x)∗p[X](-2x) p[X](-x) u[X](-x)
2
1
0 -3
-2
-1
0
1
2
3
-3
-2
-1
x
0
1
2
3
x
3
3 α = 2.0, β = 1.0 γt = 0.001
Probability density function
Probability density function
1
3 α = 2.0, β = 1.0 γt = 0.01
Probability density function
Probability density function
3
pX(x) uX(x) pS(x) 2uX|dX/dS|(x) pI(x) 2pS(x)∗p[X](-2x) p[X](-x) u[X](-x)
2
1
0
α = 1.9, β = 1.0 γt = 0.001 pX(x) uX(x) pS(x) 2uX|dX/dS|(x) pI(x) 2pS(x)∗p[X](-2x) p[X](-x) u[X](-x)
2
1
0 -3
-2
-1
0
1
2
3
-3
-2
-1
x
0
1
2
3
x
3
3 α = 2.0, β = 1.0 γt = 0.0001
Probability density function
Probability density function
0 x
pX(x) uX(x) pS(x) 2uX|dX/dS|(x) pI(x) 2pS(x)∗p[X](-2x) p[X](-x) u[X](-x)
2
1
0
α = 1.9, β = 1.0 γt = 0.0001 pX(x) uX(x) pS(x) 2uX|dX/dS|(x) pI(x) 2pS(x)∗p[X](-2x) p[X](-x) u[X](-x)
2
1
0 -3
-2
-1
0
1
2
3
-3
x
-2
-1
0
1
2
3
x
Convergence of the empirical probability densities p from 1 × 106 Monte Carlo runs (points) to the analytic probability densities u (lines) in the diffusive limit for a CTRW X(t), its Stratonovich integral S(t), its It¯o integral I(t), and its quadratic variation [X](t), with t = 1 and different choices of the index parameters α, β and of the scale parameters γx , γt , where γxα /γtβ = D = 1.
2
