S.M. Iacus, D. La Torre (Univ. Milan) Different ways of simulating - - PowerPoint PPT Presentation

S.M. Iacus, D. La Torre (Univ. Milan) Iterated function system and simulation of Brownian motion

Wien April 2006

Different ways of simulating BM paths

simulating increments B(t) − B(s) ∼ N(0, t − s) limit of the random walk Sn = Xi, with P(Xi = ±1) = 1/2 S[nt] √n , t ≥ 0

• d

→ (B(t), t ≥ 0) These implies simulation on a grid and between grid points BM path is linearly interpolated

0.0 0.2 0.4 0.6 0.8 1.0 −1.0 −0.5 0.0 0.5 1.0 t B

continuous line n = 10, dashed line n = 100, dotted line n = 1000.

Pathwise approximations

Karhunen-Loève / Kac-Siegert decomposition B(t, ω) =

∞

• i=0

Ziφi(t), 0 ≤ t ≤ T with φi(t) = 2 √ 2T (2i + 1)π sin (2i + 1)πt 2T

• φi a basis of orthogonal functions and Zi i.i.d. N(0, 1)

This approximation might be too smooth

0.0 0.2 0.4 0.6 0.8 1.0 −1.0 −0.5 0.0 0.5 1.0 t B

n = 10 (continuous line), n = 50 (dashed line) and n = 100 terms

IFS-M operator

The IFS-M operator is contractive operatore defined as T(g(x)) =

N

• k=1
• αk · g

x − ak sk

• + βk
• where (αk, βk, ak) can be determined as the solution of a

contrained Quadratic Problem given some choice of (ak, sk)’s ∆2 = ||g − Tg||2

2 = min α,β

under the constraint

N

• k=1

ck(αkg1 + βk) ≤ g1 ∆2 can be rewritten as a quadratic form ∆2 = xTAx + bTx + c where x = (α1, . . . αk, β1, . . . , βk). If g = BM then ai,i = c 1 B2(t)dt aN+i,N+i = si ai,N+i = c 1 B(t)dt bi = −2 1 B(t)B((t − ai)/si)dt bN+i = −2 ai+si

ai

B(t)dt with c = 1

0 |B(t)|dt

0.0 0.2 0.4 0.6 0.8 1.0 −1.0 −0.6 −0.2 0.2 mm1 p1 0.00 0.05 0.10 0.15 0.20 0.25 0.30 −0.2 0.0 0.2 0.4 mm1 p1

Theorem (Self-affine trajectories) Let (αk, βk) be the solution of ∆2 = minα.β then the fixed point ˜ B(t) of the operator T satisfies the self affine property ˜ B(wi(t + h)) − ˜ B(wi(t)) = αi(˜ B(t + h) − ˜ B(t)) where wi(x) = aix + si Which means that the trajectory is made of rescaled copies of itself and here comes the fractal nature of the approximation.

The IFS package

IFS’s can be built on distribution functions as well (DSC 2003) and the ifs package include both families of operators (IFS-p and IFS-M) References IFSM representation of Brownian motion with applications to simulation, submitted. A comparative simulation study on the IFS distribution function estimator, Nonlinear Analysis - Real World Applications, 6, 5, 858-873 (2005). Approximating distribution functions by iterated function systems, Journal of Applied Mathematics and Decision Sciences, 9, 1, 33-46 (2005).