Module Diffusion

Introduction

The functions in this module operate on three conceptual different objects, (although they are currently just represented as vectors and arrays.)

Stochastic processes, denoted x, y, w are arrays of values which are sampled at distance dt, ds, where dt, ds are either scalar or Vectors length(dt)=size(W)[end]. Stochastic differentials are denoted dx, dw etc., and are first differences of stochastic processes. Finally, t can denote the total time or correspond to a vector of size(W)[end] sampling time poins.

Note the following convention: In analogy with the definition of the Ito integral,

intxdw[i] = x[i]](w[i+1]-w[i]) (== x[i]dw[i])

and

length(w) = length(dw) + 1

Reference

Diffusion.brown1(u, t, n::Integer)

Compute n equally spaced samples of 1d Brownian motion in the interval [0,t], starting from point u

Diffusion.brown(u, t, d::Integer, n::Integer)

Simulate n equally spaced samples of d-dimensional Brownian motion in the interval [0,t], starting from point u

Diffusion.dW1(t, n::Integer)

Diffusion.dW(t, d::Integer, n::Integer)

Simulate a 1-dimensional (d-dimensional) Wiener differential with n values in the the interval [0,t], starting from point u

Diffusion.dW(dt::Vector, d::Integer)

Simulate a d-dimensional Wiener differential sampled at time points with distances given by the vector dt

Diffusion.ito(y, dx)

Diffusion.ito(dx)

Diffusion.cumsum0(dx)

Integrate a valued stochastic process with respect to a stochastic differential. R, R^2 (d rows, n columns), R^3.

ito(dx) is a shortcut for ito(ones(size(dx)[end], dx). So ito(dx) is just a cumsum0 function which is a inverse to dx = diff([0, x1, x2, x3,...]).

..(y, dx)

Diffusion.ydx(y, dx)

y .. dx returns the stochastic differential ydx defined by the property

ito(ydx) == ito(y, dx)

Diffusion.bb(u, v, t, n)

Simulates n equidistant samples of a Brownian bridge from point u to v in time t

Diffusion.dWcond1(v, t, n)

Simulates n equidistant samples of a “bridge noise”: that is a Wiener differential dW conditioned on W(t) = v

Diffusion.aug(dw, dt, n)

Diffusion.aug(dt, n)

Take Wiener differential sampled at dt and return Wiener differential subsampled n times between each observation with new length length(dw)*n. aug(dt,n) computes the corresponding subsample of times.

Diffusion.quvar(x)

Computes quadratic variation of x.

Diffusion.bracket(x)

Diffusion.bracket(x, y)

Computes quadratic variation process of x (of x and y).

Diffusion.euler(t0, u, b, sigma, dt, dw)

Diffusion.euler(t0, u, b, sigma, dt)

Simulates a 1-dimensional diffusion process using the Euler-Maruyama approximation with drift b(t,x) and diffusion coefficient sigma(t,x) starting in (t0, u) using dt and given Wiener differential dw.