168
Appendix E
the mistake we make to replace A(X) by A(X(t)) in Eq. E.10 is negligible for dt
infinitesimal, then we can write
dX(t) = A(X(t))dt + BdW (t)
(E.11)
We saw that the “Wiener’s increment”, dW (t), is a Gaussian SP, with σ =
√
dt.
That’s why, an each integration pass we’ll have to draw dW(t) and normalize the
result appropriately. Let’s call R G a random number, with Gaussian distribution,
centered at R G = 0 and width 1. In MATLAB/OCTAVE R G = randn. With this
convention the last term of Eq. E.11 can be written as
BdW (t) =
√
dtBR G
In case de m independent noises, dW (t) will be a column vector of m components and we have to replace randn by randn(m, 1). The more used algorithm
in the integration of stochastic equations it’s Euler’s. We divide the integration interval [0, t max ] in n intervals of size dt. We create the “vector” t =
(t 1 , t 2 , . . . t j , . . . , t n+1 ) = (0, dt, 2dt, . . . , t max ). For efficiency of the procedure,
we draw once all the “Wiener’s increments”, creating the dW vector. In MATLAB/OCTAVE we write
dW = sqrt (dt) ∗ randn(1, n)
The i mth integration pass will be,
x(i + 1) = x(i) + A(x(i)) ∗ dt + BdW (i)
When we perform several realizations of the stochastic process we have an extra
index, j corresponding to the j mth realization, namely
x(i + 1, j + 1) = x(i, j ) + A(x(i, j )) ∗ dt + BdW (i, j )
E.1.6 SDE with Multiplicative Noise
We write now the system of equations in terms of the derivatives and white noise,
letting, however, both A and B depend on X and may also have depending explicit
in t,
dX
dt
= A(X, t) + B(X, t)ξ(t)
(E.12)
Eq. E.12 is called stochastic differential equation with multiplicative noise. Tranforming it as we done in the case of aditive noise
Appendix E
the mistake we make to replace A(X) by A(X(t)) in Eq. E.10 is negligible for dt
infinitesimal, then we can write
dX(t) = A(X(t))dt + BdW (t)
(E.11)
We saw that the “Wiener’s increment”, dW (t), is a Gaussian SP, with σ =
√
dt.
That’s why, an each integration pass we’ll have to draw dW(t) and normalize the
result appropriately. Let’s call R G a random number, with Gaussian distribution,
centered at R G = 0 and width 1. In MATLAB/OCTAVE R G = randn. With this
convention the last term of Eq. E.11 can be written as
BdW (t) =
√
dtBR G
In case de m independent noises, dW (t) will be a column vector of m components and we have to replace randn by randn(m, 1). The more used algorithm
in the integration of stochastic equations it’s Euler’s. We divide the integration interval [0, t max ] in n intervals of size dt. We create the “vector” t =
(t 1 , t 2 , . . . t j , . . . , t n+1 ) = (0, dt, 2dt, . . . , t max ). For efficiency of the procedure,
we draw once all the “Wiener’s increments”, creating the dW vector. In MATLAB/OCTAVE we write
dW = sqrt (dt) ∗ randn(1, n)
The i mth integration pass will be,
x(i + 1) = x(i) + A(x(i)) ∗ dt + BdW (i)
When we perform several realizations of the stochastic process we have an extra
index, j corresponding to the j mth realization, namely
x(i + 1, j + 1) = x(i, j ) + A(x(i, j )) ∗ dt + BdW (i, j )
E.1.6 SDE with Multiplicative Noise
We write now the system of equations in terms of the derivatives and white noise,
letting, however, both A and B depend on X and may also have depending explicit
in t,
dX
dt
= A(X, t) + B(X, t)ξ(t)
(E.12)
Eq. E.12 is called stochastic differential equation with multiplicative noise. Tranforming it as we done in the case of aditive noise