Estimation of the Expectation through the Logarithm of Norm

Due to the reasons discussed above, it is required to find a new algorithm to overcome the difficulty in estimating the expectation. Since the error is caused by large variance, it is clear that how to reduce the variance of response in order to obtain a good estimation of expectation using a finite number of samples is important.

Let {Xj} be i. i.d. random vectors with the same distribution as X(t), then

Let F(£) be the distribution function of £n, then F(£) tends to N(0,1) as n ^ to according to the Central Limit Theorem, i. e. F(£) ^ Ф(£) uniformly, where Ф(£) is the probability distribution function of the standard normal variable

Ф(Є) = – E Ґ e-1-2dx. (21)

V Z’K J — to

Using the Edgeworth expansion theorem for distribution (Gnedenko and Kolmogorov, 1954), F(£) can be written as

TO

F (£) = Ф(£) + Е с*Ф<*>(£), (22)

к=3

where the coefficients are determined by the equivalence of moments on both sides.

The “tail effect” of distribution F (£) is important since it is required to solve the expectation of eEn-pain in eqUaiion (20). This means that accurate higher-order moments of are needed in order to obtain a good approximation of E. However, it is rather difficult to do so for the solution of a

general system (3) in practice. Therefore, in this paper, special systems are considered.

For a linear system with constant coefficients, i. e. the coefficient matrices in equation (3) take the

form

m(X, t) = mX, &(X, t) = aX, (23)

where m and a are constant matrice, it has been shown that the limiting distribution of p{t) = log ||X(t) || is normal as t ^ to if there is a constant h such that, for any vector Y,

(ctX, Y) = YTaX ^ h ||X||2 ||Y||2 (24)

is satisfied (Arnold, 1974, Khasminskii, 1980). In this case F(£) will be normal since the distribution of sum of independent normal distributed random variables is also normal, i. e. F(£) = Ф(£). Thus

і /■“ і—

logE [||X(t)f] = pp, + – log d®(f)

n J-x

1 , і nrStr2 1 9 9

= pu 4— log e 2np = pu 4— pzaz

n 2

Hence for the linear system with constant coefficients, by estimating the mean and variance of logarithm of norm, the moment Lyapunov exponents will be given by

Л(Р) = (pE [log ||X(T) ||] + ±pHar [log ||X(T) ||])

It is obvious that the variance of log ||X(T)|| will be much less than the variance of ||X(T)||; therefore obtaining a good estimation through the sample average is possible.

2 Algorithm for Linear Systems with Constant Coefficients

According to equation (26), the algorithm of simulating the moment Lyapunov exponents for linear system with constant coefficients

dX(t) = mX(t)dt + aX(t)dW(t), (27)

can be described below. The time step of iteration is Д, time interval of normalization = КД, and the sample size is S.

Step 1. Set the initial conditions of state vector X(t) by

||XS(0)|| = 1, a = 1,2,… ,5, (28)

where ||x|| =sjTx.

Step 2. Between every normalization operation, i. e. m =1, 2, • •• , M, and for every sample, use appropriate discrete scheme for equation (27) to perform К iterations for the Monte Carlo simulation of

Xs (t).

Step 3. Let

ps(mTN)=log ||Xs(m7V)||, (29)

and then use equation (5) to normalize the value of Xs(mTf) such that ||Xs{mT^)|| = 1.

Step 4. Repeat steps 2 and 3 until m = M, i. e. T = MT^. Then by the description in step 3,

||X(mTy )||
УХ((т – 1 )TN)||

M

E ps(mT)•

m=l

Step 5. Use

E [log ||Х(Г)||] = E [p(T)} = 1 £ ps(T) = p(T),

& S = 1

s (31)

Var [log ||X(T)||] = Var [p(T)} = £ [pS(^)2 – p(T)2] ,

to estimate the mean and variance of log ||X(T) ||.

Step 6. Use equation (26) to calculate the moment Lyapunov exponents for all values of p of interest.