## Get Almost sure invariance principles for partial sums of weakly PDF

By Walter Philipp

ISBN-10: 0821818619

ISBN-13: 9780821818619

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**Extra resources for Almost sure invariance principles for partial sums of weakly dependent random variables**

**Example text**

3 An approximation of the lognormal-cdf 37 Fig. 05 running an IEE Finally, it can be pointed out that the new IEE is an excellent method for the approximation of exercise probabilities, even if the underlying random variable is lognormal- or lognormal-like distributed and the pdf does not exist in closed-form. Hence, the IEE can be applied for the approximation of the single exercise probabilities given an underlying random variable which is composed by a sum of multiple lognormal-distributed random variables.

Nevertheless, we will see in the following that the application of the EE remains very accurate and efﬁcient. Given the mean µy and the standard deviation σy of a normal-distributed random variable y we obtain the pdf of a lognormal-distributed random variable x by (log(x)−µy )2 − 1 2σy2 √ e p(x) = , xσy 2π together with the moments α(m) = emµy + m2 2 2 σy . 2) we can compute the cumulants of a lognormal-distributed random variable x as follows 26 3 The Edgeworth Expansion 2 -pdf Fig. 4 Approximation error running an EE of a χ10 cum(n) = n!

31) together with the modiﬁed (log) strike price k˜ = log (P(t, T1 )K) . Hence, we directly obtain Q ˜ Q ˜ k − K · Πt,0 k , CBO1 (t, T0 , T1 ) = P(t, T1 )−1 Πt,1 with the probabilities Q [k] = EtQ e− Πt,a T0 t r(s)ds+aX(T0 ,T1 ) 1X(T0 ,T1 )>k˜ for a ∈ {0, 1} . Again, starting from the transform Θt (z) = EtQ e− T0 t r(s)ds+zX(T0 ,T1 ) we can show that the exponential afﬁne guess holds for Θt (z) = P(t, T0 )ezX(t,T0 )+A(t,z) . g. 1). 3 Pricing of coupon bond options 57 and ft,2 (φ ) = eiφ (X(t)− 2 Λ (t))− 1 φ2 2 Λ (t) , together with the variance N 1 i σ (T0 , T1 )2 1 − e−2βi (T0 −t) .

### Almost sure invariance principles for partial sums of weakly dependent random variables by Walter Philipp

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