Stochastic Modeling: Exercise Class 3.

Stochastic Modeling: Exercise Class 3.

Olivier Wintenberger: olivier.wintenberger@upmc.fr

Exercise 1. Assume one wants to simulate the uniform distribution over the unit ballB₁ in R^d.

• Show thatdrandom variablesU1, . . . , Ud∼ U(0,1) iid follows the uniform distribution over [0,1]^d.

• Deduce a uniform random vector over the hyper-rectangle [−1,1]^dfrom U1, . . . , Ud.

• Show that the rejection step in order to simulate the uniform distribution overB₁ has efficiencyM = 2^d/|B₁|.

• Describe the behavior ofM with respect to the dimension d.

Exercise 2. Consider the integral I =R

∆h(x)dx where ∆⊂R^d and |∆|<∞.

• Show that fromdrandom variablesU₁, . . . , U_done can simulate a uniform distribution U over a hyper-rectangleR so that ∆⊆R.

• Show thatI_n = ¹_nPn

i=1g(U_i) where g(x) =|R|h(x) I1_{x∈∆} is an unbiased approxima- tion ofI.

• Decompose the variance Var (g(U)) into two terms, one increasing with the efficiency M of the rejection step and another one independent ofM =|R|/|∆|.

• We assume thath is positive and bounded over ∆. Consider the set D=n

(x, y)∈R^d+1 :x∈∆,0≤y≤h(x)o

and the random variable Y = I1V∈D where V ∼ U(R×[0,sup_∆h]). Determine the distribution of Y.

• Show thatI_n⁰ = ^|R|sup_n^∆hPn

i=1Y_iis an alternative unbiased approximation ofI,Y₁, . . . Y_n iid versions ofY.

• Compute the variance of Y.

• Compare the variances associated toI_n and I_n⁰. Which method has the best accuracy?

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