Central Limit Theorems: The Basic Results

This chapter is in a sense the central part of the book: it contains the Central Limit Theorems associated with the Laws of Large Numbers of Chap.

3

: the one for the unnormalized functionals

V

n

(

f

,

X

) is in Sect. 5.1, whereas Sects. 5.2 and 5.3 provide the ones for the normalized functionals

V

′

n

(

f

,

X

). In both cases,

X

needs to be an Itô semimartingale, and only regular discretization schemes are considered.

Section 5.4 contains the Central Limit Theorem for the discrete approximations of the quadratic variation, again along regular schemes. Surprisingly (or perhaps, not so surprisingly), and although this amounts to studying the functionals

V

n

(

f

,

X

) or

V

′

n

(

f

,

X

) for the test function

f

(

x

)=

x

2

(the two kinds of functionals are then identical), the assumptions on

X

needed for this case are significantly weaker than they are for more general test functions.

We also provide in Sect. 5.5 a “joint Central Limit Theorem” for the pair (

V

n

(

f

,

X

),

V

′

n

(

g

,

X

)), with

f

and

g

two (possibly multi-dimensional) test functions.

Finally, Sect. 5.6 is devoted to some applications: first we pursue the estimation of the “integrated volatility” for a continuous Itô semimartingale, and the detection of jumps for an Itô semimartingale, started in Chap.

3

, and for example give confidence bounds for the integrated volatility, or tests for deciding whether the process

X

is continuous or discontinuous. Second, we show how the Central Limit Theorem for the quadratic variation can be used for studying Euler schemes for stochastic differential equations which are driven by a general Itô semimartingale

X

; in particular, we give a Central Limit Theorem for the error incurred in this method, that is for the difference

Y

n

−

Y

where

Y

is the solution of the equation and

Y

n

is its Euler approximation with (for example) step size

$\frac{1}{n}$

.