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Unformatted text preview: induction proceeds. By Blumenthal zeroone law if the
determinant is zero for a positive time, then it is so with probability one and there is a
deterministic direction in which it is degenerate. That direction is orthogonal to all the
vectors generated by all the Lie brackets.
Quantitative version.
We will estimate E [X −k ] by estimating E [e−λX ] and integrating
E [X −k ] = 1
Γ(k ) E [e−λX ]λk−1 dλ We will ﬁx M a bound on σ and b as well as the time interval [0, T ]. C (T, M ) will stand
for a constant that may depend on M and T but independent of λ.
2 Lemma 1. Let ξ (t) be a stochastic integral
t ξ ( t) = x +
0 t σ (s) · dβ (s) + b(s)ds
0 such that
σ (s) ≤ M and b(s) ≤ M Then for any λ ≥ 0,
E exp − λ2
4 t ξ 2 (s)ds +
0 λ
4M t σ 2 (s)ds
0 ≤ C ( t) Proof. Consider the function
U (t , x) = exp −
where λx2
1
1
tanh λM t + F (λM t) + t
4M
4
2 x F ( x) =
0 Then [1 − tanh x]dx = x − log cosh x ≤ log 2 λx
tanh λM t ]
2M
λ
λ2 x2
tanh2 λM t −
tanh λM t ]
Uxx = U [
2
4M
2M
λ2 x2
λM
1
Ut = U [−
sech2 λM t +
(1 − tanh...
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This note was uploaded on 03/19/2013 for the course MATH GA 2931 taught by Professor Varadhan during the Fall '06 term at NYU.
 Fall '06
 Varadhan

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