Large deviations related to the law of the iterated logarithm for Ito diffusions
aa r X i v : . [ m a t h . P R ] M a r Large deviations related to the law of the iterated logarithmfor Itô diffusions ∗ Stefan GerholdTU [email protected] Christoph GersteneckerTU [email protected] 5, 2019
Abstract
When a Brownian motion is scaled according to the law of the iterated logarithm, itssupremum converges to one as time tends to zero. Upper large deviations of the supremumprocess can be quantified by writing the problem in terms of hitting times and applying a resultof Strassen (1967) on hitting time densities. We extend this to a small-time large deviationsprinciple for the supremum of scaled Itô diffusions, using as our main tool a refinement ofStrassen’s result due to Lerche (1986).
For a standard Brownian motion W and h ( u ) := r u log log 1 u , we have lim sup t ց W t h ( t ) = lim t ց sup , ityields P (cid:18) sup
0) = σ (0 , > , ∗ We gratefully acknowledge financial support from the Austrian Science Fund (FWF) under grant P30750. X satisfies a weak form of a small-time large deviations estimate, in thesense that there are c , c > such that P (cid:18) sup
Under Assumption 1.1, the process sup
Under parts (i) – (iii) of Assumption 1.1, for ε > we have P (cid:18) sup
We can quickly see that there are positive constants γ , γ (depending on ε ) such that e − γ (log log t )(1+ o (1)) ≤ P (cid:18) sup , and thus P (cid:18) sup W u ≥ ψ a ( u ) } for some positive, in-creasing, continuously differentiable function u ψ a ( u ) , which depends on a positive parameter a .Assume that there are < t ≤ ∞ and < α < such that(i) P ( T a < t ) → as a ր ∞ ,(ii) ψ a ( u ) /u α is monotone decreasing in u for each a ,(iii) for every ε > there exists a δ > such that for all a (cid:12)(cid:12)(cid:12)(cid:12) ψ ′ a ( s ) ψ ′ a ( u ) − (cid:12)(cid:12)(cid:12)(cid:12) < ε if (cid:12)(cid:12)(cid:12) su − (cid:12)(cid:12)(cid:12) < δ, for s, u ∈ (0 , t ) .Then the density of T a satisfies p a ( u ) = Λ a ( u ) u / n (cid:16) ψ a ( u ) √ u (cid:17) (1 + o (1)) (2.2) uniformly on (0 , t ) as a ր ∞ . Here, n is the Gaussian density n ( x ) = 1 √ π e − x / , and Λ a is defined by Λ a ( u ) := ψ a ( u ) − uψ ′ a ( u ) . We can now prove the following variant of Theorem 1.3, where X is specialized to Brownianmotion, but ε is generalized to ε + o (1) . Theorem 2.2.
Let d ( t ) be a deterministic function with d ( t ) = o (1) as t ց . Then, for ε > ,P (cid:18) sup
We put q ( t ) := p ε + d ( t ) (2.4)3nd a = 1 /t, to make the notation similar to [5]. We can write the probability in (2.3) as aboundary crossing probability, P (cid:18) sup W u ≥ q (1 /a ) h ( u ) } < a (cid:19) = P (inf { au > W u ≥ q (1 /a ) h ( u ) } < P (cid:0) inf (cid:8) s > W s/a ≥ q (1 /a ) h ( s/a ) (cid:9) < (cid:1) = P (cid:0) inf (cid:8) s > √ aW s/a ≥ q (1 /a ) √ ah ( s/a ) (cid:9) < (cid:1) = P (cid:0) inf (cid:8) s > W ′ s ≥ q (1 /a ) √ ah ( s/a ) (cid:9) < (cid:1) , (2.5)where W ′ is again a Brownian motion, using the scaling property. We will verify in Lemma 2.3below that the function ψ a ( u ) := q (1 /a ) √ ah ( u/a ) (2.6)satisfies the assumptions of Theorem 2.1. By (2.5) and the uniform estimate (2.2), we thus obtain P (cid:18) sup
The function ψ a defined in (2.6) , with q defined in (2.4) , satisfies the assumptionsof Theorem 2.1.Proof. To verify condition (ii) of Theorem 2.1, it suffices to note that h ( u ) /u α decreases for small u and α ∈ ( , . The continuity condition (iii) easily follows from log( t ) ∼ log( T ) , t/T ր , t, T ր ∞ . It remains to show condition (i), i.e., that P ( T a <
1) = P (cid:0) inf (cid:8) s > W ′ s ≥ q (1 /a ) √ ah ( s/a ) (cid:9) < (cid:1) = P sup such that q (1 /a ) ≥ r ε, a ≥ a . (2.8)4y the law of the iterated logarithm for Brownian motion, we have lim s ց sup such that sup
Under parts (i) – (iii) of Assumption 1.1, we have P (cid:16) D t > √ t (cid:17) = O ( t c ) , t ց . (3.2) Proof.
The continuous function b is bounded by some constant c on [ − c , c ] × [0 , t ] , independentlyof t for t small enough. Therefore, if sup √ t, sup
Suppose that parts (i) – (iii) of Assumption 1.1 hold. Let c W be a standard Brownianmotion, and d ( t ) a deterministic function satisfying d ( t ) = o (1) as t ց . Then P sup
By (1.2), we may assume sup and small δ > , we have max ≤ x Recalling the definition of D t in (3.1), we have P (cid:18) sup √ t ) . The lower estimate now follows from Lemma 3.1 and (3.4).7 roof of Theorem 1.2. The increasing process sup be arbitrary. Wecan pick x > and δ > such that inf O < σ √ x − δ < σ √ x + δ < inf O + λ and (cid:0) σ √ x − δ, σ √ x + δ (cid:1) ⊆ O. Then, P (cid:18) sup . Then, by Theorem 1.3, lim sup t ց t log P (cid:18) sup
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