Large deviations related to the law of the iterated logarithm for Ito diffusions

TL;DR

This paper establishes a small-time large deviations principle for the supremum of scaled Ito diffusions, extending classical results on Brownian motion to more general stochastic processes using advanced hitting time analysis.

Contribution

It extends large deviations results related to the law of the iterated logarithm from Brownian motion to general Ito diffusions, utilizing refined hitting time density techniques.

Findings

Derived a small-time large deviations principle for Ito diffusions

Extended Strassen's hitting time results to diffusion processes

Provided quantitative estimates for supremum deviations

Abstract

When a Brownian motion is scaled according to the law of the iterated logarithm, its supremum converges to one as time tends to zero. Upper large deviations of the supremum process can be quantified by writing the problem in terms of hitting times and applying a result of Strassen (1967) on hitting time densities. We extend this to a small-time large deviations principle for the supremum of scaled Ito diffusions, using as our main tool a refinement of Strassen's result due to Lerche (1986).

Equations

$$h(u):=\sqrt{2u\log \log \frac{1}{u}},$$

$$\underset{t\searrow 0}{\mathrm{limsup}}\frac{W_t}{h(t)}=\underset{t\searrow 0}{\lim }\underset{0<u<t}{\sup }\frac{W_u}{h(u)}=1\text{a.s.,}$$

$$P\left(\underset{0<u<t}{\sup }\frac{W_u}{h(u)}\ge \sqrt{1+\epsilon }\right)=e^{-\epsilon (\log \log \frac{1}{t})(1+o(1))},t\searrow 0.$$

$$X_t=$$

$${\sigma }_0:=\sigma (X_0,0)=\sigma (0,0)>0,$$

$$P\left(\underset{0<u<t}{\sup }|X_u|\ge c_1\right)=O(t^{c_2}),t\searrow 0,$$

$$\underset{t\searrow 0}{\mathrm{limsup}}\frac{X_t}{h(t)}=\underset{t\searrow 0}{\lim }\underset{0<u<t}{\sup }\frac{X_u}{h(u)}={\sigma }_0,\text{a.s.}$$

$$J(x):=\{\begin{array}{ll}(x/{\sigma }_0{)}^2-1& x\ge {\sigma }_0,\\ \infty & x<{\sigma }_0.\end{array}$$

$$\underset{t\searrow 0}{\mathrm{liminf}}\frac{1}{\log \log \frac{1}{t}}\log P\left(\underset{0<u<t}{\sup }\frac{X_u}{h(u)}\in O\right)$$

$$\underset{t\searrow 0}{\mathrm{limsup}}\frac{1}{\log \log \frac{1}{t}}\log P\left(\underset{0<u<t}{\sup }\frac{X_u}{h(u)}\in C\right)$$

$$P\left(\underset{0<u<t}{\sup }\frac{X_u}{h(u)}\ge {\sigma }_0\sqrt{1+\epsilon }\right)$$

$$\displaystyle=\Big{(}\log\frac{1}{t}\Big{)}^{-\varepsilon+o(1)},\quad t\searrow 0.$$

$$e^{-{\gamma }_1(\log \log \frac{1}{t})(1+o(1))}\le P\left(\underset{0<u<t}{\sup }\frac{W_u}{h(u)}\ge \sqrt{1+\epsilon }\right)\le e^{-{\gamma }_2(\log \log \frac{1}{t})(1+o(1))},t\searrow 0.$$

$$P\left(\underset{0<u<t}{\sup }\frac{|W_u|}{h(u)}\ge \sqrt{1+\epsilon }\right)\ge P\left(\underset{0<u<t}{\sup }|W_u|\ge \sqrt{1+\epsilon }h(t)\right),t\text{ small.}$$

$$P\left(\underset{0<u<t}{\sup }\frac{W_u}{h(u)}\ge \sqrt{1+\epsilon }\right)$$

$$=P\left(\sup \{v:W_v\ge \sqrt{1+\epsilon }vh(1/v)\}\ge \frac{1}{t}\right).$$

$$D_{\varphi}(s)\sim\varphi^{\prime}(s)(2\pi s)^{-1/2}\exp\big{(}{-\varphi(s)^{2}/2s}\big{)},\quad s\nearrow\infty.$$

$$\left|\frac{{\psi }_a^{\prime }(s)}{{\psi }_a^{\prime }(u)}-1\right|<\epsilon \text{if}\left|\frac{s}{u}-1\right|<\delta ,$$

$$p_{a}(u)=\frac{\Lambda_{a}(u)}{u^{3/2}}n\Big{(}\frac{\psi_{a}(u)}{\sqrt{u}}\Big{)}\left(1+o(1)\right)$$

$$n(x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2},$$

$${\Lambda }_a(u):={\psi }_a(u)-u{\psi }_a^{\prime }(u).$$

$$P\left(\underset{0<u<t}{\sup }\frac{W_u}{h(u)}\ge \sqrt{1+\epsilon +d(t)}\right)=e^{-\epsilon (\log \log \frac{1}{t})(1+o(1))},t\searrow 0.$$

$$q(t):=\sqrt{1+\epsilon +d(t)}$$

$$P\left(\underset{0<u<t}{\sup }\frac{W_u}{h(u)}\ge q(t)\right)$$

$$=P\left(\inf \left\{au>0:W_u\ge q(1/a)h(u)\right\}<1\right)$$

$$=P\left(\inf \left\{s>0:W_{s/a}\ge q(1/a)h(s/a)\right\}<1\right)$$

$$=P\left(\inf \left\{s>0:\sqrt{a}{W}_{s/a}\ge q(1/a)\sqrt{a}h(s/a)\right\}<1\right)$$

$$=P\left(\inf \left\{s>0:W_s^{\prime }\ge q(1/a)\sqrt{a}h(s/a)\right\}<1\right),$$

$${\psi }_a(u):=q(1/a)\sqrt{a}h(u/a)$$

$$P\left(\sup_{0<u<t}\frac{W_{u}}{h(u)}\geq q(t)\right)\sim\int_{0}^{1}\frac{\Lambda_{a}(u)}{u^{3/2}}n\Big{(}\frac{\psi_{a}(u)}{\sqrt{u}}\Big{)}du,\quad a=\frac{1}{t}\nearrow\infty.$$

$${\Lambda }_a(u)\sim const\cdot \sqrt{u\log \log \frac{a}{u}},$$

$$\displaystyle\int_{0}^{1}\frac{\Lambda_{a}(u)}{u^{3/2}}n\Big{(}\frac{\psi_{a}(u)}{\sqrt{u}}\Big{)}du$$

$$=const\cdot {\int }_a^{\infty }\frac{1}{x}\sqrt{\log \log x}(\log x{)}^{-(1+\epsilon +d(t))}dx$$

$$=const\cdot {\int }_a^{\infty }\frac{1}{x}(\log x{)}^{-(1+\epsilon +o(1))}dx$$

Full text

1 Introduction and main results

For a standard Brownian motion $W$ and

[TABLE]

we have

[TABLE]

by Khinchin’s law of the iterated logarithm, and there are extensions to the diffusion case. In this note we are not interested in a.s. convergence, but rather in small-time large deviations of the process $\sup_{0<u<t}X_{u}/h(u)$ for an Itô diffusion $X$. For Brownian motion, a large deviations estimate follows from a result of Strassen [8], which gives precise tail asymptotics for the last (or, by time inversion, first) time at which a Brownian motion hits a smooth curve. For fixed $\varepsilon>0$, it yields

[TABLE]

See Section 2 for details. In Theorem 2.1 below, we cite an extension of Strassen’s result due to Lerche [5], which we will use when extending the estimate (1.1) to Itô diffusions. We make the following mild assumptions on our diffusion process.

Assumption 1.1**.**

1. (i)

The continuous one-dimensional stochastic process $X=(X_{t})_{t\geq 0}$ satisfies the SDE

[TABLE] 2. (ii)

the coefficients $b$ and $\sigma$ are continuous functions from $\mathbb{R}\times[0,\infty)$ to $\mathbb{R}$ with

[TABLE] 3. (iii)

the supremum of $X$ satisfies a weak form of a small-time large deviations estimate, in the sense that there are $c_{1},c_{2}>0$ such that

[TABLE] 4. (iv)

the process $X$ satisfies the small-time law of the iterated logarithm, i.e.,

[TABLE]

By inspecting our proofs (see Lemma 3.1 and (3.7)), it is not hard to see that the continuity assumption (ii) can be slightly weakened. As for part (iii), note that it is much weaker than a classical large deviations estimate, with exponential decay rate. The latter holds, e.g., under the conditions of the small-noise LDP in [1], by applying Brownian scaling and the contraction principle. For sufficient conditions for the law of the iterated logarithm, we refer to p.57 in [6] and p.11 in [2].

Theorem 1.2**.**

Under Assumption 1.1, the process $\,\sup_{0<u<t}X_{u}/h(u)$ satisfies a small-time large deviations principle with speed $\log\log(1/t)$ and rate function

[TABLE]

This means that

[TABLE]

for any open set $O$ and

[TABLE]

for any closed set $C$, where $J(M):=\inf_{x\in M}J(x).$

Obviously, $J$ is a good rate function in the sense of [3], i.e. the level sets $\{J\leq c\},$ $c\in\mathbb{R},$ are compact. The main estimate needed to prove Theorem 1.2 is contained in the following result.

Theorem 1.3**.**

Under parts (i)–(iii) of Assumption 1.1, for $\varepsilon>0$ we have

[TABLE]

After some preparations, the proofs of Theorems 1.2 and 1.3 are given at the end of Section 3.

2 Brownian motion

We can quickly see that there are positive constants $\gamma_{1},\gamma_{2}$ (depending on $\varepsilon$) such that

[TABLE]

As for the lower estimate, note that $h(u)$ increases for small $u>0$, and thus

[TABLE]

From this and the reflection principle, it is very easy to see that we can take $\gamma_{1}=\varepsilon+1$ in (2.1). The upper estimate in (2.1) follows from applying the Borell inequality (Theorem D.1 in [7]) to the centered Gaussian process $(W_{u}/h(u))_{0<u<t},$ but neither of these estimates is sharp. To get the optimal constants $\gamma_{1}=\gamma_{2}=\varepsilon$, we use a result of Strassen [8]. By time inversion, we have

[TABLE]

Define $\varphi(v)=\sqrt{1+\varepsilon}\,vh(1/v)$. Then, by Theorem 1.2 of [8], the random variable $\sup\{v:W_{v}\geq\varphi(v)\}$ has a density $D_{\varphi}(s)$ (except possibly for some mass at zero, which is irrelevant for our asymptotic estimates), which satisfies

[TABLE]

From this, the estimate (1.1) easily follows, very similarly as in the proof of Theorem 2.2 below. That theorem strengthens (1.1), replacing $\varepsilon$ by some quantity that converges to $\varepsilon$. To prove it, we apply the following theorem due to Lerche:

Theorem 2.1** (Theorem 4.1 in [5], p.60).**

Let $T_{a}:=\inf\{u>0:W_{u}\geq\psi_{a}(u)\}$ for some positive, increasing, continuously differentiable function $u\mapsto\psi_{a}(u),$ which depends on a positive parameter $a$. Assume that there are $0<t_{1}\leq\infty$ and $0<\alpha<1$ such that

1. (i)

$P(T_{a}<t_{1})\to 0$* as $a\nearrow\infty$,* 2. (ii)

$\psi_{a}(u)/u^{\alpha}$* is monotone decreasing in $u$ for each $a$,* 3. (iii)

for every $\varepsilon>0$ there exists a $\delta>0$ such that for all $a$

[TABLE]

for $s,u\in(0,t_{1})$.

Then the density of $T_{a}$ satisfies

[TABLE]

uniformly on $(0,t_{1})$ as $a\nearrow\infty$. Here, $n$ is the Gaussian density

[TABLE]

and $\Lambda_{a}$ is defined by

[TABLE]

We can now prove the following variant of Theorem 1.3, where $X$ is specialized to Brownian motion, but $\varepsilon$ is generalized to $\varepsilon+o(1)$.

Theorem 2.2**.**

Let $d(t)$ be a deterministic function with $d(t)=o(1)$ as $t\searrow 0$. Then, for $\varepsilon>0,$

[TABLE]

Proof.

We put

[TABLE]

and $a=1/t,$ to make the notation similar to [5]. We can write the probability in (2.3) as a boundary crossing probability,

[TABLE]

where $W^{\prime}$ is again a Brownian motion, using the scaling property. We will verify in Lemma 2.3 below that the function

[TABLE]

satisfies the assumptions of Theorem 2.1. By (2.5) and the uniform estimate (2.2), we thus obtain

[TABLE]

An easy calculation shows that

[TABLE]

uniformly in $u\in(0,1),$ and so

[TABLE]

As for the third line, note that

[TABLE]

and that the exponent is $o(1)$ for $x\geq a$ and $a\nearrow\infty$. ∎

Lemma 2.3**.**

The function $\psi_{a}$ defined in (2.6), with $q$ defined in (2.4), satisfies the assumptions of Theorem 2.1.

Proof.

To verify condition (ii) of Theorem 2.1, it suffices to note that $h(u)/u^{\alpha}$ decreases for small $u$ and $\alpha\in(\tfrac{1}{2},1)$. The continuity condition (iii) easily follows from

[TABLE]

It remains to show condition (i), i.e., that

[TABLE]

converges to zero as $a\nearrow\infty$. Choose $a_{0}>0$ such that

[TABLE]

By the law of the iterated logarithm for Brownian motion, we have

[TABLE]

From this we get that there exists an $s_{0}>0$ such that

[TABLE]

By monotonicity w.r.t. $a$, we obtain

[TABLE]

For $s\in[s_{0},1]$, note that the first factor of

[TABLE]

is bounded pathwise, and that the second factor satisfies

[TABLE]

uniformly on $[s_{0},1]$. From this and (2.9), we get

[TABLE]

and together with (2.8) this implies that (2.7) converges to zero. ∎

3 Itô diffusions

We now show that our results about Itô diffusions can be reduced to the case of Brownian motion, which was handled in the preceding section. The drift of $X$ can be easily controlled by continuity and part (iii) of Assumption 1.1. Define

[TABLE]

Lemma 3.1**.**

Under parts (i)–(iii) of Assumption 1.1, we have

[TABLE]

Proof.

The continuous function $b$ is bounded by some constant $c$ on $[-c_{1},c_{1}]\times[0,t],$ independently of $t$ for $t$ small enough. Therefore, if $\sup_{0<u<t}|X_{u}|<c_{1},$ then

[TABLE]

which implies

[TABLE]

The assertion thus follows from (1.2). ∎

Note that the decay rate in (3.2) is $e^{-c_{2}\log(1/t)}$, and thus negligible in comparison to (1.1). The next step in the proof of Theorem 1.3 is contained in Lemma 3.3, which allows us to deal with the local martingale part, after expressing it as a time-changed Brownian motion. We will require the following well-known result:

Theorem 3.2** (Lévy modulus of continuity, Theorem 2.9.25 in [4]).**

For $f(\delta):=\sqrt{2\delta\log(1/\delta)},$ we have

[TABLE]

Lemma 3.3**.**

Suppose that parts (i)–(iii) of Assumption 1.1 hold. Let $\widehat{W}$ be a standard Brownian motion, and $d(t)$ a deterministic function satisfying $d(t)=o(1)$ as $t\searrow 0$. Then

[TABLE]

Proof.

By (1.2), we may assume $\sup_{0<u<t}|X_{u}|<c_{1}.$ Define

[TABLE]

Since

[TABLE]

we have

[TABLE]

by the mean value theorem. For arbitrary $u>0$ and small $\delta>0$, we have

[TABLE]

by Theorem 3.2 and Brownian scaling. From (3.5) and (3.6), we obtain

[TABLE]

on the event $\sup_{0<u<t}|X_{u}|<c_{1}.$ This implies

[TABLE]

where $\widetilde{W}$ is again a Brownian motion. Now the upper estimate in (3.3) follows from Theorem 2.2. To complete the proof of the lemma, a lower estimate for the left hand side of (3.4) is needed. We have

[TABLE]

and thus, by (3.7),

[TABLE]

using $P(A\cap B)\geq P(A)-P(B^{c})$. The first probability in (3.8) can be estimated by Theorem 2.2, and the second probability in (3.8) is asymptotically smaller by (1.2). ∎

We now conclude the paper by proving our main results, Theorem 1.3 and its consequence, Theorem 1.2.

Proof of Theorem 1.3.

Recalling the definition of $D_{t}$ in (3.1), we have

[TABLE]

By the Dambis-Dubins-Schwarz theorem (Theorem 3.4.6 and Problem 3.4.7 in [4]), the local martingale can be written as

[TABLE]

with a Brownian motion $\widehat{W}$. The upper estimate thus follows from applying (1.2), Lemma 3.1, and (3.3) to (3.9). We proceed with the lower estimate in Theorem 1.3. From

[TABLE]

and (3.10), we get

[TABLE]

Since we need a lower bound, we can intersect with the event $D_{t}\leq\sqrt{t}.$ Using $P(A\cap B)\geq P(A)-P(B^{c}),$ we obtain

[TABLE]

The lower estimate now follows from Lemma 3.1 and (3.4). ∎

Proof of Theorem 1.2.

The increasing process $\sup_{0<u<t}X_{u}/h(u)$ converges to $\sigma_{0}$ as $t\searrow 0$ by part (iv) of Assumption 1.1, and thus its values are $\geq\sigma_{0}$ a.s. Hence, there are no lower deviations, and it suffices to consider subsets of $[\sigma_{0},\infty).$ First, let $O\neq\emptyset$ be open, and $\lambda>0$ be arbitrary. We can pick $x>1$ and $\delta>0$ such that

[TABLE]

and

[TABLE]

Then,

[TABLE]

by Theorem 1.3. Therefore,

[TABLE]

Then taking $\lambda\searrow 0$ yields (1.3). Now let $C$ be closed. Recall that we may assume $C\subseteq[\sigma_{0},\infty).$ If $\inf C=\sigma_{0},$ then $J(C)=0$ as $C$ is closed, and it suffices to estimate the probability in (1.4) by $1$. Otherwise, let $\sigma_{0}\sqrt{1+\kappa}:=\inf C$ with $\kappa>0$. Then, by Theorem 1.3,

[TABLE]

∎