## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 25 (2020), paper no. 16, 11 pp.

### Large deviations related to the law of the iterated logarithm for Itô diffusions

Stefan Gerhold and Christoph Gerstenecker

#### 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 Itô diffusions, using as our main tool a refinement of Strassen’s result due to Lerche (1986).

#### Article information

**Source**

Electron. Commun. Probab., Volume 25 (2020), paper no. 16, 11 pp.

**Dates**

Received: 12 March 2019

Accepted: 9 February 2020

First available in Project Euclid: 18 February 2020

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1581995086

**Digital Object Identifier**

doi:10.1214/20-ECP297

**Subjects**

Primary: 60F10: Large deviations 60J60: Diffusion processes [See also 58J65] 60J65: Brownian motion [See also 58J65]

**Keywords**

large deviations principle law of the iterated logarithm boundary crossings Itô diffusion

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Gerhold, Stefan; Gerstenecker, Christoph. Large deviations related to the law of the iterated logarithm for Itô diffusions. Electron. Commun. Probab. 25 (2020), paper no. 16, 11 pp. doi:10.1214/20-ECP297. https://projecteuclid.org/euclid.ecp/1581995086