## Bernoulli

• Bernoulli
• Volume 20, Number 3 (2014), 1165-1209.

### Small-time expansions for local jump-diffusion models with infinite jump activity

#### Abstract

We consider a Markov process $X$, which is the solution of a stochastic differential equation driven by a Lévy process $Z$ and an independent Wiener process $W$. Under some regularity conditions, including non-degeneracy of the diffusive and jump components of the process as well as smoothness of the Lévy density of $Z$ outside any neighborhood of the origin, we obtain a small-time second-order polynomial expansion for the tail distribution and the transition density of the process $X$. Our method of proof combines a recent regularizing technique for deriving the analog small-time expansions for a Lévy process with some new tail and density estimates for jump-diffusion processes with small jumps based on the theory of Malliavin calculus, flow of diffeomorphisms for SDEs, and time-reversibility. As an application, the leading term for out-of-the-money option prices in short maturity under a local jump-diffusion model is also derived.

#### Article information

Source
Bernoulli, Volume 20, Number 3 (2014), 1165-1209.

Dates
First available in Project Euclid: 11 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.bj/1402488937

Digital Object Identifier
doi:10.3150/13-BEJ518

Mathematical Reviews number (MathSciNet)
MR3217441

Zentralblatt MATH identifier
1304.60083

#### Citation

Figueroa-López, José E.; Luo, Yankeng; Ouyang, Cheng. Small-time expansions for local jump-diffusion models with infinite jump activity. Bernoulli 20 (2014), no. 3, 1165--1209. doi:10.3150/13-BEJ518. https://projecteuclid.org/euclid.bj/1402488937

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