## The Annals of Probability

### On asymptotic errors in discretization of processes

#### Abstract

We study the rate at which the difference $X^n_t=X_t-X_{[nt]/n}$ between a process $X$ and its time-discretization converges. When $X$ is a continuous semimartingale it is known that, under appropriate assumptions, the rate is $\sqrt{n}$, so we focus here on the discontinuous case. Then $\alpha_nX^n$ explodes for any sequence $\alpha_n$ going to infinity, so we consider "integrated errors'' of the form $Y^n_t=\int_0^tX^n_s\,ds$ or $Z^{n,p}_t=\int_0^t|X^n_s|^p\,ds$ for $p\in(0,\infty)$: we essentially prove that the variables $\sup_{s\leq t}|nY^n_s|$ and $\sup_{s\leq t}nZ^{n,p}_s$ are tight for any finite $t$ when $X$ is an arbitrary semimartingale, provided either $p\geq2$ or\break $p\in(0,2)$ and $X$ has no continuous martingale part and the sum $\sum_{s\leq t}|\Delta X_s|^p$ converges a.s. for all $t<\infty$, and in addition $X$ is the sum of its jumps when $p<1$. Under suitable additional assumptions, we even prove that the discretized processes $nY^n_{[nt]/n}$ and $nZ^{n,p}_{[nt]/n}$\vadjust{\vspace{1pt}} converge in law to nontrivial processes which are explicitly given.

As a by-product, we also obtain a generalization of Itö's formula for functions that are not twice continuously differentiable and which may be of interest by itself.

#### Article information

Source
Ann. Probab., Volume 31, Number 2 (2003), 592-608.

Dates
First available in Project Euclid: 24 March 2003

https://projecteuclid.org/euclid.aop/1048516529

Digital Object Identifier
doi:10.1214/aop/1048516529

Mathematical Reviews number (MathSciNet)
MR1964942

Zentralblatt MATH identifier
1058.60020

#### Citation

Jacod, J.; Jakubowski, A.; Mémin, J. On asymptotic errors in discretization of processes. Ann. Probab. 31 (2003), no. 2, 592--608. doi:10.1214/aop/1048516529. https://projecteuclid.org/euclid.aop/1048516529

#### References

• [1] DELLACHERIE, C. and MEy ER, P. A. (1978). Probabilités et Potentiel II. Hermann, Paris.
• [2] JACOD, J. (2003). On processes with conditional independent increments and stable convergence in law. Séminaire de Probabilités XXXVI. Lecture Notes in Math. 1801 383-401. Springer, New York.
• [3] JACOD, J. and PROTTER, P. (1998). Asy mptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267-307.
• [4] JACOD, J. and SHIRy AEV, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
• [5] JUREK, Z. J. (1977). Limit distributions for sums of shrunken random variables. In Second Vilnius Conference on Probability Theory and Mathematical Statists, Abstracts of Communications 3 95-96. Vilnius.
• [6] JUREK, Z. J. (1985). Relations between the s-selfdecomposable and selfdecomposable measures. Ann. Probab. 13 592-609.
• [7] PROTTER, P. (1990). Stochastic Integration and Differential Equations. A New Approach. Springer, Berlin.
• [8] RENy I, A. (1963). On stable sequences of events. Sanky¯a 25 293-302.