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In probability theory, a stochastic process is said to be continuous in probability or stochastically continuous if its distributions converge whenever the values in the index set converge.

Definition

Let ${\displaystyle X=(X_{t})_{t\in T}}$ be a stochastic process in ${\displaystyle \mathbb {R} ^{n}}$. The process ${\displaystyle X}$ is continuous in probability when ${\displaystyle X_{r}}$ converges in probability to ${\displaystyle X_{s}}$ whenever ${\displaystyle r}$ converges to ${\displaystyle s}$.

Examples and Applications

Feller processes are continuous in probability at ${\displaystyle t=0}$. Continuity in probability is a sometimes used as one of the defining property for Lévy process. Any process that is continuous in probability and has independent increments has a version that is càdlàg. As a result, some authors immediately define Lévy process as being càdlàg and having independent increments.

References

1. ^ Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53.
2. ^ Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 286.
3. Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 290.

• Stochastic processes