An increasing process is a

stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...

... :

(X_t)_

...where the

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...

X_t

which make up the process are increasing

almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur ha ...

and adapted: :

0=X_0 \leq X_ \leq \cdots .

A continuous increasing process is such a process where the set

M

is continuous. Consider a stochastic process

(\Chi_t)

satisfying

X_t \leq X_s

a.s. for all

t \leq s

My question is: Does there exist a modification

\breve

of ,

X

which almost surely has increasing sample paths

t \mapsto \breve_t(\omega)

References

Stochastic processes {{probability-stub