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Are stochastic processes differentiable?
In contrast to Hölder continuity of stochastic processes and random fields, sample differen- tiability has hardly been studied so far, partly because the Brownian motion and the related stochastic processes are almost surely nowhere differentiable.
What makes a derivative not differentiable?
A function is not differentiable at a if its graph has a vertical tangent line at a. The tangent line to the curve becomes steeper as x approaches a until it becomes a vertical line. Since the slope of a vertical line is undefined, the function is not differentiable in this case.
How do you know if an equation is not differentiable?
Since function f is defined using different formulas, we need to find the derivative at x = 0 using the left and the right limits. The limits to the left and to the right of x = 0 are not equal therefore f ‘(0) is undefined and function f in not differentiable at x = 0.
Why is Brownian motion not differentiable?
As we have seen, even though Brownian motion is everywhere continuous, it is nowhere differentiable. The randomness of Brownian motion means that it does not behave well enough to be integrated by traditional methods.
Is every continuous function differentiable?
We have the statement which is given to us in the question that: Every continuous function is differentiable. Therefore, the limits do not exist and thus the function is not differentiable. But we see that f(x)=|x| is continuous because limx→cf(x)=limx→c|x|=f(c) exists for all the possible values of c.
How are Stochastic Processes classified in different ways?
A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the cardinality of the index set and the state space.
When is the stochastic process said to be in discrete time?
When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in discrete time.
How is a stochastic process represented in probability theory?
In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables. Many stochastic processes can be represented by time series. However, a stochastic process is by nature continuous while a time series is a set of observations indexed by integers.
How is a sample function formed in a stochastic process?
Sample function. A sample function is a single outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process. More precisely, if is a stochastic process, then for any point , the mapping is called a sample function, a realization, or,…