17. Stochastic Processes II

What people said

Recommends qwerty_uiop · 2019-02 · Mathematics Stack Exchange · on “Expressing $\mathbb{P} \left( \sup_{s \leq t} B_s>a \right)$ in terms of stopping times”

In this video lecture [this video] the professor is proving the theorem that For a Brownian motion $(B_t)_{t \geq 0}$ it holds that $$P(M(t)>a) = 2 P(B(t)>a)$$ where $M_t := \sup_{s: s \leq t} B(s)$ . The proof uses the following axiom that $$P(M(t)>a) = P (\tau_a<t)$$ where $\tau_a =$ stopping time. My confusion is that this axiom doesn't appear correct because, all those paths which just touch $B(t)=a$ , but never cross $a$ , are a part of the RHS of the above equation, but for such paths the LHS doesn't hold true. Hence the RHS is greater than LHS in this case. So according to me the LHS should be equal to intersection of RHS with one more condition, which would make sure that the motion goes

Cites Kian · 2015-03 · Mathematics Stack Exchange · on “Mean Value Theorem and an Inequality”

First of all, I would like to say I am following MIT's video [this video] at 52:00. Assume $f$ is continuously differentiable. They say that $$ \sum_{i=1}^{n}(f(t_{i+1})-f(t_{i}))^{2}\leq\sum_{i=1}^{n}((f'(s_{i})(t_{i+1}-t_{i}))^{2} $$ by the mean value theorem where $s_{i}\in[t_{i},t_{i+1}].$ Why? I see that the mean value theorem gives us $$ f'(s_{i})(t_{i+1}-t_{i})=f(t_{i+1})-f(t_{i}) $$ for some $s_{i}\in[t_{i},t_{i+1}].$ But why is the inequality there?

Cites Christian · 2021-11 · Mathematics Stack Exchange · on “Intuitively and graphically what does the inequality of the mean value mean?”

Let $f: [a, b] \to \mathbb{R}$ $f$ is continuous on $[a, b]$ $f$ is differentiable in $(a, b)$ $f'$ (the derivative) is continuous on $[a, b]$ If all this is true, the mean value theorem can be expressed as follows: $$|f(b)−f(a)|\le (b−a)M$$ (Where $M$ is the bound for the derivative of $f$ ( $| f' | \le M$ )). Well. What I want to know is, graphically and intuitively, what does this mean? Why am I asking the question? In this youtube video [this video] (min: $43:35$ ) An attempt is made to give an intuitive explanation, but it is not rigorous at all and is therefore confusing, since the lecturer treats $M$ as $f'(x)$ and not as a bound of $f'(x)$ Thank you.