Geometric Brownian Motion as the limit of Binomial Tree

I know that GBM can be discretely approximated by methods such as Euler-Maruyama, and it can be shown that Binomial tree converges to GBM at the continuous time limit.

However I’m having a hard time to understand the intuition. If a GBM process behaves like a binomial tree, then wouldn’t the log diffrences have only two possible outcomes?

My understanding is that time step is the key. Suppose the infetestimal step is hourly, but when we observe daily data points, there are already $$2^{24}$$ possible outcomes and these daily observations are log-normal. Comments?

Reference request: (very specific) old (French?) calculus/real analysis book with a geometric orientation?

This is a spiritual followup to my question from a while back.

Reference request: Oldest calculus, real analysis books with exercises?

I remember a few months ago coming across an old (French?) calculus/real analysis book with a very visual, geometric orientation. Indeed, there was a review for it (maybe in the Bulletin of the AMS?) that described the book as having a geometric emphasis relative to the other well-known French analysis books like Cauchy, Goursat, for instance. Does anyone know what this book might be?

Approximation argument in geometric flows

I’m studying by myself Mean Curvature Flow and I’m reading the paper “Interior estimates for hypersurfaces moving by mean curvature” by Klaus Ecker and Gerhard Huisken and I’m stuck in the following theorem:

$$\textbf{Theorem 5.1}$$: $$M_0 \equiv F_0(\mathbb{R}^n)$$ be a locally Lipschitz continuous entire graph over $$\mathbb{R}^n$$, then the initial value problem $$(1)$$ has a smooth solution $$M_t = F_t(\mathbb{R}^n)$$ for all $$t > 0$$. Moreover, each $$M_t$$ is an entire graph over $$\mathbb{R}^n$$.

I didn’t understand what approximation argument exactly was used in the the last paragraph of the proof:

From Corollary $$3.5 (ii)$$ we then conclude for any integer $$m \geq 0$$

$$\sup_{B_{R_0}(0) \times [0,T]} \left| D^m w_R \right| \leq c_m,$$

where $$c_m = c_m(m,n,R_0,c_0, c_1)$$.

We can therefore select a sequence of solutions $$(w_{R_k})$$ for $$R_k \rightarrow \infty$$ ($$R_k > R_1$$ for any $$k \geq 2$$) s.t. $$w_{R_k} \rightarrow w$$ in $$\mathcal{C}^{\infty}$$ uniformly on $$\Omega \times [0,T]$$. Since $$\Omega$$ and $$T > 0$$ were arbitrary this establishes the existence of a family of entire graphs $$M_t = \text{graph} \ w(\cdot, t)$$ solving (1) where $$w \in \mathcal{C}^{\infty}(\mathbb{R}^n \times (0,\infty))$$. As the second and higher order derivative estimates for $$w$$ on each compact subset of $$\mathbb{R}^n$$ depend only on the initial height and gradient on a slightly larger subset, an approximation argument yields a smooth solution of (1) also for locally Lipschitz initial data.

This last paragraph leads me to think that I need some corollary of Arzela-Ascoli theorem and proceed as the section “4.4 – Curvature explodes” of this thesis, which teach how construct a global solution. This thesis construct a global solution for the curve shortening flow and the author of thesis states that uniform limit of the subsequence there it’s a smooth map and it’s clear for me, because this result:

$$\textbf{Theorem A.4.2}$$ Let $$\Omega \subset \mathbb{R}^s$$ be a closed bounded set. There is a function $$u \in \mathcal{C}^{\infty}(\Omega)$$ and a sequence $$m_j \rightarrow \infty$$ with

$$\max_{x,t} \left| \frac{\partial^{p+q}}{\partial x^p \partial t^q} u_{m_j}(x,t) – \frac{\partial^{p+q}}{\partial x^p \partial t^q} u(x,t) \right| \rightarrow 0 \ \text{as} \ m_j \rightarrow \infty.$$

This theorem can be found on Appendix $$4$$ of the book “Initial-Boundary Value Problems and the Navier-Stokes Equations” by Heinz-Otto Kreiss and Jens Lorenz, but the problem is that I don’t have uniform bounds for $$\frac{\partial^{p+q}}{\partial x^p \partial t^q} w_{R_k}$$, so I don’t know how to proceed here. Is the approximation argument commented by authors really an argument based on Arzela-Ascoli theorem? Anyone can give some details about this argument?

Thanks in advance!

$T_1, … , T_n$ time of life of instruments with geometric distribution of parameter p independent

$$T_1, … , T_n$$ time of life of instruments with geometric distribution of parameter p independents. I define S as the first n instants of failure and U as the last n instant of failure. I want to find the law of S and the density of U.

This is what I did so far:

To find the law of S I start by calculating: $$\mathbb{P}(S = k) = \mathbb{P}(S \geq k) – \mathbb{P}(S \geq k + 1)$$ $$\mathbb{P}(S = k) = \mathbb{P}(S < k+1) – \mathbb{P}(S < k)$$ But here I am stuck on what I should do because I still do not know the distribution.

I do the same reasoning with U as follows: $$\mathbb{U}(S = k) = \mathbb{U}(U \leq k) – \mathbb{U}(S \leq k + 1)$$ But I still get nowhere.

Any suggestions?