Stochastic Modeling: Need Guidance on Measures & SDEs

Question

I'm embarking on a journey to master measure theory and integration, with the ultimate goal of comprehending stochastic differential equations, which are key to understanding various formalizations in stochastic modeling. To give you an idea, I'm currently working to grasp expressions like:

$\lim_{{t \to +\infty}} \sup \mathbb{E}[(1+u(t))^{p}] \leq \frac{pH}{k} := H_{0} \text{ a.s.} $

$\mathbb{E}\left[\int_{0}^{t_{n}\wedge t}V(i(\xi),s(\xi))\,\mathrm{d}\xi\right]$

$\mathbb{E}\left[ \sup_{{k\delta \leq t \leq (k+1)\delta}}(1+u(t))^{p} \right] $

$\mathbb{E}\left[ \sup_{{k\delta \leq t \leq (k+1)\delta}} \left| \int_{k\delta}^{t}p(1+u(\xi))^{p-2}(\mu-\mu u^{2}(\xi))\,\mathrm{d}\xi \right| \right]$

Could you kindly recommend detailed, step-by-step books for beginners in measure theory and stochastic differential equations?

Should I start with a dedicated measure theory resource, or dive directly into stochastic differential equations? I'm eager to acquire the necessary foundation to effectively utilize these formalizations in stochastic modeling.

I'd greatly appreciate your guidance and suggestions.

Warm regards,

Answer 1

Could you kindly recommend detailed, step-by-step books for beginners in measure theory and stochastic differential equations?

• For a first introduction to "beginner" measure theory, I think that The Elements of Integration and Lebesgue Measure by Bartle is a good fit, as it is fairly short (less than 200 pages), and builds measure theory and Lebesgue integration from the ground up with very little prerequisites beyond undergraduate calculus and linear algebra.
• Once you have the basics of measure theory down, you may look at more advanced real analysis texts. Rudin's Real and Complex Analysis is fantastic (see this answer), but it's quite a tough read. I suggest instead going through either Real Analysis by Royden and Fitzpatrick, and/or Folland's Real Analysis. Both of these books build rigorous analysis from measure theory, so they will be a good transition from Bartle's book. Also they will provide some theory of Banach/Hilbert spaces and most notably $L^p$ spaces which will be very useful for SDE theory. Folland's book covers a wider range of subjects, some of which will probably not be of interest to you, and it's shorter so the exposition is not as detailed. The first 6 chapters are still a valuable read, and as a bonus chapter 10 is a (very) short and sweet introduction to measure-theoretic probability.
• With this out of the way, assuming you already know some probability theory (if not, Billingsley's book is a good one for measure-theoretic probability), you can get started with a SDE textbook. I recommend Oksendal's Stochastic Differential Equations : An Introduction with Applications as a first introduction, as it quickly gets into what kind of problems SDEs can be useful for and how to work with them for a variety of problems. I also recommend Schilling and Partzsch' Brownian Motion : An Introduction to Stochastic Processes which builds the theory of stochastic integration from the ground up.

Hope this helps !