# Øksendal Exercise Solutions: SDEs, Ch. 5 – part a

Having completed the exercises of Chapter 4 of Øksendal’s book “Stochastic Differential Equations,” we now move on to Chapter 5 on SDEs.

Continue reading →# A few stochastic integrals and their variances

Here’s a list of a few stochastic integrals and their variances.

Continue reading →# Øksendal Exercise Solutions: SDEs, Ch. 4 – part b

Solutions of exercises in Øksendal’s book “Stochastic Differential Equations,” Chapter 4 (The Itô formula and the Martingale Representation Theorem) – part b (Exercises 4.9 and 4.12-4.17).

Continue reading →# Øksendal Exercise Solutions: SDEs, Ch. 4

Solutions of exercises in Øksendal’s book “Stochastic Differential Equations,” Chapter 4 (The Itô formula and the Martingale Representation Theorem).

# Probability cookbook: brand new version and open source on github

The second and updated version of the probability cookbook is now available. The content has almost doubled in length (the PDF is now 55 pages and will keep expanding). The new material includes a section on the Brownian motion, a section on copulas and a chapter on uncertainty quantification using polynomial chaos methods.

I created a separate page where I will be announcing new versions of the cookbook. The LaTeX source code of the document can be found on github and it is licensed under the Creative Commons Attribution 4.0 International License.

**Download** the PDF **here**.

# Strict and strong convexity of maximum of two functions

This is a brief note on the strict and strong convexity properties of the maximum of two functions. The question is whether the maximum of two strictly/strongly convex functions is strintly/strongly convex. The proofs are very simple.

# Stochastic constraints

Where in this post we discuss various types of constraints on stochastic optimal control problems starting from the classic uniform constraints to chance/probabilistic constraints and more elaborate risk constraints. We point out some caveats and propose a few remedies. Continue reading →

# Conditional risk mappings: robust representations

Coherent risk measures can be written as the support function of a set of random variables or as a worst-case expectation over a set of probability measures. This is the so-called *dual* or *robust* representation of risk measures. These representations extend to the *conditional* variants of risk measures. In this post we revisit the construction and properties of conditional risk mappings and derive the dual representation of compositions of conditional risk mappings. Continue reading →

# Probability Theory Cookbook

I have been working on a probability theory cookbook which is intended to serve as a collection of important results in general probability theory. It can be used for a quick brush up, as a quick reference or cheat sheet, but not as primary tutorial material. This is a first draft version, but I will keep adding material

Download it here: **probability cookbook v0.1**

Feedback is more than welcome!

# Mean Square Stable MJLS, but higher moments diverge

It is well known that MJLS can be mean-stable (the expected value of the state’s norm converges to 0), but not mean-square stable. It makes sense to assume that an MJLS can be mean-square stable, but some higher-order norms do not converge to 0. Despite the fact that mean-square stability conditions for MJLS are well studied and are easy to check, this is not always the case for moments of other order. In this post we present a counterexample and state conditions for p-order mean stability.

Continue reading →