# Course on Stochastic Differential Equations, Course materials

Pietro Faccioli

LECT. 1: Stochastic dynamics of bio-molecules.

The Langevin Eq. is often used to model the dynamics of a macro-molecule in its solvent.
In this first lecture, I will begin by mentioning the connection between the Langevin and the Hamiltonian dynamics and I will discuss the advantages and disadvantages of both approaches. Then, we shall see why, in both cases, the direct integration of the equation of motion provides an extremely inefficient method to investigate rare thermally activated molecular reactions, such as protein folding and protein-protein aggregation. In this case, the stochastic nature of the Langevin Eq. allows to rigorously formulate the dynamics in terms of a path integral. Based on the path integral formalism it is possible to develop a much more efficient approach to the kinetics of rare transitions. I will show some applications to protein folding and to chemical reactions.

LECT. 2: Stochastic dynamics at variable time resolution power

The numerical difficulties arising in the integration of the Eq. of motion of a molecular system are intimately related to the existence of a variety of widely separated time scales, in the internal dynamics. In this lecture, I shall discuss how, using the stochastic path integrals formalism, it is possible to construct effective theories which lead by construction to the same long-time dynamics, but have a different time resolution power. Using such effective theories one can generate directly the long-time evolution, without worrying about simulating the short-time oscillations. We shall discuss applications of these ideas to molecular systems.
Faccioli's lectures in pdf format

Stefan Geiss

In the first step the course introduces the Brownian motion $W=(W_t)_{t \ge 0}$ and stochastic integration with respect to the Brownian motion. Next we turn to Itô’s formula which is the stochastic counterpart of Taylor's formula. Our final goal is to consider stochastic differential equations of the form
X_t = x_0 + \int_0^t \sigma(s,X_s) dW_s + \int_0^t a(s,X_s) ds
and some applications. The course is partly based on the script
S. Geiss: Ten lectures about stochastic differential equations.
http://users.jyu.fi/~geiss/scripts.html

Andreas Neuenkirch

Numerics of Stochastic Differential Equations: A Short Introduction

I will give an introduction into the strong and weak approximation of stochastic differential equations, focusing mainly on the Euler method. For the strong approximation problem I will also discuss higher order methods,  while for weak approximation I will present the plain Monte-Carlo method and the Multi-level approach in detail.

I discuss some applications of stochastic differential equations in
population biology. I start with non-spatial models that have been used to
study e.g. the nature of population fluctuations and to predict the risk of
population extinction in a single well-mixing local population [1].
Stochasticity can arise from different types of sources, representing e.g.
demographic stochasticity (random variation in the number of births and
deaths) and environmental stochasticity (random variation in the
environmental conditions influencing the death and birth rates of all
individuals simultaneously). I then move to spatial and stochastic models,
including models that can be interpreted both [2] in terms of animal
movement [3] and in terms of evolution by natural selection and genetic
drift, and models of spatio-temporal population dynamics [4].

Teemu Laurila

Laurila's slides are found here