Dissertation: Analysis of mean-field stochastic differential equations (Nykänen)
Irregularity in the coefficient functions affects the solvability
Jani Nykänen's doctoral thesis in the field of mathematics studies mean-field SDEs that have discontinuity in the so-called diffusion coefficient function, which determines how strong the random movement is.
- When some quantity depending on the distribution of the solution process hits a given threshold, the magnitude of the random movement changes abruptly. One can think of a particle system where the interaction between the particles suddenly decays if the density of the particles gets too low, for instance when matter changes its state, illustrates university teacher Jani Nykänen from the Ģֱ.
In his thesis, Nykänen focuses on questions regarding the existence and uniqueness of a solution.
- Sufficiently regular coefficient functions guarantee that there is a unique solution, but as a consequence of the discontinuity there might exist multiple solutions or no solution at all, says Nykänen.
The thesis provides examples of the phenomena described above, and introduces adequate conditions that ensure the existence of a unique solution.
Simulation with the help of random walk
The second part of the doctoral thesis concerns mean-field backward SDEs, which are SDEs with a terminal condition. These equations arise naturally in the study of stochastic control problems, which are used in various fields, especially in finance.
Brownian motion, which is used to model the random noise, can be approximated with a simple, symmetric random walk. This provides a straightforward yet powerful method to simulate backward SDEs.
- The primary objective is to establish a convergence rate estimate for the approximation scheme, but the thesis also introduces the main principles on how to implement the algorithm, Nykänen explains.
The public examination of MSc Jani Nykänen's dissertation “Analysis of mean-field SDEs and BSDEs: irregular coefficients and approximation” is held on Friday 6.6.2025 at 12.00 in the lecture hall YAA303 in Ylistönrinne. Opponent is professor Gunther Leobacher (University of Graz)) and custos is professor Stefan Geiss (Ģֱ). The language of the examination is English.
The dissertation “Analysis of mean-field SDEs and BSDEs: irregular coefficients and approximation” is available in the JYX publication archive:
