Disputas: Hannes Hagen Haferkorn

M.Sc. Hannes Hagen Haferkorn ved Matematisk institutt vil forsvare sin avhandling for graden ph.d.:

The effect of noise in the modelling and the analysis of random systems

 

Hannes Hagen Haferkorn

Tid og sted for prøveforelesning

13. desember 2017 kl. 10.15, Ullevål Stadion: Gates of Eden

Bedømmelseskomité

  • Professor Francesco Russo, Ecole Nationale Supérieure des Techniques Avancées, ENSTA-ParisTech
  • Professor Peter Imkeller, Humboldt-Universität zu Berlin
  • Associate Professor Ingrid Hobæk Haff, Universitetet i Oslo

Leder av disputas

Instituttleder Geir Dahl, Matematisk institutt, Universitet i Oslo

Veiledere

Sammendrag

During the last decades, mathematical models incorporating randomness became increasingly popular in order to model the uncertainty that is inherent in nature. Often, one is interested in describing how a certain entity develops over time. Typically this is done via so called differential equations, which describe the change dX(t) of the “state” or “position” X of the observed entity in any “infinitely small” time interval (t,t+dt) as a function f (t,X(t)) multiplied with the length dt of the interval: dX(t)=f(t,X(t))dt.

If the change of state additionally incorporates a random component, which we refer to as “(stochastic) noise”, one speaks of a stochastic differential equation: dX(t)=f(t,X(t))dt+”noise”.

This thesis aims at improving our understanding of the effects that this stochastic noise and its particular choice has on differential equations.

The thesis consists of three articles. In the first article, we consider a so called “mean-field stochastic differential equation” where we manipulate the noise by a time-change. Time-change allows for extremely flexible models and is particularly interesting for financial applications as an intuitive way to introduce stochastic volatility. In this setup we investigate an optimal control problem such as the problem of finding an optimal investment strategy. The time-change makes it necessary to develop a new technique based on “conditioning” and “enlargement of filtration” in order to solve the optimization problem.

The second and the third article both deal with so-called “delay equations”. Here, the change of the state of the observed entity in the interval (t,t+dt) does not only depend on the state at time t but on the entire path that the entity described from time t-r to time t: dX(t)=f(t,”path of X from t-r to t”)dt+”noise”.

The second article considers evaluations based on such models (e.g. the price of an option on a stock whose price dynamics is described by a stochastic delay equation). We investigate how sensitive the evaluations behave for small changes in the initial path of the underlying dynamic. Here, it turns out very helpful to randomize the initial path by introducing additional noise.

In the final article, we study delay equations with very singular coefficients. This means that the aforementioned function f(t,”path of X from t-r to t”) is very “rough”. In the deterministic setup, such equations can very easily have several solutions or no solution at all. However, we prove that adding a noise which is “rough enough” inside the function f can insure the existence and the uniqueness of a solution. This might have implications on our view of the world: Often, stochastic noise is seen as the result of measurement error or some physical effect that is impossible to explain or predict but that one cannot get rid of either. However, our results suggest that it might just be the randomness of the world that allows such equations to have solutions, meaning that effects described by such equations would not possibly exist in a world without randomness.  

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Publisert 28. nov. 2017 10:59 - Sist endret 28. nov. 2017 11:07