| 1. |
- Gabrysch, Katja
(författare)
-
On Directed Random Graphs and Greedy Walks on Point Processes
- 2016
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of an introduction and five papers, of which two contribute to the theory of directed random graphs and three to the theory of greedy walks on point processes. We consider a directed random graph on a partially ordered vertex set, with an edge between any two comparable vertices present with probability p, independently of all other edges, and each edge is directed from the vertex with smaller label to the vertex with larger label. In Paper I we consider a directed random graph on ℤ2 with the vertices ordered according to the product order and we show that the limiting distribution of the centered and rescaled length of the longest path from (0,0) to (n, [na] ), a<3/14, is the Tracy-Widom distribution. In Paper II we show that, under a suitable rescaling, the closure of vertex 0 of a directed random graph on ℤ with edge probability n−1 converges in distribution to the Poisson-weighted infinite tree. Moreover, we derive limit theorems for the length of the longest path of the Poisson-weighted infinite tree. The greedy walk is a deterministic walk on a point process that always moves from its current position to the nearest not yet visited point. Since the greedy walk on a homogeneous Poisson process on the real line, starting from 0, almost surely does not visit all points, in Paper III we find the distribution of the number of visited points on the negative half-line and the distribution of the index at which the walk achieves its minimum. In Paper IV we place homogeneous Poisson processes first on two intersecting lines and then on two parallel lines and we study whether the greedy walk visits all points of the processes. In Paper V we consider the greedy walk on an inhomogeneous Poisson process on the real line and we determine sufficient and necessary conditions on the mean measure of the process for the walk to visit all points.
|
|
| 2. |
- Görgens, Maik, 1984-
(författare)
-
Gaussian Bridges : Modeling and Inference
- 2014
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of a summary and five papers, dealing with the modeling of Gaussian bridges and membranes and inference for the α-Brownian bridge.In Paper I we study continuous Gaussian processes conditioned that certain functionals of their sample paths vanish. We deduce anticipative and non-anticipative representations for them. Generalizations to Gaussian random variables with values in separable Banach spaces are discussed. In Paper II we present a unified approach to the construction of generalized Gaussian random fields. Then we show how to extract different Gaussian processes, such as fractional Brownian motion, Gaussian bridges and their generalizations, and Gaussian membranes from them.In Paper III we study a simple decision problem on the scaling parameter in α-Brownian bridges. We generalize the Karhunen-Loève theorem and obtain the distribution of the involved likelihood ratio based on Karhunen-Loève expansions and Smirnov's formula. The presented approach is applied to a simple decision problem for Ornstein-Uhlenbeck processes as well. In Paper IV we calculate the bias of the maximum likelihood estimator for the scaling parameter and propose a bias-corrected estimator. We compare it with the maximum likelihood estimator and two alternative Bayesian estimators in a simulation study. In Paper V we solve an optimal stopping problem for the α-Brownian bridge. In particular, the limiting behavior as α tends to zero is discussed.
|
|
| 3. |
|
|
| 4. |
- Renlund, Henrik, 1979-
(författare)
-
Recursive Methods in Urn Models and First-Passage Percolation
- 2011
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This PhD thesis consists of a summary and four papers which deal with stochastic approximation algorithms and first-passage percolation. Paper I deals with the a.s. limiting properties of bounded stochastic approximation algorithms in relation to the equilibrium points of the drift function. Applications are given to some generalized Pólya urn processes. Paper II continues the work of Paper I and investigates under what circumstances one gets asymptotic normality from a properly scaled algorithm. The algorithms are shown to converge in some other circumstances, although the limiting distribution is not identified. Paper III deals with the asymptotic speed of first-passage percolation on a graph called the ladder when the times associated to the edges are independent, exponentially distributed with the same intensity. Paper IV generalizes the work of Paper III in allowing more edges in the graph as well as not having all intensities equal.
|
|
| 5. |
- Wanntorp, Henrik, 1975-
(författare)
-
Optimal Stopping and Model Robustness in Mathematical Finance
- 2008
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Optimal stopping and mathematical finance are intimately connected since the value of an American option is given as the solution to an optimal stopping problem. Such a problem can be viewed as a game in which we are trying to maximize an expected reward. The solution involves finding the best possible strategy, or equivalently, an optimal stopping time for the game. Moreover, the reward corresponding to this optimal time should be determined. It is also of interest to know how the solution depends on the model parameters. For example, when pricing and hedging an American option, the volatility needs to be estimated and it is of great practical importance to know how the price and hedging portfolio are affected by a possible misspecification.The first paper of this thesis investigates the performance of the delta hedging strategy for a class of American options with non-convex payoffs. It turns out that an option writer who overestimates the volatility will obtain a superhedge for the option when using the misspecified hedging portfolio.In the second paper we consider the valuation of a so-called stock loan when the lender is allowed to issue a margin call. We show that the price of such an instrument is equivalent to that of an American down-and-out barrier option with a rebate. The value of this option is determined explicitly together with the optimal repayment strategy of the stock loan.The third paper considers the problem of how to optimally stop a Brownian bridge. A finite horizon optimal stopping problem like this can rarely be solved explicitly. However, one expects the value function and the optimal stopping boundary to satisfy a time-dependent free boundary problem. By assuming a special form of the boundary, we are able to transform this problem into one which does not depend on time and solving this we obtain candidates for the value function and the boundary. Using stochastic calculus we then verify that these indeed satisfy our original problem.In the fourth paper we consider an investor wanting to take advantage of a mispricing in the market by purchasing a bull spread, which is liquidated in case of a market downturn. We show that this can be formulated as an optimal stopping problem which we then, using similar techniques as in the third paper, solve explicitly.In the fifth and final paper we study convexity preservation of option prices in a model with jumps. This is done by finding a sufficient condition for the no-crossing property to hold in a jump-diffusion setting.
|
|
| 6. |
|
|
| 7. |
- Holmgren, Cecilia, 1984-
(författare)
-
Split Trees, Cuttings and Explosions
- 2010
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis is based on four papers investigating properties of split trees and also introducing new methods for studying such trees. Split trees comprise a large class of random trees of logarithmic height and include e.g., binary search trees, m-ary search trees, quadtrees, median of (2k+1)-trees, simplex trees, tries and digital search trees. Split trees are constructed recursively, using “split vectors”, to distribute n “balls” to the vertices/nodes. The vertices of a split tree may contain different numbers of balls; in computer science applications these balls often represent “key numbers”. In the first paper, it was tested whether a recently described method for determining the asymptotic distribution of the number of records (or cuts) in a deterministic complete binary tree could be extended to binary search trees. This method used a classical triangular array theorem to study the convergence of sums of triangular arrays to infinitely divisible distributions. It was shown that with modifications, the same approach could be used to determine the asymptotic distribution of the number of records (or cuts) in binary search trees, i.e., in a well-characterized type of random split trees. In the second paper, renewal theory was introduced as a novel approach for studying split trees. It was shown that this theory is highly useful for investigating these types of trees. It was shown that the expected number of vertices (a random number) divided by the number of balls, n, converges to a constant as n tends to infinity. Furthermore, it was demonstrated that the number of vertices is concentrated around its mean value. New results were also presented regarding depths of balls and vertices in split trees. In the third paper, it was tested whether the methods of proof to determine the asymptotic distribution of the number of records (or cuts) used in the binary search tree, could be extended to split trees in general. Using renewal theory it was demonstrated for the overall class of random split trees that the normalized number of records (or cuts) has asymptotically a weakly 1-stable distribution. In the fourth paper, branching Markov chains were introduced to investigate split trees with immigration, i.e., CTM protocols and their generalizations. It was shown that there is a natural relationship between the Markov chain and a multi-type (Galton-Watson) process that is well adapted to study stability in the corresponding tree. A stability condition was presented to describe a phase transition deciding when the process is stable or unstable (i.e., the tree explodes). Further, the use of renewal theory also proved to be useful for studying split trees with immigration. Using this method it was demonstrated that when the tree is stable (i.e., finite), there is the same type of expression for the number of vertices as for normal split trees.
|
|
| 8. |
- Petersson, Niclas, 1977-
(författare)
-
The Maximum Displacement for Linear Probing Hashing
- 2009
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis we study the standard probabilistic model for hashing with linear probing. The main purpose is to determine the asymptotic distribution for the maximum displacement. Depending on the ratio between the number of items and the number of cells, there are several cases to consider. Paper I solves the problem for the special case of almost full hash tables. That is, hash tables where every cell but one is occupied. Paper II completes the analysis by solving the problem for all remaining cases. That is, for every case where the number of items divided by the number of cells lies in the interval [0,1].The last two papers treat quite different topics. Paper III studies the area covered by the supremum process of Brownian motion. One of the main theorems in Paper I is expressed in terms of the Laplace transform of this area. Paper IV provides a new sufficient condition for a collection of independent random variables to be negatively associated when conditioned on their total sum. The condition applies to a collection of independent Borel-distributed random variables, which made it possible to prove a Poisson approximation that where essential for the completion of Paper II.
|
|
| 9. |
|
|
| 10. |
- Thörnblad, Erik
(författare)
-
Degrees in Random Graphs and Tournament Limits
- 2018
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of an introduction and six papers on the topics of degree distributions in random graphs and tournaments and their limits.The first two papers deal with a dynamic random graph, evolving in time through duplication and deletion of vertices and edges. In Paper I we study the degree densities of this model. We show that these densities converge almost surely and determine their limiting values exactly as well as asymptotically for large degrees. In Paper II we study the evolution of the maximum degree and provide a precise growth rate thereof.Paper III deals with a dynamic random tree model known as the vertex-splitting tree model. We show that the degree densities converge almost surely and find an infinite linear system of equations which they must satisfy. Unfortunately we are not able to show that this system has a unique solution except in special cases.Paper IV is about self-converse generalised tournaments. A self-converse generalised tournament can be seen as a matrix whose entries take values in [0,1] and whose diagonally opposite elements sum to 1. We characterise completely the marginals of such a matrix, and show that such marginals can always be realised by a self-converse generalised tournament.In Paper V, we define and develop the theory of tournament limits and tournament kernels. We characterise transitive and irreducible tournament limits and kernels, and prove that any tournament limit and kernel has an essentially unique decomposition into irreducible tournament limits or kernels interlaced by a transitive part.In Paper VI, we study the degree distributions of tournament limits, or equivalently, the marginals of tournament kernels. We describe precisely which distributions on [0,1] which may appear as degree distributions of tournament limits and which functions from [0,1] to [0,1] may appear as the marginals of tournament kernels. Moreover, we show that any distribution or marginal on this form may be realised by a tournament limit or tournament kernel. We also study those distributions and marginals which can be realised by a unique tournament limit or kernel, and find that only the transitive tournament limit/kernel gives rise to a degree distribution or marginal with this property.
|
|
