Rough paths and 1d sde with a time dependent distributional drift. Application to polymers

Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar-Parisi-Zhang (KPZ) equation, we reconsider the unique solvability of one-dimensional stochastic differential equations, the drift of which is a distribution, by means of rough paths theory. Existence and uniqueness are established in the weak sense when the drift reads as the derivative of a H{\"o}lder continuous function. Regularity of the drift part is investigated carefully and a related stochastic calculus is also proposed, which makes the structure of the solutions more explicit than within the earlier framework of Dirichlet processes

Instabilities in quasi-efficient markets

This thesis studies ways of modelling instabilities in quasi-efficient markets. We consider quasi-efficient markets where arbitrage is possible, but is relatively small and short lived. Under such a assumption we derive optimal arbitrage strategy of one agent and consider possible ways of funding optimal strategy under stop-loss constraint. Optimal strategy is used to build a multi-agent model which defines the arbitrage dynamics, i.e. its mean- reverting behaviour. The influence of agents on the asset prices is modelled by means of permanent price impact function. Multi-agent model is self-consistent as it creates mean-reverting term of the same type under which the optimal strategy for one agent was derived. As we show adding stop-loss constraint creates possibility for market instabilitie

A continuous-time search model with job switch and jumps

We study a new search problem in continuous time. In the traditional approach, the basic formulation is to maximize the expected (discounted) return obtained by taking a job, net of search cost incurred until the job is taken. Implicitly assumed in the traditional modeling is that the agent has no job at all during the search period or her decision on a new job is independent of the job situation she is currently engaged in. In contrast, we incorporate the fact that the agent has a job currently and starts searching a new job. Hence we can handle more realistic situation of the search problem. We provide optimal decision rules as to both quitting the current job and taking a new job as well as explicit solutions and proofs of optimality. Further, we extend to a situation where the agent's current job satisfaction may be affected by sudden downward jumps (e.g., de-motivating events), where we also find an explicit solution; it is rather a rare case that one finds explicit solutions in control problems using a jump diffusion

Truncated Euler Maruyama numerical method for stochastic differential (delay) equations models

In this thesis, our focus has been on enhancing the applicability and reliability of the truncated Euler-Maruyama (EM) numerical method for stochastic differential equations (SDEs) and stochastic delay differential equations (SDDEs), initially introduced by Mao [21]. Building upon this method, our contributions span several chapters. In Chapter 3, we pointed out its limitations in determining the convergence rate over a finite time interval and established a new result for SDEs whose diffusion coefficients may not satisfy the global Lipschitz condition. We extended our exploration to include time delays in Chapter 4, allowing for varying delays over time. The chapter also introduces additional lemmas to ensure the convergence rates of the method to the solution at specific time points and over finite intervals. However, the global Lipschitz condition on the diffusion coefficient is currently required. In Chapter 5, we focused on the Lotka-Volterra model, introducing modifications such as the Positive Preserving Truncated EM (PPTEM) and Nonnegative Preserving Truncated EM (NPTEM) methods to handle instances where the truncated EM method generated nonsensical negative solutions. The proposed adjustments, guided by Assumption 5.1.1, ensure that the numerical solutions remain meaningful and interpretable. Chapter 6 extends these concepts to the stochastic delay Lotka-Volterra model with a variable time delay, demonstrating the adaptability and applicability of our methods. Despite we also assume the stronger condition 6.1.1 to prove the convergence of numerical solutions, future research aims to explore relaxed conditions, broadening the applicability of these numerical methods. Overall, this thesis contributes to establishing convergence rates for SDEs under local Lipschitz diffusion coefficients, extending the methodology to address time delays and modifying the truncated EM method to ensure positive and nonnegative numerical solutions. These advancements are demonstrated through applications to the stochastic variable time delay Lotka-Volterra model, emphasizing the meaningfulness and interpretability of the solutions.In this thesis, our focus has been on enhancing the applicability and reliability of the truncated Euler-Maruyama (EM) numerical method for stochastic differential equations (SDEs) and stochastic delay differential equations (SDDEs), initially introduced by Mao [21]. Building upon this method, our contributions span several chapters. In Chapter 3, we pointed out its limitations in determining the convergence rate over a finite time interval and established a new result for SDEs whose diffusion coefficients may not satisfy the global Lipschitz condition. We extended our exploration to include time delays in Chapter 4, allowing for varying delays over time. The chapter also introduces additional lemmas to ensure the convergence rates of the method to the solution at specific time points and over finite intervals. However, the global Lipschitz condition on the diffusion coefficient is currently required. In Chapter 5, we focused on the Lotka-Volterra model, introducing modifications such as the Positive Preserving Truncated EM (PPTEM) and Nonnegative Preserving Truncated EM (NPTEM) methods to handle instances where the truncated EM method generated nonsensical negative solutions. The proposed adjustments, guided by Assumption 5.1.1, ensure that the numerical solutions remain meaningful and interpretable. Chapter 6 extends these concepts to the stochastic delay Lotka-Volterra model with a variable time delay, demonstrating the adaptability and applicability of our methods. Despite we also assume the stronger condition 6.1.1 to prove the convergence of numerical solutions, future research aims to explore relaxed conditions, broadening the applicability of these numerical methods. Overall, this thesis contributes to establishing convergence rates for SDEs under local Lipschitz diffusion coefficients, extending the methodology to address time delays and modifying the truncated EM method to ensure positive and nonnegative numerical solutions. These advancements are demonstrated through applications to the stochastic variable time delay Lotka-Volterra model, emphasizing the meaningfulness and interpretability of the solutions

Sharp High-dimensional Central Limit Theorems for Log-concave Distributions

Let $X_1,\dots,X_n$ be i.i.d. log-concave random vectors in $\mathbb R^d$ with mean 0 and covariance matrix $\Sigma$. We study the problem of quantifying the normal approximation error for $W=n^{-1/2}\sum_{i=1}^nX_i$ with explicit dependence on the dimension $d$. Specifically, without any restriction on $\Sigma$, we show that the approximation error over rectangles in $\mathbb R^d$ is bounded by $C(\log^{13}(dn)/n)^{1/2}$ for some universal constant $C$. Moreover, if the Kannan-Lov\'asz-Simonovits (KLS) spectral gap conjecture is true, this bound can be improved to $C(\log^{3}(dn)/n)^{1/2}$. This improved bound is optimal in terms of both $n$ and $d$ in the regime $\log n=O(\log d)$. We also give $p$-Wasserstein bounds with all $p\geq2$ and a Cram\'er type moderate deviation result for this normal approximation error, and they are all optimal under the KLS conjecture. To prove these bounds, we develop a new Gaussian coupling inequality that gives almost dimension-free bounds for projected versions of $p$-Wasserstein distance for every $p\geq2$. We prove this coupling inequality by combining Stein's method and Eldan's stochastic localization procedure.Comment: 37 pages. Some typos are correcte