This thesis proposes a novel and efficient method (Method of Evolving Junctions) for solving optimal control problems with path constraints, and whose optimal paths are separable. A path is separable if it is the concatenation of finite number of subarcs that are optimal and either entirely constraint active or entirely constraint inactive. In the case when the subarcs can be computed efficiently, the search for the optimal path boils down to determining the junctions that connect those subarcs. In this way, the original infinite dimensional problem of finding the entire path is converted into a finite dimensional problem of determine the optimal junctions. The finite dimensional optimization problem is then solved by a recently developed global optimization strategy, intermittent diffusion. The idea is to add perturbations (noise) to the gradient flow intermittently, which essentially converts the ODE's (gradient descent) into a SDE's problem. It can be shown that the probability of finding the globally optimal path can be arbitrarily close to one. Comparing to existing methods, the method of evolving junctions is fundamentally faster and able to find the globally optimal path as well as a series of locally optimal paths. The efficiency of the algorithm will be demonstrated by solving path planning problems, more specifically, finding the optimal path in cluttered environments with static or dynamic obstacles.

In the work we estimate the rate of convergence of the Wong-Zakai type of approximations for SDEs and SPDEs. Two cases are studied: SDEs in finite dimensional settings and evolution stochastic systems (SDEs in the infinite dimensional case). The latter result is applied to the second order SPDEs of parabolic type and the filtering problem. Roughly, the result is the following. Let Wn be a sequence of continuous stochastic processes of finite variation on an interval [0, T]. Assume that for some a > 0 the processes Wn converge almost surely in the supremum norm in [0, T] to W with the rate n-k for each k < a. Then the solutions Un of the differential equations with Wn converge almost surely in the supremum norm in [0, T] to the solution u of the "Stratonovich" SDE with W with the same rate of convergence, n-k for each k < a, in the case of SDEs and with the rate of convergence n-k/2 for each k < a, in the case of evolution systems and SPDEs. In the final chapter we verify that the two most common approximations of the Wiener process, smoothing and polygonal approximation, satisfy the assumptions made in the previous chapters.

We study a new method for the strong approximate solution of stochastic differential equations using coupling and we prove order one error bounds for the new scheme in Lp space assuming the invertibility of the diffusion matrix. We introduce and implement two couplings called the exact and approximate coupling for this scheme obtaining good agreement with the theoretical bound. Also we describe a method for non-invertibility case (Combined method) and we investigate its convergence order which will give O(h3/4 √log(h)j) under some conditions. Moreover we compare the computational results for the combined method with its theoretical error bound and we have obtained a good agreement between them. In the last part of this thesis we work out the performance of the multilevel Monte Carlo method using the new scheme with the exact coupling and we compare the results with the trivial coupling for the same scheme.

An accelerated pace of discovery in biological sciences is made possible by a new generation of computational biology and bioinformatics tools. In this dissertation we develop novel computational, analytical, and high performance simulation techniques for biological problems, with applications to the yeast cell division cycle, and to the RNA-Sequencing of the yellow fever mosquito. Cell cycle system evolves stochastic effects when there are a small number of molecules react each other. Consequently, the stochastic effects of the cell cycle are important, and the evolution of cells is best described statistically. Stochastic simulation algorithm (SSA), the standard stochastic method for chemical kinetics, is often slow because it accounts for every individual reaction event. This work develops a stochastic version of a deterministic cell cycle model, in order to capture the stochastic aspects of the evolution of the budding yeast wild-type and mutant strain cells. In order to efficiently run large ensembles to compute statistics of cell evolution, the dissertation investigates parallel simulation strategies, and presents a new probabilistic framework to analyze the performance of dynamic load balancing algorithms. This work also proposes new accelerated stochastic simulation algorithms based on a fully implicit approach and on stochastic Taylor expansions. Next Generation RNA-Sequencing, a high-throughput technology to sequence cDNA in order to get information about a sample's RNA content, is becoming an efficient genomic approach to uncover new genes and to study gene expression and alternative splicing. This dissertation develops efficient algorithms and strategies to find new genes in Aedes aegypti, which is the most important vector of dengue fever and yellow fever. We report the discovery of a large number of new gene transcripts, and the identification and characterization of genes that showed male-biased expression profiles. This basic information may open important avenues to control mosquito borne infectious diseases. / Ph. D.

In this thesis, we study the existence, uniqueness, and regularity of systems of degenerate linear stochastic integro-differential equations (SIDEs) of parabolic type with adapted coefficients in the whole space. We also investigate explicit and implicit finite difference schemes for SIDEs with non-degenerate diffusion. The class of equations we consider arise in non-linear filtering of semimartingales with jumps. In Chapter 2, we derive moment estimates and a strong limit theorem for space inverses of stochastic flows generated by Lévy driven stochastic differential equations (SDEs) with adapted coefficients in weighted Hölder norms using the Sobolev embedding theorem and the change of variable formula. As an application of some basic properties of flows of Weiner driven SDEs, we prove the existence and uniqueness of classical solutions of linear parabolic second order stochastic partial differential equations (SPDEs) by partitioning the time interval and passing to the limit. The methods we use allow us to improve on previously known results in the continuous case and to derive new ones in the jump case. Chapter 3 is dedicated to the proof of existence and uniqueness of classical solutions of degenerate SIDEs using the method of stochastic characteristics. More precisely, we use Feynman-Kac transformations, conditioning, and the interlacing of space inverses of stochastic flows generated by SDEs with jumps to construct solutions. In Chapter 4, we prove the existence and uniqueness of solutions of degenerate linear stochastic evolution equations driven by jump processes in a Hilbert scale using the variational framework of stochastic evolution equations and the method of vanishing viscosity. As an application, we establish the existence and uniqueness of solutions of degenerate linear stochastic integro-differential equations in the L2-Sobolev scale. Finite difference schemes for non-degenerate SIDEs are considered in Chapter 5. Specifically, we study the rate of convergence of an explicit and an implicit-explicit finite difference scheme for linear SIDEs and show that the rate is of order one in space and order one-half in time.

Diese Arbeit beginnt mit einer Analyse von stochastischen Rückwärtsdifferentialgleichungen (BSDEs) mit Sprüngen, getragen von zufälligen Maßen mit ggf. unendlicher Aktivität und zeitlich inhomogenem Kompensator. Unter konkreten, in Anwendungen leicht verifizierbaren Bedingungen liefern wir Existenz-, Eindeutigkeits- und Vergleichsergebnisse beschränkter Lösungen für eine Klasse von Generatorfunktionen, die nicht global Lipschitz-stetig im Sprungintegranden sein brauchen. Der übrige Teil der Arbeit behandelt robuste Bewertung und Hedging in unvollständigen Märkten. Wir verfolgen den No-Good-Deal-Ansatz, der Good-Deal-Grenzen liefert, indem nur eine Teilmenge der risikoneutralen Maße mit ökonomischer Bedeutung betrachtet wird (z.B. Grenzen für instantanen Sharpe-Ratio, optimale Wachstumsrate oder erwarteten Nutzen). Durchweg untersuchen wir ein Konzept des Good-Deal-Hedgings für welches Hedgingstrategien als Minimierer geeigneter dynamischer Risikomaße auftreten, was optimale Risikoteilung mit der Markt erlaubt. Wir zeigen, dass Hedging mindestens im-Mittel-selbstfinanzierend ist, also, dass Hedgefehler unter geeigneten A-priori-Bewertungsmaßen eine Supermartingaleigenschaft haben. Wir leiten konstruktive Ergebnisse zu Good-Deal-Bewertung und -Hedging im Rahmen von Prozessen mit Sprüngen durch BSDEs mit Sprüngen, sowie im Brown''schen Fall mit Driftunsicherheit durch klassische BSDEs und mit Volatilitätsunsicherheit durch BSDEs zweiter Ordnung her. Wir liefern neue Beispiele, die insbesondere für versicherungs- und finanzmathematische Anwendungen von Bedeutung sind. Bei Ungewissheit des Real-World-Maßes führt ein Worst-Case-Ansatz bei Annahme mehrerer Referenzmaße zu Good-Deal-Hedging, welches robust bzgl. Unsicherheit, im Sinne von gleichmäßig über alle Referenzmaße mindestens im-Mittel-selbstfinanzierend, ist. Daher ist bei hinreichend großer Driftunsicherheit Good-Deal-Hedging zur Risikominimierung äquivalent. / This thesis starts by an analysis of backward stochastic differential equations (BSDEs) with jumps driven by random measures possibly of infinite activity with time-inhomogeneous compensators. Under concrete conditions that are easy to verify in applications, we prove existence, uniqueness and comparison results for bounded solutions for a class of generators that are not required to be globally Lipschitz in the jump integrand. The rest of the thesis deals with robust valuation and hedging in incomplete markets. The focus is on the no-good-deal approach, which computes good-deal valuation bounds by using only a subset of the risk-neutral measures with economic meaning (e.g. bounds on instantaneous Sharpe ratios, optimal growth rates, or expected utilities). Throughout we study a notion of good-deal hedging consisting in minimizing some dynamic risk measures that allow for optimal risk sharing with the market. Hedging is shown to be at least mean-self-financing in that hedging errors satisfy a supermartingale property under suitable valuation measures. We derive constructive results on good-deal valuation and hedging in a jump framework using BSDEs with jumps, as well as in a Brownian setting with drift uncertainty using classical BSDEs and with volatility uncertainty using second-order BSDEs. We provide new examples which are particularly relevant for actuarial and financial applications. Under ambiguity about the real-world measure, a worst-case approach under multiple reference priors leads to good-deal hedging that is robust w.r.t. uncertainty in that it is at least mean-self-financing uniformly over all priors. This yields that good-deal hedging is equivalent to risk-minimization if drift uncertainty is sufficiently large.

Chemical reactions involving small number of reacting molecules are noisy processes. They are simulated using stochastic simulation algorithms like the Gillespie SSA, which are valid when the reaction environment is well-mixed. This is not the case in reactions occuring on biological media like cell membranes, where alternative simulation methods have to be used to account for the crowded nature of the reacting environment. Ion channels, which are membrane proteins controlling the flow of ions into and out of the cell, offer excellent single molecule conditions to test stochastic simulation schemes in crowded biological media. Single molecule reactions are of great importance in determining the functions of biological molecules. Access to their experimental data have increased the scope of com-putational modeling of biological processes. Recently, single molecule experiments have revealed the non-Markovian nature of chemical reactions, due to a phenomenon called `dynamic disorder', which makes the rate constants a deterministic function of time or a random process. This happens when there are additional slow scale conformational transitions, giving the molecule a memory of its previous states. In a previous work, the hTREK1 two pore domain potassium channel was revealed to have long term memory in its kinetics, prompting alternate non-Markovian schemes to analyze its gating. Traditionally, ion channel gating is modeled as Markovian transitions between fixed states. In this work, we have used single channel data from hTREK1 ion channel and have provided a simple diffusion model for its gating. The main assumption of this model is that the ion channel diffuses through a continuum of states on its potential energy landscape, which is derived from the steady state probability distribution of ionic current recorded from patch clamp experiments. A stochastic differential equation (SDE) driven by Gaussian white noise is proposed to model this motion in an asymmetric double well potential. The method is computationally very simple and efficient and reproduces the amplitude histogram very well. For the case when ligands are added, leading to incorporation of long term memory in the kinetics, the SDE is modified to run on coloured noise. This has been done by introducing an auxiliary variable into the equation. It has been shown that increasing the noise correlation with ligand concentration improves the fits to the experimental data. This has been validated for several datasets. These methods are more advantageous for simulation than the Markovian models as they are true to the physical picture of gating and also computationally very efficient. Reproducing the whole raw data trace takes no more than a few seconds with our scheme, with the only input being the amplitude histogram and four parameters. Finally a quantitative model based on a modified version of the Chemical Langevin equation is given, which works on random rate parameters. This model is computationally simple to implement and reproduces the catalytic activity of the channel as a function of time. From the computational analysis undertaken in this work, we can infer that ion channel activity can be modeled using the framework of non-Markovian processes, lending credence to the recent understanding that single molecule reactions are basically processes with long-term memory. Since the ion channel is basically a protein, we can also hypothesize that the some of the properties that make proteins so vital to living organ-isms could be attributed to long-term memory in their folding kinetics, giving them the ability to sample specific regions of their conformation space, which are of interest to biological functions.

Cette thèse introduit une nouvelle notion de solution pour des équationsd'évolution non-linéaires déterministes, appellées solutionsmild découplées.Nous revisitons les liens entre équations différentielles rétrogrades(EDSRs) markoviennes browniennes et EDPsparaboliques semilinéaires en montrant que, sous de très faibles hypothèses,les EDSRs produisent une unique solution mild découplée d'une EDP.Nous étendons ce résultat à de nombreuses autres équations déterministestelles que des Pseudo-EDPs, des Equations Intégrales aux Dérivées Partielles(EIDPs), des EDPs à drift distributionnel, ou des E(I)DPs à dépendancetrajectorielle. Les solutions de ces équations sont représentées via des EDSRs qui peuvent être sans martingale de référence, ou dirigées par des martingales cadlag. En particulier, cette thèse résout le problème d'identification,qui consiste, dans le cas classique d'une EDSR markovienne brownienne, à donner un sens analytique au processus Z, second membre de la solution (Y,Z) de l'EDSR. Dans la littérature, Y détermine en général une solution de viscosité de l'équation déterministe et ce problème d'identification n'est résolu que quand cette solution de viscosité a un minimum de régularité. Notre méthode permet de résoudre ce problème même dans le cas général d'EDSRs à sauts (non nécéssairement markoviennes). / This thesis introduces a new notion of solution for deterministic non-linear evolution equations, called decoupled mild solution.We revisit the links between Markovian Brownian Backward stochastic differential equations (BSDEs) and parabolic semilinear PDEs showing that under very mild assumptions, the BSDEs produce a unique decoupled mild solution of some PDE.We extend this result to many other deterministic equations such asPseudo-PDEs, Integro-PDEs, PDEs with distributional drift or path-dependent(I)PDEs. The solutions of those equations are represented throughBSDEs which may either be without driving martingale, or drivenby cadlag martingales. In particular this thesis solves the so calledidentification problem, which consists, in the case of classical Markovian Brownian BSDEs, to give an analytical meaning to the second component Z ofthe solution (Y,Z) of the BSDE. In the literature, Y generally determinesa so called viscosity solution and the identification problem is only solved when this viscosity solution has a minimal regularity.Our method allows to treat this problem even in the case of general (even non-Markovian) BSDEs with jumps.