<p dir="ltr">There is a growing trend towards the development of parsimonious surrogate models for studying physical phenomena. While they typically offer less accuracy, these models bypass the computational costs of numerical methods, usually by multiple orders of magnitude, allowing statistical applications such as sensitivity analysis, stochastic treatments, parametric problems, and uncertainty quantification. Researchers have explored generalized surrogate frameworks leveraging Gaussian processes, various basis function expansions, support vector machines, and neural networks. Dynamical fields, represented through time-dependent partial differential equation, pose a particular hardship for existing frameworks due to their high dimensional representation, and possibly multi-scale solutions.</p><p dir="ltr">In this work, we present a novel architecture for solving time-dependent partial differential equations using co-ordinate neural networks and time-marching updates through hyper-networks. We show that it provides a temporally meshed and spatially mesh-free solution which are causally coherent as justified through a theoretical treatment of Lie groups. We showcase results on some benchmark problems in computational physics while discussing their performance against similar physics-informed approaches like physics-informed DeepOnets and Physics informed neural networks.</p>

In this work, the influence of the second messenger cAMP on the functional plasticity of M-DC8+ dendritic cells (DC) was examined. The marker M-DC8 defines a population of native DC first described in blood. After their isolation, M-DC8+ DC acquire a mature CD83+ phenotype during a short culture ex vivo. After a challenge with LPS and IFN-g, M-DC8+ DC secrete large amounts of the proinflammatory cytokines IL-12(p70) and TNF-a surpassing by far other DC populations and monocytes. Due to their preferential induction of TH1-dominated T cell responses, M-DC8+ DC might play a role in the pathogenesis of inflammatory diseases. Different cAMP-elevating agents suppressed the proinflammatory cytokine production and enhanced the secretion of anti-inflammatory IL-10. Activity of phosphodiesterase (PDE) 4, the most important cAMP-hydrolysing enzyme in immune cells, was detected and RT-PCR revealed the expression of PDE4 subtypes 4A, 4B and 4D in M-DC8+ DC, whereas 4C was not detectable. The PDE4-specific inhibitors AWD12-281 and Roflumilast were then used to elevate cAMP concentrations. These substances have been proven to be efficient in anti-inflammatory therapies. In the presence of PDE4 inhibitors, the LPS/IFN-g-induced production of IL-12 and TNF-a was decreased by 90 % and 60 %, respectively, whereas the IL-10-release was doubled. These effects were only observed, if the PDE4 inhibitors where present from the beginning of the culture. The inhibition of the IL-12 secretion was reverted using an a-IL-10-receptor antibody. PDE4 inhibitor-treated M-DC8+ DC showed a reduced capacity to polarize TH1-cells, which was demonstrated analysing culture supernatants by ELISA and by single-cell analysis detecting intracellular IFN-g und IL-4. These results suggest that PDE4 inhibitors may not only be useful in the therapy of TH2-mediated diseases but also in TH1-dominated indications such as multiple sclerosis and Crohn´s disease. Despite the shift of the cytokine profile, the in vitro maturation of M-DC8+ DC was not affected by PDE4 inhibitors. The expression of CD83, CD80, CD86, MHC-molecules as well as CD54 and CD58, was assessed by FACS analysis. Correspondingly, in the presence of AWD12-281, M-DC8+ DC efficiently stimulated the proliferation of allogeneic CD4+CD45RA+ T-cells. In the second part of this study, the effects of an inhibition of cAMP-synthesis in M-DC8+ DC were analyzed. Two adenylyl cyclase (AC) inhibitors, 2,5-Dideoxyadenosine and SQ22536, clearly hampered the in vitro maturation of M-DC8+ DC. The expression of the DC maturation marker CD83 could be reconstituted using the stable cAMP-analogon 8-Br-cAMP. Measuring the intracellular cAMP concentration in M-DC8+ DC, initially low cAMP-levels were observed, but within 30 min the concentration raised and returned to original levels within 2 hrs. Blocking the cAMP synthesis by AC inhibitors, the LPS/IFN-g-induced production of IL-12, TNF-a and IL-10 was strongly reduced. Furthermore, it was demonstrated that M-DC8+ DC can only release IL-12 after a transient elevation of cAMP, i.e. they acquire a &amp;quot;license&amp;quot;. Such a regulation of the IL-12 production has not been described before. Protein kinase A is an important effector molecule of cAMP. Inhibiting its activity resulted in a reduced expression of the DC maturation marker CD83 and a lower cytokine production underlining the importance of cAMP-signalling for the activation of M-DC8+ DC. In conclusion, this study provides evidence for a new concept of the immune-regulatory function of cAMP. Here, cAMP is essentially involved in the initial activation and maturation of DC and enables them to secrete large amounts of IL-12 and TNF-a upon stimulation with a TLR ligand. Conversely, a long-term elevation of cAMP-concentrations inhibits the proinflammatory effector functions of M-DC8+ DC and can induce anti-inflammatory responses by enhancing the secretion of IL-10. / In dieser Arbeit wurde der Einfluss des second messengers cAMP auf die funktionelle Plastizität von M-DC8+ dendritischen Zellen (DC) untersucht. Der Oberflächenmarker M-DC8 definiert eine zunächst im Blut beschriebene Population nativer DC. Nach ihrer Isolation erlangen M-DC8+ DC während einer kurzen Kultur einen maturen CD83+ Phänotyp. Nach Stimulation mit LPS und IFN-g produzieren native M-DC8+ DC deutlich höhere Mengen der proinflammatorischen Zytokine IL-12(p70) und TNF-a als andere DC-Populationen oder Monozyten. Dies resultiert in einer Programmierung TH1-dominierter T-Zellantworten. M-DC8+ DC könnten daher an der Pathogenese entzündlicher Krankheiten beteiligt sein. Unterschiedliche cAMP-erhöhende Substanzen supprimierten die proinflammatorische Zytokinproduktion und verstärkten gleichzeitig die Sekretion des anti-inflammatorischen IL-10. In M-DC8+ DC konnte die Aktivität von Phosphodiesterase (PDE) 4, dem wichtigsten cAMP-hydrolysierenden Enzym in Immunzellen, nachgewiesen werden. Durch RT-PCR wurde die Expression der PDE4-Subtypen 4A, 4B und 4D gezeigt, nicht aber 4C. Zur Erhöhung der cAMP-Konzentration wurden dann die PDE4-spezifischen Inhibitoren AWD12-281 und Roflumilast eingesetzt, deren klinische Effizienz bei anti-inflammatorischen Therapien belegt ist. Auch diese Substanzen verringerten die LPS/IFN-g-induzierte Produktion von IL-12 und TNF-a durch M-DC8+ DC um 90 % bzw. 60 %, während die IL-10-Freisetzung etwa verdoppelt wurde. Diese starken Effekte konnten nur erzielt werden, wenn die PDE4-Inhibitoren von Beginn der Kultur an eingesetzt wurden. Die Hemmung der IL-12-Sekretion wurde in Gegenwart eines a-IL-10-Rezeptor-Antikörpers aufgehoben. Unter dem Einfluss von PDE4-Inhibitoren war die TH1-Programmierung durch M-DC8+ DC deutlich reduziert, was sowohl durch die Analyse der Zellüberstände mittels ELISA als auch auf Einzelzell-Ebene durch intrazelluläre Detektion von IFN-g und IL-4 nachgewiesen wurde. Diese Ergebnisse legen nahe, dass PDE4-Inhibitoren nicht nur für TH2-vermittelte Erkrankungen sondern auch für TH1-dominierte Indikationen wie Multiple Sklerose oder Morbus Crohn von Nutzen sein könnten. Trotz der starken Modulation des Zytokinprofils blieb die in vitro-Ausreifung M-DC8+ DC unbeeinflusst von PDE4-Inhibitoren. Untersucht wurde die Expression von CD83, CD80, CD86, MHC-Molekülen, CD54 und CD58 mittels FACS-Analyse. Entsprechend induzierten M-DC8+ DC auch in Anwesenheit von AWD12-281 die Proliferation allogener CD4+CD45RA+ T-Zellen. Im zweiten Teil der Arbeit wurde untersucht, wie sich die Blockade der cAMP-Synthese auf M-DC8+ DC auswirkt. Zwei Adenylatcyclase-Inhibitoren, 2,5-Dideoxyadenosine und SQ22536, hemmten die in vitro-Maturation von M-DC8+ DC deutlich. Die CD83-Expression wurde mit 8-Br-cAMP rekonstituiert. Messungen der intrazellulären cAMP-Konzentration in unbehandelten M-DC8+ DC zeigten initial niedrige cAMP-Spiegel, die innerhalb von 30 min anstiegen und nach 2 h wieder auf das Ausgangsniveau abfielen. Die LPS/IFN-g-induzierte Produktion von IL-12, TNF-a und IL-10 wurde durch AC-Inhibitoren deutlich vermindert. M-DC8+ DC erhalten nur nach einer transienten cAMP-Erhöhung die &amp;quot;Lizenz&amp;quot; IL-12 freizusetzen. Eine derartige Regulation der IL-12-Sekretion ist bisher nicht beschrieben. Eine Hemmung des cAMP-Effektormoleküls Proteinkinase A resultierte in der reduzierten Expression des DC-Maturationsmarkers CD83 und einer verringerten Zytokinproduktion. Dies unterstreicht die Bedeutung von cAMP für die Aktivierung M-DC8+ DC. Zusammenfassend gibt diese Arbeit am Beispiel nativer humaner DC Anhalt für ein neues Konzept der immunregulatorischen Funktion von cAMP. Hierbei ist cAMP wesentlich an der Ausreifung von M-DC8+ DC beteiligt, woraufhin diese große Mengen IL-12 und TNF-a sekretieren können. Dagegen wirkt eine langfristige cAMP-Erhöhung durch die Induktion von IL-10 anti-inflammatorisch.

Les substances pharmaceutiques font partie du groupe des contaminants émergents du fait de leur intérêt récent dans les études environnementales comparativement à des polluants étudiés depuis plus longtemps tels que les pesticides. Elles correspondent aux principes actifs des médicaments et, à ce titre, sont responsables des propriétés pharmacologiques des médicaments. Ce sont donc des molécules biologiquement actives qui peuvent agir sur les organismes vivants présents dans les écosystèmes impactés. L’origine des substances pharmaceutiques dans l’environnement est variable mais les principales sources sont liées à leur utilisation en médecine humaine ou vétérinaire. Une fois consommées, les substances pharmaceutiques sont excrétées dans les urines ou les fèces et se retrouvent dans les eaux usées (consommation humaine) ou dans les déchets d’élevage (consommation vétérinaire). Dans le premier cas, elles peuvent être rejetées directement dans le milieu, ou indirectement, avec les eaux usées traitées ou les boues résiduaires, après traitement dans les stations d’épuration (STEP). Dans le deuxième cas, elles atteignent directement le milieu lorsque les animaux sont élevés en prairie ou indirectement lorsque les déchets d’élevage sont épandus sur les sols agricoles pour les fertiliser. Ces travaux de thèse se sont attachés à étudier l’origine et le devenir de ces substances dans ces 2 cas de figure. Ainsi en se basant sur des critères de consommation, de présence dans l’environnement par rapport à des études antérieures, de toxicité et d’écotoxicité, d’originalité et de disponibilité des composés standards de référence, 32 puis 78 molécules appartenant aux classes thérapeutiques des antibiotiques, des anticancéreux, des béta-bloquants, des anti-VIH et des inhibiteurs de phosphodiestérase de type 5 (PDE 5) ont été étudiées dans 2 continuums : i) effluents hospitaliers - eaux usées brutes et traitées – eaux de surface, et ii) eaux usées brutes et traitées - eaux de surface - eaux de captage souterraines. En s’appuyant sur les mêmes critères de sélection, le devenir de 7 antibiotiques a été étudié dans des lisiers porcins dans des filières simples de traitement du lisier (fosse de stockage), dans des filières complexes de traitement du lisier (système de traitement ressemblant à des mini STEP) et dans des mésocosmes en conditions contrôlées. Pour pouvoir réaliser l’ensemble de ces études, des protocoles analytiques mettant en œuvre une étape d’extraction par SPE (Solide Phase Extraction) ou d’extraction ASE (Extraction Accélérée par Solvant) puis de purification par SPE et d’analyse par LC/MS/MS (Chromatographie en phase liquide couplée à la spectrométrie de masse en tandem) ont été développés. Ces protocoles, en remplissant des critères de qualité tels que des limites de détection et de quantification compatibles avec des analyses environnementales (de l’ordre du ng/l à la dizaine de ng/l), une bonne linéarité, précision, justesse et performance, ont permis d’analyser la phase dissoute des échantillons d’eaux et la phase dissoute et solide des échantillons de lisiers. Il ressort des analyses des échantillons aqueux que : i) les béta-bloquants, les anti-VIH et les antibiotiques appartenant aux familles des macrolides, des fluoroquinolones et des sulfonamides, sont les molécules les plus représentatifs de la contamination du milieu naturel parmi les classes étudiées ; ii) les rejets de STEP sont une source majeure de la contamination des systèmes aquatiques ; iii) les eaux usées sont davantage contaminées en hiver qu’en été ; et iv) les eaux de surface sont davantage contaminées en été qu’en hiver. / Pharmaceutical substances belong to the group of emerging contaminants due to their recent interest in environmental studies in comparison with pollutants who have been studied for a longer time like pesticides. They correspond to the active ingredient of drugs and by this mean are responsible for their pharmacological properties. Consequently they are biologically active molecules that can act on living organisms present in impacted ecosystems. The origin of pharmaceuticals in the environment is variable but the main sources are related to their use in human and veterinary medicine. Once consumed, pharmaceutical substances are excreted in urine or feces and are found in wastewater (human consumption) or animal manure (veterinary consumption). In the first case, they can be discharged directly in the environment, or indirectly, with treated wastewater or sludge from sewage treatment plants (SWTP). In the second case, they directly reach the environment when animals are bred on grassland or indirectly when livestock wastes are spread on agricultural soils as fertilizer. This PhD work has been focused on the study of the origin and fate of pharmaceutical substances in these 2 cases. Thus according to consumption data, occurrence in the environment reported in previous studies, toxicity and ecotoxicity data, originality and availability of reference standard compounds, 32 then 78 molecules belonging to 5 different therapeutic classes (antibiotics, antineoplastics, beta-blockers, anti-HIV, phosphodiesterase type 5 inhibitors (PDE 5 inhibitors)) were studied in 2 continuums : i) hospital wastewater effluents – raw and treated wastewater – surface water, and ii) raw and treated wastewater – surface water – ground water. Based on the same selection criteria, the fate of 7 antibiotics was studied in pig manure in simple manure storage facilities (storage tank), in aerobic manure treatment facilities (treatment system like in small SWTP) and in mesocosms under controlled conditions. In order to achieve all these studies, analytical protocols implementing an extraction step by SPE (Solid Phase Extraction) or an ASE extraction (Accelerated Solvent Extraction) followed by a SPE purification and an analytical step by LC / MS / MS (liquid chromatography tandem mass spectrometry) have been developed. These protocols, by filling out quality criteria such as limits of detection and quantification compatible with environmental analysis (ng/l to dozen of ng/l), good linearity, precision, accuracy and performance, were used to analyze the dissolved phase of water samples and dissolved and solid phases of pig manure samples. The water samples analysis shows : i) beta-blockers, anti-HIV and antibiotic belonging to the families of macrolides, fluoroquinolones and sulfonamides are the most representative molecules of the environmental contamination from the classes studied; ii) SWTP releases are a major source of aquatic systems’ contamination; iii) wastewaters are more contaminated in winter than in summer; and iv) surface water are more contaminated in summer than in winter. The pig manure samples analysis shows : i) the levels of contamination of manure by antibiotics are high, from a few µg/l to mg/l; ii) the manure level of contamination is not related to the physiological stage of pigs; iii) the interest to store manure before spreading in order to reduce the antibiotics contamination is not highlighted; iv) oxytetracycline, tetracycline, tylosin and marbofloxacin are mainly present in the solid phase whereas sulfadiazine, lincomycin and monensin are mainly present in the liquid phase of manure; v) the separation of solid and liquid phases reduce manure contamination in aerobic treatment facilities; and vi) antibiotics degradation is mainly aerobic.Key words: ,

Traditional methods in optical design like ray tracing suffer from slow convergence and are not constructive, i.e., each minimal perturbation of input parameters might lead to “chaotic” changes in the output. However, so-called inverse methods can be helpful in designing optical systems of reflectors and lenses. The equations in R2 become ordinary differential equations, while in R3 the equations become partial differential equations. These equations are then used to transform source distributions into target distributions, where the distributions are arbitrary, though assumed to be positive and integrable. In this project, we derive the governing equations and solve them numerically, for the systems presented by our instructor Martijn Anthonissen [Anthonissen et al. 2021]. Additionally, we show how point sources can be derived as a special case of a interval source with di- rected source interval, i.e., with each point in the source interval there is also an associated unit direction vector which could be derived from a system of two interval sources in R2. This way, it is shown that connecting source distributions with target distributions can be classified into two instead of three categories. The resulting description of point sources as a source along an interval with directed rays could potentially be extended to three dimensions, leading to interpretations of point sources as directed sources on convex or star-shaped sets.:1 Abstract 4 2 Notation And Conventions 4 3 Introduction 5 4 ECMI Modeling Week Challenges 5 4.1 Problem 1 - Parallel to Near-Field Target 5 4.1.1 Description 5 4.1.2 Deriving The Equations 5 4.2 Problem 2 - Parallel Source To Two Targets 8 4.3 Problem 3 - Point Source To Near-Field Target 9 4.3.1 Deriving The Equations 9 4.4 Problem 4 - Point Source To Two Targets 11 5 Validation - Ray tracing 13 5.1 Splines 13 5.1.1 Piece-Wise Affine Reflectors 13 5.1.2 Piece-Wise Cubic Reflectors 14 5.2 Error Estimates For Spline Reflectors 14 5.2.1 Lemma: A Priori Feasibility Of Starting Values For Near-Field Problems 15 5.2.2 Estimates for single reflector, near-field targets 16 5.3 Ray Tracing Errors - Illumination Errors 17 5.3.1 Definition: Axioms For Errors 18 5.3.2 Extrapolated Ray Tracing Error (ERTE) 18 5.3.3 Definition: Minimal Distance Ray Tracing Error (MIRTE) 19 5.3.4 Lemma: Continuity Of The Ray Traced Reflection Projection Of Smooth Reflectors 19 5.3.5 Theorem: Convergence Of The MIRTE 20 5.3.6 Convergence Of The ERTE 21 5.3.7 Application 21 6 Numerical Implementation 21 6.1 The DOPTICS Library 21 6.2 Pseudocode Of The Implementation 21 6.2.1 Solutions Of The Problems 22 6.2.2 Ray Tracing And Ray Tracing Error 22 6.3 ERTE Implementation 25 7 Results 26 7.1 Problem 1: Results 26 7.2 Problem 2: Results 26 7.3 Problem 3: Results 27 7.4 Problem 4: Results 27 8 Generalizations In Two Dimensions 29 8.1 Directed Densities 29 8.2 Generalized, Orthogonally Emitting Sources in R2 30 8.2.1 Point Light Sources As Orthogonally Emitting Sources 30 9 Conclusion and Future Research 32 10 Group Dynamic 32 References 32

We discuss a full-waveform based material profile reconstruction in two-dimensional heterogeneous semi-infinite domains. In particular, we try to image the spatial variation of shear moduli/wave velocities, directly in the time-domain, from scant surficial measurements of the domain's response to prescribed dynamic excitation. In addition, in one-dimensional media, we try to image the spatial variability of elastic and attenuation properties simultaneously. To deal with the semi-infinite extent of the physical domains, we introduce truncation boundaries, and adopt perfectly-matched-layers (PMLs) as the boundary wave absorbers. Within this framework we develop a new mixed displacement-stress (or stress memory) finite element formulation based on unsplit-field PMLs for transient scalar wave simulations in heterogeneous semi-infinite domains. We use, as is typically done, complex-coordinate stretching transformations in the frequency-domain, and recover the governing PDEs in the time-domain through the inverse Fourier transform. Upon spatial discretization, the resulting equations lead to a mixed semi-discrete form, where both displacements and stresses (or stress histories/memories) are treated as independent unknowns. We propose approximant pairs, which numerically, are shown to be stable. The resulting mixed finite element scheme is relatively simple and straightforward to implement, when compared against split-field PML techniques. It also bypasses the need for complicated time integration schemes that arise when recent displacement-based formulations are used. We report numerical results for 1D and 2D scalar wave propagation in semi-infinite domains truncated by PMLs. We also conduct parametric studies and report on the effect the various PML parameter choices have on the simulation error. To tackle the inversion, we adopt a PDE-constrained optimization approach, that formally leads to a classic KKT (Karush-Kuhn-Tucker) system comprising an initial-value state, a final-value adjoint, and a time-invariant control problem. We iteratively update the velocity profile by solving the KKT system via a reduced space approach. To narrow the feasibility space and alleviate the inherent solution multiplicity of the inverse problem, Tikhonov and Total Variation (TV) regularization schemes are used, endowed with a regularization factor continuation algorithm. We use a source frequency continuation scheme to make successive iterates remain within the basin of attraction of the global minimum. We also limit the total observation time to optimally account for the domain's heterogeneity during inversion iterations. We report on both one- and two-dimensional examples, including the Marmousi benchmark problem, that lead efficiently to the reconstruction of heterogeneous profiles involving both horizontal and inclined layers, as well as of inclusions within layered systems. / text

<p>This thesis considers the problem of making dynamic models for industrial processes by combining physical modelling with experimental data. The focus is on distributed parameter systems, that is, systems for which the model structure involves partial differential equations (PDE). Distributed parameter systems are important in many applications, e.g., in chemical process systems and in intracellular biochemical processes, and involve for instance all forms of transport and transfer phenomena. For such systems, the postulated model structure usually requires a finite dimensional approximation to enable identification and validation using experimental data. The finite dimensional approximation involves translating the PDE model into a set of ordinary differential equations, and is termed model reduction.</p><p>The objective of the thesis is two-fold. First, general PDE model reduction methods which are efficient in terms of model order for a given level of accuracy are studied. The focus here is on a class of methods called moving mesh methods, in which the discretization mesh is considered a dynamic degree of freedom that can be used for reducing the model reduction error. These methods are potentially highly efficient for model reduction of PDEs, but often suffer from stability and robustness problems. In this thesis it is shown that moving mesh methods can be cast as standard feedback control problems. Existing moving mesh methods are analyzed based on tools and results available from control theory, and plausible explanations to the robustness problems and parametric sensitivity experienced with these methods are provided. Possible remedies to these problems are also proposed. A novel moving finite element method, Orthogonal Collocation on Moving Finite Elements (OCMFE), is proposed based on a simple estimate of the model reduction error combined with a low order linear feedback controller. The method is demonstrated to be robust, and hence puts only small demands on the user.</p><p>In the second part of the thesis, the integration of PDE model reduction methods with grey-box modelling tools available for finite dimensional models is considered. First, it is shown that the standard approach based on performing model reduction using some ad hoc discretization method and model order, prior to calibrating and validating the reduced model, has a number of potential pitfalls and can easily lead to falsely validated PDE models. To overcome these problems, a systematic approach based on separating model reduction errors from discrepancies between postulated model structures and measurement data is proposed. The proposed approach is successfully demonstrated on a challenging chromatography process, used for separation in biochemical production, for which it is shown that data collected at the boundaries of the process can be used to clearly distinguish between two model structures commonly used for this process.</p>

保險人因提供最低保證給付之投資型商品，使公司亦涉入投資風險。本研究旨在探討最低保證給付人壽保險附約之風險分析。首先利用隨機模型建構投資者帳戶價值的動態過程，進而推導出在未來時點帳戶發生餘額不足之機率及其所符合的偏微分方程式。並藉由數值方法－有限差分法，求出投資帳戶餘額不足之機率。最終，以不同的參數選取之下，進行敏感度分析，探討參數值的設定對於帳戶發生餘額不足之機率的影響。本研究結果可以提供保險公司與監理機關，作為日後發行保證給付商品時，一項風險管理上的考慮因素。 研究結果可以歸納為兩點結論： 1. 在市場因素中，投資帳戶連結之標的報酬率與帳戶餘額不足機率呈現反向變動，而波動度則是與帳戶餘額不足機率呈現正向變動。在兩因素同時考慮下，當報酬率愈高且波動度愈低，投資帳戶發生餘額不足的機率會愈低。當波動度愈高且報酬率愈低時，帳戶餘額不足機率則會愈高。其兩者的力量會相互抵銷，對投資帳戶餘額不足之機率的影響需視何者的力量較強而定。 2. 在條款設計的因素中，保證附約相關費用率、保證提領比率與保證提領期間對於投資帳戶發生餘額不足機率的影響皆呈現正向的關係。而投資帳戶期初的價值則與帳戶餘額不足機率呈現反向變動。其中保證提領比率對於投資帳戶的價值影響最大，其帳戶餘額不足機率之變動百分比相較於其他因素而言，變動幅度較大，範圍皆大於4%以上，甚至高達37.11%。 / Insurers have investment risks because they issue the guaranteed minimum benefit life insurance riders. Therefore, the purpose of this thesis is analyzing the risk for the riders. In the context, we implement numerical PDE solution to compute the ruin probability of separate account which is the probability that guaranteed minimum benefit life insurance riders will lead to financial insolvency under stochastic investment returns. Moreover, we will do sensitivity analyses to discuss the two aspects, market factors and contract designs, how to influence the ruin probability. Finally, we conclude two main results: 1. For market factors, the rate of investment return is negatively related to ruin probability; however, the volatility is positive correlation. 2. For contract designs, the results show negative correlation between ruin probability and insurance fee, withdrawals, and withdrawal period. But the initial account value shows positive correlation.

This thesis is devoted to the study of the applications of the error theory using Dirichlet forms. Our work is split into three parts. The first one deals with the models described by stochastic differential equations. After a short technical chapter, an innovative model for order books is proposed. We assume that the bid-ask spread is not an imperfection, but an intrinsic property of exchange markets instead. The uncertainty is carried by the Brownian motion guiding the asset. We find that spread evolutions can be evaluated using closed formulae and we estimate the impact of the underlying uncertainty on the related contingent claims. Afterwards, we deal with the PBS model, a new model to price European options. The seminal idea is to distinguish the market volatility with respect to the parameter used by traders for hedging. We assume the former constant, while the latter volatility being an erroneous subjective estimation of the former. We prove that this model anticipates a bid-ask spread and a smiled implied volatility curve. Major properties of this model are the existence of closed formulae for prices, the impact of the underlying drift and an efficient calibration strategy. The second part deals with the models described by partial differential equations. Linear and non-linear PDEs are examined separately. In the first case, we show some interesting relations between the error and wavelets theories. When non-linear PDEs are concerned, we study the sensitivity of the solution using error theory. Except when exact solution exists, two possible approaches are detailed: first, we analyze the sensitivity obtained by taking "derivatives" of the discrete governing equations. Then, we study the PDEs solved by the sensitivity of the theoretical solutions. In both cases, we show that sharp and bias solve linear PDE depending on the solution of the former PDE itself and we suggest algorithms to evaluate numerically the sensitivities. Finally, the third part is devoted to stochastic partial differential equations. Our analysis is split into two chapters. First, we study the transmission of an uncertainty, present on starting conditions, on the solution of SPDE. Then, we analyze the impact of a perturbation of the functional terms of SPDE and the coefficient of the related Green function. In both cases, we show that the sharp and bias verify linear SPDE depending on the solution of the former SPDE itself

We study the convergence of a nonlinear approximation method introduced in the engineering literature for the numerical solution of a high-dimensional Fokker--Planck equation featuring in Navier--Stokes--Fokker--Planck systems that arise in kinetic models of dilute polymers. To do so, we build on the analysis carried out recently by Le~Bris, Leli\`evre and Maday (Const. Approx. 30: 621--651, 2009) in the case of Poisson's equation on a rectangular domain in $\mathbb{R}^2$, subject to a homogeneous Dirichlet boundary condition, where they exploited the connection of the approximation method with the greedy algorithms from nonlinear approximation theory explored, for example, by DeVore and Temlyakov (Adv. Comput. Math. 5:173--187, 1996). We extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le~Bris, Leli\`evre and Maday to the technically more complicated situation of the elliptic Fokker--Planck equation, where the role of the Laplace operator is played out by a high-dimensional Ornstein--Uhlenbeck operator with unbounded drift, of the kind that appears in Fokker--Planck equations that arise in bead-spring chain type kinetic polymer models with finitely extensible nonlinear elastic potentials, posed on a high-dimensional Cartesian product configuration space $\mathsf{D} = D_1 \times \dotsm \times D_N$ contained in $\mathbb{R}^{N d}$, where each set $D_i$, $i=1, \dotsc, N$, is a bounded open ball in $\mathbb{R}^d$, $d = 2, 3$. We exploit detailed information on the spectral properties and elliptic regularity of the Ornstein--Uhlenbeck operator to give conditions on the true solution of the Fokker--Planck equation which guarantee certain rates of convergence of the greedy algorithms. We extend the analysis to discretized versions of the greedy algorithms.

This thesis treats a range of stochastic methods with various applications, most notably in finance. It is comprised of five articles, and a summary of the key concepts and results these are built on. The first two papers consider a jump-to-default model, which is a model where some quantity, e.g. the price of a financial asset, is represented by a stochastic process which has continuous sample paths except for the possibility of a sudden drop to zero. In Paper I prices of European-type options in this model are studied together with the partial integro-differential equation that characterizes the price. In Paper II the price of a perpetual American put option in the same model is found in terms of explicit formulas. Both papers also study the parameter monotonicity and convexity properties of the option prices. The third and fourth articles both deal with valuation problems in a jump-diffusion model. Paper III concerns the optimal level at which to exercise an American put option with finite time horizon. More specifically, the integral equation that characterizes the optimal boundary is studied. In Paper IV we consider a stochastic game between two players and determine the optimal value and exercise strategy using an iterative technique. Paper V employs a similar iterative method to solve the statistical problem of determining the unknown drift of a stochastic process, where not only running time but also each observation of the process is costly.