Yim Man Yi = 亞擴散雙穩分數福克-普朗克系統的線性響應及隨機共振和有色噪音對雙穩系統所引起的效應 / 嚴敏儀. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 85-89). / Text in English; abstracts in English and Chinese. / Yim Man Yi = Ya kuo san shuang wen fen shu Fuke-Pulangke xi tong de xian xing xiang ying ji sui ji gong zhen he you se zao yin dui shuang wen xi tong suo yin qi de xiao ying / Yan Minyi. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Brownian motion and anomalous dynamics --- p.1 / Chapter 1.2 --- White and colored noises --- p.3 / Chapter 2 --- Linear response theory of sub diffusive Fokker-Planck systems --- p.6 / Chapter 2.1 --- Introduction to subdiffusive Fokker-Planck systems --- p.6 / Chapter 2.2 --- Spectral density --- p.8 / Chapter 2.3 --- Linear response --- p.11 / Chapter 2.4 --- Signal-to-noise ratio (SNR) --- p.13 / Chapter 2.5 --- Stochastic energy --- p.14 / Chapter 3 --- Perturbation due to a sinusoidal signal --- p.16 / Chapter 3.1 --- Presence of an external sinusoidal forcing --- p.16 / Chapter 3.1.1 --- PDF and linear reponse --- p.16 / Chapter 3.1.2 --- Phase lag --- p.19 / Chapter 3.1.3 --- SNR --- p.19 / Chapter 3.1.4 --- Stochastic energy --- p.23 / Chapter 3.2 --- Presence of a sinusoidally time-varying diffusion coefficient --- p.23 / Chapter 4 --- Perturbation due to a rectangular signal --- p.26 / Chapter 4.1 --- Linear response to a rectangular pulse --- p.26 / Chapter 4.2 --- Perturbation due to a periodic rectangular signal --- p.28 / Chapter 4.2.1 --- Linear response --- p.29 / Chapter 4.2.2 --- SNR --- p.31 / Chapter 4.2.3 --- Stochastic energy --- p.33 / Chapter 4.3 --- Comparison with the response to a sinusoidal driving force --- p.35 / Chapter 5 --- The Effects of Colored Noise on the Stationary Probability Distribution --- p.38 / Chapter 5.1 --- Formulation of a general colored noises-driven system --- p.39 / Chapter 5.2 --- Approximation schemes for a colored noise --- p.42 / Chapter 5.2.1 --- Decoupling approximation --- p.42 / Chapter 5.2.2 --- UCNA --- p.45 / Chapter 5.2.3 --- Small r approximation . --- p.46 / Chapter 5.2.4 --- Presence of an additive noise: g(x) = 1 --- p.47 / Chapter 5.2.5 --- Presence of a multiplicative noise: g(x) = x --- p.52 / Chapter 5.3 --- Approximation scheme for two colored noises --- p.53 / Chapter 5.3.1 --- Presence of two additive noises --- p.55 / Chapter 5.3.2 --- Presence of an additive noise and a multiplicative noise --- p.55 / Chapter 5.4 --- Unimodal-bimodal transitions --- p.61 / Chapter 6 --- A stochastic genetic regulatory transcription model with colored noises --- p.68 / Chapter 6.1 --- Biological background --- p.69 / Chapter 6.2 --- Effect of a single noise --- p.72 / Chapter 6.3 --- Effects of two noises --- p.75 / Chapter 6.4 --- Biological implications of colored noises in the model --- p.81 / Chapter 7 --- Conclusions --- p.82 / Bibliography --- p.85 / Chapter A --- Mittag-Leffler function --- p.90 / Chapter B --- H-function (Fox function) --- p.92 / Chapter C --- Operator algebra --- p.94 / Chapter D --- Mean first passage time --- p.97

Chow Cheuk Wang = 含粒子阱之福克-普朗克方程及亞擴散分數福克-普朗克方程之研究 / 周卓宏. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 70-72). / Text in English; abstracts in English and Chinese. / Chow Cheuk Wang = Han li zi jing zhi Fuke-Pulangke fang cheng ji ya kuo san fen shu Fuke-Pulangke fang cheng zhi yan jiu / Zhou Zhuohong. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Derivation of the Fokker-Planck equation --- p.5 / Chapter 2.1 --- Diffusion equation --- p.5 / Chapter 2.2 --- Kramers-Moyal Equation --- p.7 / Chapter 2.3 --- Fokker-Planck Equation --- p.9 / Chapter 2.3.1 --- Eigenfunction Expansion --- p.10 / Chapter 2.3.2 --- Mapping FPE to a pseudo-Schrodinger equation --- p.11 / Chapter 3 --- Conventional Fokker-Planck equation with sinks --- p.15 / Chapter 3.1 --- Propagator with sinks --- p.16 / Chapter 3.1.1 --- One-sink propagator --- p.17 / Chapter 3.1.2 --- Two-sink propagator --- p.18 / Chapter 3.2 --- Survival probability --- p.19 / Chapter 3.3 --- Expectation value of the position --- p.22 / Chapter 3.4 --- Mean survival time --- p.25 / Chapter 4 --- Fractional Fokker-Planck equation with sinks --- p.38 / Chapter 4.1 --- Fractional diffusion equation --- p.39 / Chapter 4.2 --- Propagator of the subdiffusive system --- p.41 / Chapter 4.3 --- Survival probability and expectation value of the position --- p.43 / Chapter 4.4 --- Mean survival-time distribution --- p.47 / Chapter 5 --- Boundary value problems for diffusion and subdiffusion --- p.53 / Chapter 5.1 --- Diffusion in a linear potential U(x) = Fx --- p.54 / Chapter 5.2 --- Two absorbing boundaries --- p.55 / Chapter 5.3 --- One absorbing boundary and one reflecting boundary --- p.58 / Chapter 5.4 --- Two reflecting boundaries --- p.59 / Chapter 6 --- Summary --- p.68 / Bibliography --- p.70 / Chapter A --- Laplace transform and the method of Abate and Whitt --- p.73 / Chapter B --- Mittag-Leffler function and its two-point Pade approximant --- p.77

University, Diss.--Paderborn, 1985.

A stochastic process or sometimes called random process is the counterpart to a deterministic process in theory. A stochastic process is a random field, whose domain is a region of space, in other words, a random function whose arguments are drawn from a range of continuously changing values. In this case, Instead of dealing only with one possible 'reality' of how the process might evolve under time (as is the case, for example, for solutions of an ordinary differential equation), in a stochastic or random process there is some indeterminacy in its future evolution described by probability distributions. This means that even if the initial condition (or starting point) is known, there are many possibilities the process might go to, but some paths are more probable and others less. However, in discrete time, a stochastic process amounts to a sequence of random variables known as a time series. Over the past decades, the problems of synergetic are concerned with the study of macroscopic quantitative changes of systems belonging to various disciplines such as natural science, physical science and electrical engineering. When such transition from one state to another take place, fluctuations i.e. (random process) may play an important role. Fluctuations in its sense are very common in a large number of fields and nearly every system is subjected to complicated external or internal influences that are often termed noise or fluctuations. Fokker-Planck equation has turned out to provide a powerful tool with which the effects of fluctuation or noise close to transition points can be adequately be treated. For this reason, in this thesis work analytical and numerical methods of solving Fokker-Planck equation, its derivation and some of its applications will be carefully treated. Emphasis will be on both for one variable and N- dimensional cases.

Orientador: Paulo H. Sakanaka / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Fisica "Gleb Wataghin" / Made available in DSpace on 2018-07-23T12:45:45Z (GMT). No. of bitstreams: 1 Samogin_Euniced'Avila_D.pdf: 1282510 bytes, checksum: 4e68b6e2ad2de72e9046aa9187545b31 (MD5) Previous issue date: 1997 / Resumo: A geração de corrente por onda híbrida inferior LH ("lower hybrid"), é a técnica mais promissora para a obtenção de corrente contínua para se chegar à fusão termonuclear controlada. A exitência da "brecha espectral" ("spectral gap") não permite a conciliação satisfatória entre resultados teóricos e medidas experimentais da corrente gerada. A teoria tem previsto correntes sempre menores que as medidas. Neste trabalho apresentamos uma possível solução para a "brecha espectral", através de um código numérico que calcula a corrente gerada por ondas LH, usando a equação de Fokker-Planck Quasilinear em duas dimensões, levando em conta detalhes da ressonância das partículas com as ondas longitudinais (LH) e sua distribuição espectral / Abstract: Current drive by lower hybrid wave LH, is the most promising technique to obtain continuos current for the controlled thermonuclear fusion. The existence of "Spectral gap" forbid satisfactory match between theoretical results and experimental measurements of the generated current. The theory has always predicted smaller values. In this Thesis we show a possible solution for the spectral gap, by a numerical code which calculate the generated current by LH waves, using the "Quasi1inear Fokker-Planck" equation in two dimensions, taking in to account detai1s of ressonance between particles and the longitudinal LH waves and its spectral distribution / Doutorado / Física / Doutor em Ciências

Firman So. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves [159]-160). / Text in English; abstracts in English and Chinese. / Firman So. / Abstract --- p.i / Acknowledgement --- p.iii / Contents --- p.iv / List of Figures --- p.vii / List of Tables --- p.xii / Chapter Chapter 1. --- Introduction --- p.1 / Chapter Chapter 2. --- Derivation of the Fokker-Planck Equation --- p.4 / Chapter 2.1 --- Brownian Motion --- p.4 / Chapter 2.2 --- Non-Linear Langevin Equation --- p.7 / Chapter 2.3 --- Conditional Probability Density --- p.9 / Chapter 2.4 --- Kramers-Moyal Expansion --- p.11 / Chapter 2.5 --- Fokker-Planck Equation --- p.13 / Chapter Chapter 3. --- Method & Solution of the One-variable Fokker-Planck Equation with Time-Independent Coefficients --- p.15 / Chapter 3.1 --- Stationary Solution --- p.16 / Chapter 3.2 --- Ornstein-Ulhenbeck Process: An Exactly Solvable Fokker-Planck Equation --- p.17 / Chapter 3.3 --- Eigenfunction Expansion --- p.19 / Chapter 3.4 --- Ornstein-Ulhenbeck process by Eigenfunction Expansion --- p.29 / Chapter 3.5 --- Eigenfunctions and Eigenvalues of Inverted Potentials --- p.30 / Chapter 3.6 --- Kramers' Escape Rate --- p.32 / Chapter Chapter 4. --- Diffusion in Potential Wells --- p.36 / Chapter 4.1 --- Symmetric Double-Well Potential --- p.36 / Chapter 4.2 --- Asymmetric Bistable Potential --- p.61 / Chapter Chapter 5. --- Stochastic Resonance --- p.100 / Chapter 5.1 --- Introduction --- p.100 / Chapter 5.2 --- Probability Density........................... --- p.101 / Chapter 5.3 --- Power Spectrum of the Autocorrelation Function of x --- p.113 / Chapter 5.4 --- Stochastic Resonance --- p.120 / Chapter Chapter 6. --- Colored Noise --- p.124 / Chapter 6.1 --- Introduction --- p.124 / Chapter 6.2 --- Approximation Schemes for the Colored Noise Problem --- p.125 / Chapter 6.3 --- Stationary Probability Density of the Colored Noise Driven Bistable System --- p.132 / Chapter 6.4 --- Escape Rate in the Presence of Colored Noise --- p.140 / Chapter Chapter 7. --- Conclusion --- p.146 / Appendix A --- p.149 / Chapter A.1 --- State-Dependent Diagonalization Method --- p.149 / Chapter A.2 --- Infinite-Square-Well Basis Diagnalization --- p.153 / Chapter A.3 --- Solving the Fokker-Planck equation --- p.156 / References

The Fokker-Planck (FP) equation is a fundamental model in thermodynamic kinetic theories and statistical mechanics. In general, the FP equation appears in a number of different fields in natural sciences, for instance in solid-state physics, quantum optics, chemical physics, theoretical biology, and circuit theory. These equations also provide a powerful mean to define robust control strategies for random models. The FP equations are partial differential equations (PDE) describing the time evolution of the probability density function (PDF) of stochastic processes. These equations are of different types depending on the underlying stochastic process. In particular, they are parabolic PDEs for the PDF of Ito processes, and hyperbolic PDEs for piecewise deterministic processes (PDP). A fundamental axiom of probability calculus requires that the integral of the PDF over all the allowable state space must be equal to one, for all time. Therefore, for the purpose of accurate numerical simulation, a discretized FP equation must guarantee conservativeness of the total probability. Furthermore, since the solution of the FP equation represents a probability density, any numerical scheme that approximates the FP equation is required to guarantee the positivity of the solution. In addition, an approximation scheme must be accurate and stable. For these purposes, for parabolic FP equations on bounded domains, we investigate the Chang-Cooper (CC) scheme for space discretization and first- and second-order backward time differencing. We prove that the resulting space-time discretization schemes are accurate, conditionally stable, conservative, and preserve positivity. Further, we discuss a finite difference discretization for the FP system corresponding to a PDP process in a bounded domain. Next, we discuss FP equations in unbounded domains. In this case, finite-difference or finite-element methods cannot be applied. By employing a suitable set of basis functions, spectral methods allow to treat unbounded domains. Since FP solutions decay exponentially at infinity, we consider Hermite functions as basis functions, which are Hermite polynomials multiplied by a Gaussian. To this end, the Hermite spectral discretization is applied to two different FP equations; the parabolic PDE corresponding to Ito processes, and the system of hyperbolic PDEs corresponding to a PDP process. The resulting discretized schemes are analyzed. Stability and spectral accuracy of the Hermite spectral discretization of the FP problems is proved. Furthermore, we investigate the conservativity of the solutions of FP equations discretized with the Hermite spectral scheme. In the last part of this thesis, we discuss optimal control problems governed by FP equations on the characterization of their solution by optimality systems. We then investigate the Hermite spectral discretization of FP optimality systems in unbounded domains. Within the framework of Hermite discretization, we obtain sparse-band systems of ordinary differential equations. We analyze the accuracy of the discretization schemes by showing spectral convergence in approximating the state, the adjoint, and the control variables that appear in the FP optimality systems. To validate our theoretical estimates, we present results of numerical experiments. / Die Fokker-Planck (FP) Gleichung ist ein grundlegendes Modell in thermodynamischen kinetischen Theorien und der statistischen Mechanik. Die FP-Gleichungen sind partielle Differentialgleichungen (PDE), welche die zeitliche Entwicklung der Wahrscheinlichkeitsdichtefunktion (PDF) von stochastischen Prozessen beschreiben. Diese Gleichungen sind von verschiedenen Arten, abhängig von dem zugrunde liegenden stochastischen Prozess. Insbesondere sind dies parabolische PDEs für die PDF von Ito Prozessen, und hyperbolische PDEs für teilweise deterministische Prozesse (PDP). Ein grundlegendes Axiom der Wahrscheinlichkeitsrechnung verlangt, dass das Integral der PDF über den ganzen Raum für alle Zeit muss gleich sein muss. Daher muss eine diskretisierte FP Gleichung Konservativität der Gesamtwahrscheinlichkeit garantieren. Da die Lösung der FP Gleichung eine Wahrscheinlichkeitsdichte darstellt, muss das numerische Verfahren, das die FP-Gleichung approximiert, die Positivität der Lösung gewährleisten. Darüber hinaus muss ein Approximationsschema genau und stabil sein. Für FP-Gleichungen auf begrenzte Bereiche untersuchen wir das Chang-Cooper (CC) Schema. Wir beweisen, dass die Diskretisierungsmethoden genau, bedingt stabil und konservativ sind, und Positivität bewahren. Als nächstes diskutieren wir FP Gleichungen in unbeschränkten Gebieten und wir wählen die Hermite spektrale Diskretisierung. Die resultierenden diskretisierten Schemata werden analysiert. Stabilität und spektrale Genauigkeit der Hermiten spektralen Diskretisierung ist bewiesen. Darüber hinaus untersuchen wir die Konservativität der numerischen Lösungen der FP Gleichungen. Im letzten Teil dieser Arbeit diskutieren wir Probleme der optimalen Steuerung, die von FP Gleichungen geregelt werden. Wir untersuchen dann die Hermite spektrale Diskretisierung von FP Optimalitätssystemen in unbeschränkten Gebieten. Wir erhalten spärliche Band-Systeme gewöhnlicher Differentialgleichungen. Wir analysieren die Genauigkeit der Diskretisierungsmethoden, indem wir spektrale Konvergenz bei der Annäherung des zustandes, das Adjoint, und die Stellgrößen, die in den FP Optimalitätssystemen erscheinen, aufzeigen. Um unsere theoretischen Schätzungen zu bestätigen, präsentieren wir Ergebnisse von numerischen Experimenten.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1983. / Typescript. Vita. Description based on print version record. Includes bibliographical references.

In problems involving highly forward-peaked scattering, the Boltzmann transport equation can be simplified using the Fokker-Planck model. The purpose of this project was to develop an analytical solution to the resulting Fokker-Planck equation. This analytical solution can then be used to benchmark numerical transport codes. A numerical solution to the Fokker-Planck equation was also developed. The analytical solution found is a generalized function. It satisfies the purpose of the project with two limitations. The first limitation is that the solution can only be evaluated for certain sources. The second limitation is that the solution can only be evaluated for small times. The moments of the Fokker-Planck equation can be evaluated for any time. The numerical solution developed works for all sources and all times. The analytical solution, then, provides an accurate and precise benchmark under certain conditions. The numerical solution provides a less accurate benchmark under all conditions.

The topic of this thesis is the theoretical and numerical analysis of optimal control problems, whose differential constraints are given by Fokker-Planck models related to jump-diffusion processes. We tackle the issue of controlling a stochastic process by formulating a deterministic optimization problem. The key idea of our approach is to focus on the probability density function of the process, whose time evolution is modeled by the Fokker-Planck equation. Our control framework is advantageous since it allows to model the action of the control over the entire range of the process, whose statistics are characterized by the shape of its probability density function. We first investigate jump-diffusion processes, illustrating their main properties. We define stochastic initial-value problems and present results on the existence and uniqueness of their solutions. We then discuss how numerical solutions of stochastic problems are computed, focusing on the Euler-Maruyama method. We put our attention to jump-diffusion models with time- and space-dependent coefficients and jumps given by a compound Poisson process. We derive the related Fokker-Planck equations, which take the form of partial integro-differential equations. Their differential term is governed by a parabolic operator, while the nonlocal integral operator is due to the presence of the jumps. The derivation is carried out in two cases. On the one hand, we consider a process with unbounded range. On the other hand, we confine the dynamic of the sample paths to a bounded domain, and thus the behavior of the process in proximity of the boundaries has to be specified. Throughout this thesis, we set the barriers of the domain to be reflecting. The Fokker-Planck equation, endowed with initial and boundary conditions, gives rise to Fokker-Planck problems. Their solvability is discussed in suitable functional spaces. The properties of their solutions are examined, namely their regularity, positivity and probability mass conservation. Since closed-form solutions to Fokker-Planck problems are usually not available, one has to resort to numerical methods. The first main achievement of this thesis is the definition and analysis of conservative and positive-preserving numerical methods for Fokker-Planck problems. Our SIMEX1 and SIMEX2 (Splitting-Implicit-Explicit) schemes are defined within the framework given by the method of lines. The differential operator is discretized by a finite volume scheme given by the Chang-Cooper method, while the integral operator is approximated by a mid-point rule. This leads to a large system of ordinary differential equations, that we approximate with the Strang-Marchuk splitting method. This technique decomposes the original problem in a sequence of different subproblems with simpler structure, which are separately solved and linked to each other through initial conditions and final solutions. After performing the splitting step, we carry out the time integration with first- and second-order time-differencing methods. These steps give rise to the SIMEX1 and SIMEX2 methods, respectively. A full convergence and stability analysis of our schemes is included. Moreover, we are able to prove that the positivity and the mass conservation of the solution to Fokker-Planck problems are satisfied at the discrete level by the numerical solutions computed with the SIMEX schemes. The second main achievement of this thesis is the theoretical analysis and the numerical solution of optimal control problems governed by Fokker-Planck models. The field of optimal control deals with finding control functions in such a way that given cost functionals are minimized. Our framework aims at the minimization of the difference between a known sequence of values and the first moment of a jump-diffusion process; therefore, this formulation can also be considered as a parameter estimation problem for stochastic processes. Two cases are discussed, in which the form of the cost functional is continuous-in-time and discrete-in-time, respectively. The control variable enters the state equation as a coefficient of the Fokker-Planck partial integro-differential operator. We also include in the cost functional a $L^1$-penalization term, which enhances the sparsity of the solution. Therefore, the resulting optimization problem is nonconvex and nonsmooth. We derive the first-order optimality systems satisfied by the optimal solution. The computation of the optimal solution is carried out by means of proximal iterative schemes in an infinite-dimensional framework. / Die vorliegende Arbeit beschäftigt sich mit der theoretischen und numerischen Analyse von Optimalsteuerungsproblemen, deren Nebenbedingungen die Fokker-Planck-Gleichungen von Sprung-Diffusions-Prozessen sind. Unsere Strategie baut auf der Formulierung eines deterministischen Problems auf, um einen stochastischen Prozess zu steuern. Der Ausgangspunkt ist, die Wahrscheinlichkeitsdichtefunktion des Prozesses zu betrachten, deren zeitliche Entwicklung durch die Fokker-Planck-Gleichung modelliert wird. Dieser Ansatz ist vorteilhaft, da er es ermöglicht, den gesamten Bereich des Prozesses durch die Wirkung der Steuerung zu beeinflussen. Zuerst beschäftigen wir uns mit Sprung-Diffusions-Prozessen. Wir definieren Ausgangswertprobleme, die durch stochastische Differentialgleichungen beschrieben werden, und präsentieren Ergebnisse zur Existenz und Eindeutigkeit ihrer Lösungen. Danach diskutieren wir, wie numerische Lösungen stochastischer Probleme berechnet werden, wobei wir uns auf die Euler-Maruyama-Methode konzentrieren. Wir wenden unsere Aufmerksamkeit auf Sprung-Diffusions-Modelle mit zeit- und raumabhängigen Koeffizienten und Sprüngen, die durch einen zusammengesetzten Poisson-Prozess modelliert sind. Wir leiten die zugehörigen Fokker-Planck-Glei-chungen her, die die Form von partiellen Integro-Differentialgleichungen haben. Ihr Differentialterm wird durch einen parabolischen Operator beschrieben, während der nichtlokale Integraloperator Spr\"{u}nge modelliert. Die Ableitung wird auf zwei unterschiedlichen Arten ausgef\"{u}hrt, je nachdem, ob wir einen Prozess mit unbegrenztem oder beschränktem Bereich betrachten. In dem zweiten Fall muss das Verhalten des Prozesses in der Nähe der Grenzen spezifiziert werden; in dieser Arbeit setzen wir reflektierende Grenzen. Die Fokker-Planck-Gleichung, zusammen mit einem Anfangswert und geeigneten Randbedingungen, erzeugt das Fokker-Planck-Problem. Die Lösbarkeit dieses Pro-blems in geeigneten Funktionenräumen und die Eigenschaften dessen Lösung werden diskutiert, nämlich die Positivität und die Wahrscheinlichkeitsmassenerhaltung. Da analytische Lösungen von Fokker-Planck-Problemen oft nicht verfügbar sind, m\"{u}ssen numerische Methoden verwendet werden. Die erste bemerkenswerte Leistung dieser Arbeit ist die Definition und Analyse von konservativen numerischen Verfahren, die Fokker-Planck-Probleme lösen. Unsere SIMEX1 und SIMEX2 (Splitting-Implizit-Explizit) Schemen basieren auf der Linienmethode. Der Differentialoperator wird durch das Finite-Volumen-Schema von Chang und Cooper diskretisiert, während der Integraloperator durch eine Mittelpunktregel angenähert wird. Dies führt zu einem großen System von gewöhnlichen Differentialgleichungen, das mit der Strang-Marchuk-Splitting-Methode gelöst wird. Diese Technik teilt das ursprüngliche Problem in eine Folge verschiedener Teilprobleme mit einer einfachen Struktur, die getrennt gelöst werden und danach durch deren Anfangswerte miteinander verbunden werden. Dank der Splitting-Methode kann jedes Teilproblem implizit oder explizit gelöst werden. Schließlich wird die numerische Integration des Anfangswertsproblems mit zwei Verfahren durchgeführt, n\"{a}mlich dem Euler-Verfahren und dem Predictor-Corrector-Verfahren. Eine umfassende Konvergenz- und Stabilitätsanalyse unserer Systeme ist enthalten. Darüber hinaus können wir beweisen, dass die Positivität und die Massenerhaltung der Lösung von Fokker-Planck-Problemen auf diskreter Ebene durch die numerischen Lösungen erfüllt werden, die mit den SIMEX-Schemen berechnet wurden. Die zweite bemerkenswerte Leistung dieser Arbeit ist die theoretische Analyse und die numerische Behandlung von Optimalsteuerungsproblemen, deren Nebenbedingungen die Fokker-Planck-Probleme von Sprung-Diffusions-Prozessen sind. Der Bereich der optimalen Steuerung befasst sich mit der Suche nach einer optimalen Funktion, die eine gegebene Zielfunktion minimiert. Wir zielen auf die Minimierung des Unterschieds zwischen einer bekannten Folge von Werten und dem ersten Moment eines Sprung-Diffusions-Prozesses. Auf diese Weise kann unsere Formulierung auch als ein Parameterschätzungsproblem für stochastische Prozesse angesehen werden. Zwei Fälle sind erläutert, in denen die Zielfunktion zeitstetig beziehungsweise zeitdiskret ist. Da die Steuerung ein Koeffizient des Integro-Differentialoperators der Zustandsglei-chung ist und die Zielfunktion einen $ L^1 $-Term beinhaltet, der die dünne Besetzung der Lösung erhöht, ist das Optimierungsproblem nichtkonvex und nichtglatt. Die von der optimalen L\"{o}sung erf\"{u}llten notwendigen Bedingungen werden hergeleitet, die man mit einem System beschreiben kann. Die Berechnung optimaler Lösungen wird mithilfe von Proximal-Methoden durchgeführt, die entsprechend um den unendlichdimensionalen Fall erweitert wurden.