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Deep Backward Stochastic Differential Equation Method
Deep backward stochastic differential equation method is a numerical method that combines deep learning with Backward stochastic differential equation (BSDE). This method is particularly useful for solving high-dimensional problems in financial derivatives pricing and risk management. By leveraging the powerful function approximation capabilities of deep neural networks, deep BSDE addresses the computational challenges faced by traditional numerical methods in high-dimensional settings. History Backwards stochastic differential equations BSDEs were first introduced by Pardoux and Peng in 1990 and have since become essential tools in stochastic control and financial mathematics. In the 1990s, Étienne Pardoux and Shige Peng established the existence and uniqueness theory for BSDE solutions, applying BSDEs to financial mathematics and control theory. For instance, BSDEs have been widely used in option pricing, risk measurement, and dynamic hedging. Deep learning Deep Learning is a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deep BSDE Method
Deep or The Deep may refer to: Places United States * Deep Creek (Appomattox River tributary), Virginia * Deep Creek (Great Salt Lake), Idaho and Utah * Deep Creek (Mahantango Creek tributary), Pennsylvania * Deep Creek (Mojave River tributary), California * Deep Creek (Pine Creek tributary), Pennsylvania * Deep Creek (Soque River tributary), Georgia * Deep Creek (Texas), a tributary of the Colorado River (Texas), Colorado River * Deep Creek (Washington), a tributary of the Spokane River * Deep River (Indiana), a tributary of the Little Calumet River * Deep River (Iowa), a minor tributary of the English River * Deep River (North Carolina) * Deep River (Washington), a minor List of rivers of Washington (state)#Lower Columbia Basin, tributary of the Columbia River * Deep Voll Brook, New Jersey, also known as Deep Brook Elsewhere * Deep Creek (Bahamas) * Deep Creek (Melbourne, Victoria), Australia, a tributary of the Maribyrnong River * Deep River (Western Australia) People * Dee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Image Processing
An image or picture is a visual representation. An image can be two-dimensional, such as a drawing, painting, or photograph, or three-dimensional, such as a carving or sculpture. Images may be displayed through other media, including a projection on a surface, activation of electronic signals, or digital displays; they can also be reproduced through mechanical means, such as photography, printmaking, or photocopying. Images can also be animated through digital or physical processes. In the context of signal processing, an image is a distributed amplitude of color(s). In optics, the term ''image'' (or ''optical image'') refers specifically to the reproduction of an object formed by light waves coming from the object. A ''volatile image'' exists or is perceived only for a short period. This may be a reflection of an object by a mirror, a projection of a camera obscura, or a scene displayed on a cathode-ray tube. A ''fixed image'', also called a hard copy, is one that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fully Connected Network
Network topology is the arrangement of the elements ( links, nodes, etc.) of a communication network. Network topology can be used to define or describe the arrangement of various types of telecommunication networks, including command and control radio networks, industrial fieldbusses and computer networks. Network topology is the topological structure of a network and may be depicted physically or logically. It is an application of graph theory wherein communicating devices are modeled as nodes and the connections between the devices are modeled as links or lines between the nodes. Physical topology is the placement of the various components of a network (e.g., device location and cable installation), while logical topology illustrates how data flows within a network. Distances between nodes, physical interconnections, transmission rates, or signal types may differ between two different networks, yet their logical topologies may be identical. A network's physical topology i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Financial Modeling
Financial modeling is the task of building an abstract representation (a model) of a real world financial situation. This is a mathematical model designed to represent (a simplified version of) the performance of a financial asset or portfolio of a business, project, or any other investment. Typically, then, financial modeling is understood to mean an exercise in either asset pricing or corporate finance, of a quantitative nature. It is about translating a set of hypotheses about the behavior of markets or agents into numerical predictions. At the same time, "financial modeling" is a general term that means different things to different users; the reference usually relates either to accounting and corporate finance applications or to quantitative finance applications. Accounting In corporate finance and the accounting profession, ''financial modeling'' typically entails financial statement forecasting; usually the preparation of detailed company-specific models used for deci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Image Recognition
Computer vision tasks include methods for acquiring, processing, analyzing, and understanding digital images, and extraction of high-dimensional data from the real world in order to produce numerical or symbolic information, e.g. in the form of decisions. "Understanding" in this context signifies the transformation of visual images (the input to the retina) into descriptions of the world that make sense to thought processes and can elicit appropriate action. This image understanding can be seen as the disentangling of symbolic information from image data using models constructed with the aid of geometry, physics, statistics, and learning theory. The scientific discipline of computer vision is concerned with the theory behind artificial systems that extract information from images. Image data can take many forms, such as video sequences, views from multiple cameras, multi-dimensional data from a 3D scanner, 3D point clouds from LiDaR sensors, or medical scanning devices. The t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brownian Motion
Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical sources. This motion pattern typically consists of Randomness, random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). More specifically, the fluid's overall Linear momentum, linear and Angular momentum, angular momenta remain null over time. The Kinetic energy, kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Differential Equations
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to model various behaviours of stochastic models such as stock prices,Musiela, M., and Rutkowski, M. (2004), Martingale Methods in Financial Modelling, 2nd Edition, Springer Verlag, Berlin. random growth models or physical systems that are subjected to thermal fluctuations. SDEs have a random differential that is in the most basic case random white noise calculated as the distributional derivative of a Brownian motion or more generally a semimartingale. However, other types of random behaviour are possible, such as jump processes like Lévy processes or semimartingales with jumps. Stochastic differential equations are in general neither differential equations nor random differential equations. Random differential equations a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pi Monte Carlo All
The number (; spelled out as pi) is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter. It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining , to avoid relying on the definition of the length of a curve. The number is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as \tfrac are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an algebraic equation involving only finite sums, products, powers, and integers. The transcendence of implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of appear to be randomly distributed, but no proof of thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Difference Method
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating Derivative, derivatives with Finite difference approximation, finite differences. Both the spatial domain and time domain (if applicable) are Discretization, discretized, or broken into a finite number of intervals, and the values of the solution at the end points of the intervals are approximated by solving algebraic equations containing finite differences and values from nearby points. Finite difference methods convert ordinary differential equations (ODE) or partial differential equations (PDE), which may be Nonlinear partial differential equation, nonlinear, into a system of linear equations that can be solved by matrix algebra techniques. Modern computers can perform these linear algebra computations efficiently, and this, along with their relative ease of implementation, has led to the widespread use of FDM in modern numerical analysi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monte Carlo Method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. The name comes from the Monte Carlo Casino in Monaco, where the primary developer of the method, mathematician Stanisław Ulam, was inspired by his uncle's gambling habits. Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration, and generating draws from a probability distribution. They can also be used to model phenomena with significant uncertainty in inputs, such as calculating the risk of a nuclear power plant failure. Monte Carlo methods are often implemented using computer simulations, and they can provide approximate solutions to problems that are otherwise intractable or too complex to analyze mathematically. Monte Carlo methods are widely used in va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runge–Kutta Method (SDE)
In mathematics of stochastic systems, the Runge–Kutta method is a technique for the approximate numerical solution of a stochastic differential equation. It is a generalisation of the Runge–Kutta method for ordinary differential equations to stochastic differential equations (SDEs). Importantly, the method does not involve knowing derivatives of the coefficient functions in the SDEs. Most basic scheme Consider the Itō diffusion X satisfying the following Itō stochastic differential equation dX_t = a(X_t) \, dt + b(X_t) \, dW_t, with initial condition X_0=x_0, where W_t stands for the Wiener process, and suppose that we wish to solve this SDE on some interval of time ,T/math>. Then the basic Runge–Kutta approximation to the true solution X is the Markov chain Y defined as follows: * partition the interval ,T/math> into N subintervals of width \delta=T/N > 0: 0 = \tau_ < \tau_ < \dots < \tau_ = T; * set ; * recursively compute |
Milstein Method
In mathematics, the Milstein method is a technique for the approximate numerical analysis, numerical solution of a stochastic differential equation. It is named after Grigori Milstein who first published it in 1974. Description Consider the autonomous equation, autonomous Itō calculus, Itō stochastic differential equation: \mathrm X_t = a(X_t) \, \mathrm t + b(X_t) \, \mathrm W_t with initial condition X_ = x_, where W_ denotes the Wiener process, and suppose that we wish to solve this SDE on some interval of time [0,T]. Then the Milstein approximation to the true solution X is the Markov chain Y defined as follows: * Partition the interval [0,T] into N equal subintervals of width \Delta t>0: 0 = \tau_0 < \tau_1 < \dots < \tau_N = T\text\tau_n:=n\Delta t\text\Delta t = \frac * Set * Recursively define for by: |
