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Basic Affine Jump Diffusion
In mathematics probability theory, a basic affine jump diffusion (basic AJD) is a stochastic process Z of the form : dZ_t=\kappa (\theta -Z_t)\,dt+\sigma \sqrt\,dB_t+dJ_t,\qquad t\geq 0, Z_\geq 0, where B is a standard Brownian motion, and J is an independent compound Poisson process with constant jump intensity l and independent exponentially distributed jumps with mean \mu . For the process to be well defined, it is necessary that \kappa \theta \geq 0 and \mu \geq 0 . A basic AJD is a special case of an affine process and of a jump diffusion. On the other hand, the Cox–Ingersoll–Ross (CIR) process is a special case of a basic AJD. Basic AJDs are attractive for modeling default times in credit risk applications, since both the moment generating function : m\left( q\right) =\operatorname \left( e^\right) ,\qquad q\in \mathbb, and the characteristic function : \varphi \left( u\right) =\operatorname \left( e^\right) ,\qquad u\in \mathbb, are known in close ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brownian Motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of 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 and angular momenta remain null over time. The 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 internal energy (the equipartition theorem). This motion is named after the botanist Robert Brown, who first described the phenomenon in 1827, while looking throu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compound Poisson Process
A compound Poisson process is a continuous-time (random) stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution. A compound Poisson process, parameterised by a rate \lambda > 0 and jump size distribution ''G'', is a process \ given by :Y(t) = \sum_^ D_i where, \ is a counting of a Poisson process with rate \lambda, and \ are independent and identically distributed random variables, with distribution function ''G'', which are also independent of \.\, When D_i are non-negative integer-valued random variables, then this compound Poisson process is known as a stuttering Poisson process which has the feature that two or more events occur in a very short time. Properties of the compound Poisson process The expected value of a compound Poisson process can be calculated using a result known as Wald's equation as: :\operatorname E(Y(t)) = \operatorname E(D_1 + \cdo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Process
An affine term structure model is a financial model that relates zero-coupon bond prices (i.e. the discount curve) to a spot rate model. It is particularly useful for deriving the yield curve – the process of determining spot rate model inputs from observable bond market data. The affine class of term structure models implies the convenient form that log bond prices are linear functions of the spot rate (and potentially additional state variables). Background Start with a stochastic short rate model r(t) with dynamics: : dr(t)=\mu(t,r(t)) \, dt + \sigma(t,r(t)) \, dW(t) and a risk-free zero-coupon bond maturing at time T with price P(t,T) at time t. The price of a zero-coupon bond is given by:P(t,T) = \mathbb^\left\where T=t+\tau, with \tau being is the bond's maturity. The expectation is taken with respect to the risk-neutral probability measure \mathbb. If the bond's price has the form: :P(t,T)=e^ where A and B are deterministic functions, then the short rate model i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jump Diffusion
Jump diffusion is a stochastic process that involves jumps and diffusion. It has important applications in magnetic reconnection, coronal mass ejections, condensed matter physics, option pricing, and pattern theory and computational vision. In physics In crystals, atomic diffusion typically consists of jumps between vacant lattice sites. On time and length scales that average over many single jumps, the net motion of the jumping atoms can be described as regular diffusion. Jump diffusion can be studied on a microscopic scale by inelastic neutron scattering and by Mößbauer spectroscopy. Closed expressions for the autocorrelation function have been derived for several jump(-diffusion) models: * Singwi, Sjölander 1960: alternation between oscillatory motion and directed motion * Chudley, Elliott 1961: jumps on a lattice * Sears 1966, 1967: jump diffusion of rotational degrees of freedom * Hall, Ross 1981: jump diffusion within a restricted volume In economics and finance I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Credit Risk
A credit risk is risk of default on a debt that may arise from a borrower failing to make required payments. In the first resort, the risk is that of the lender and includes lost principal and interest, disruption to cash flows, and increased collection costs. The loss may be complete or partial. In an efficient market, higher levels of credit risk will be associated with higher borrowing costs. Because of this, measures of borrowing costs such as yield spreads can be used to infer credit risk levels based on assessments by market participants. Losses can arise in a number of circumstances, for example: * A consumer may fail to make a payment due on a mortgage loan, credit card, line of credit, or other loan. * A company is unable to repay asset-secured fixed or floating charge debt. * A business or consumer does not pay a trade invoice when due. * A business does not pay an employee's earned wages when due. * A business or government bond issuer does not make a payment on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment Generating Function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables. However, not all random variables have moment-generating functions. As its name implies, the moment-generating function can be used to compute a distribution’s moments: the ''n''th moment about 0 is the ''n''th derivative of the moment-generating function, evaluated at 0. In addition to real-valued distributions (univariate distributions), moment-generating functions can be defined for vector- or matrix-valued random variables, and can even be extended to more general cases. The moment-generating func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Function (probability Theory)
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables. In addition to univariate distributions, characteristic functions can be defined for vector- or matrix-valued random variables, and can also be extended to more generic cases. The characteristic function always exists when treated as a function of a real-valued argument, unlike the moment-generating function. There are relations between the behavior of the characteristic function of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed-form Expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit, differentiation, or integration. The set of operations and functions may vary with author and context. Example: roots of polynomials The solutions of any quadratic equation with complex coefficients can be expressed in closed form in terms of addition, subtraction, multiplication, division, and square root extraction, each of which is an elementary function. For example, the quadratic equation :ax^2+bx+c=0, is tractable since its solutions can be expressed as a closed-form expression, i.e. in terms of elementary functions: :x=\frac. Similarly, solutions of cubic and quartic (third and fourth degree) equations can be expresse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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