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Doob Decomposition Theorem
In the theory of stochastic processes in discrete time, a part of the mathematical theory of probability, the Doob decomposition theorem gives a unique decomposition of every adapted and integrable stochastic process as the sum of a martingale and a predictable process (or "drift") starting at zero. The theorem was proved by and is named for Joseph L. Doob. The analogous theorem in the continuous-time case is the Doob–Meyer decomposition theorem. Statement Let (\Omega, \mathcal, \mathbb) be a probability space, with N \in \N or I = \N_0 a finite or an infinite index set, (\mathcal_n)_ a filtration of \mathcal, and an adapted stochastic process with for all . Then there exist a martingale and an integrable predictable process starting with such that for every . Here predictable means that is \mathcal_-measurable for every . This decomposition is almost surely unique. Remark The theorem is valid word by word also for stochastic processes taking values in the -dim ... [...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] |
Almost Surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space, there is no difference between ''almost surely'' and ''surely'' (since having a probability of 1 often entails including all the sample points). However, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, and the continuity of the paths of Brownian motion. The terms almost certainly (a.c.) and almost always (a.a.) are also used. Almost never describes the opposite of ''almost surely'': an event that h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martingale Measure
In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or ''equivalent martingale measure'') is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure. This is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing, which implies that in a complete market, a derivative's price is the discounted expected value of the future payoff under the unique risk-neutral measure. Such a measure exists if and only if the market is arbitrage-free. The easiest way to remember what the risk-neutral measure is, or to explain it to a probability generalist who might not know much about finance, is to realize that it is: # The probability measure of a transformed random variable. Typically this transformation is the utility function of the payoff. The risk-neutral measure would be the measure corresponding to an expectation of the payoff with a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence (measure Theory)
In mathematics, and specifically in measure theory, equivalence is a notion of two measures being qualitatively similar. Specifically, the two measures agree on which events have measure zero. Definition Let \mu and \nu be two measures on the measurable space (X, \mathcal A), and let :\mathcal_\mu := \ and :\mathcal_\nu := \ be the sets of \mu-null sets and \nu-null sets, respectively. Then the measure \nu is said to be absolutely continuous in reference to \mu iff \mathcal N_\nu \supseteq \mathcal N_\mu. This is denoted as \nu \ll \mu. The two measures are called equivalent iff \mu \ll \nu and \nu \ll \mu, which is denoted as \mu \sim \nu. That is, two measures are equivalent if they satisfy \mathcal N_\mu = \mathcal N_\nu. Examples On the real line Define the two measures on the real line as : \mu(A)= \int_A \mathbf 1_(x) \mathrm dx : \nu(A)= \int_A x^2 \mathbf 1_(x) \mathrm dx for all Borel sets A . Then \mu and \nu are equivalent, since all sets outside of ,1 have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discounted Cash Flow
The discounted cash flow (DCF) analysis is a method in finance of valuing a security, project, company, or asset using the concepts of the time value of money. Discounted cash flow analysis is widely used in investment finance, real estate development, corporate financial management and patent valuation. It was used in industry as early as the 1700s or 1800s, widely discussed in financial economics in the 1960s, and became widely used in U.S. courts in the 1980s and 1990s. Application To apply the method, all future cash flows are estimated and discounted by using cost of capital to give their present values (PVs). The sum of all future cash flows, both incoming and outgoing, is the net present value (NPV), which is taken as the value of the cash flows in question; see aside. For further context see valuation overview; and for the mechanics see valuation using discounted cash flows, which includes modifications typical for startups, private equity and venture capital, co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Option
In finance, the style or family of an option is the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American (style) options. These options—as well as others where the payoff is calculated similarly—are referred to as "vanilla options". Options where the payoff is calculated differently are categorized as "exotic options". Exotic options can pose challenging problems in valuation and hedging. American and European options The key difference between American and European options relates to when the options can be exercised: * A European option may be exercised only at the expiration date of the option, i.e. at a single pre-defined point in time. * An American option on the other hand may be exercised at any time before the expiration date. For both, the payoff—when it occurs—is given by * \max\, for a call option * \max\, for a put option where K is the strike pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and risk and portfolio management on the other. Mathematical finance overlaps heavily with the fields of computational finance and financial engineering. The latter focuses on applications and modeling, often by help of stochastic asset models, while the former focuses, in addition to analysis, on building tools of implementation for the models. Also related is quantitative investing, which relies on statistical and numerical models (and lately machine learning) as opposed to traditional fundamental analysis when managing portfolios. French mathematician Louis Bachelier's doctoral thesis, defended in 1900, is considered the first scholarly work on mathematical fina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doob's Optional Stopping Theorem
In probability theory, the optional stopping theorem (or sometimes Doob's optional sampling theorem, for American probabilist Joseph Doob) says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial expected value. Since martingales can be used to model the wealth of a gambler participating in a fair game, the optional stopping theorem says that, on average, nothing can be gained by stopping play based on the information obtainable so far (i.e., without looking into the future). Certain conditions are necessary for this result to hold true. In particular, the theorem applies to doubling strategies. The optional stopping theorem is an important tool of mathematical finance in the context of the fundamental theorem of asset pricing. Statement A discrete-time version of the theorem is given below: Let be a discrete-time martingale and a stopping time with values in , both with respect to a filtration . Assume that one of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Integrability
In mathematics, uniform integrability is an important concept in real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ..., functional analysis and measure theory, and plays a vital role in the theory of Martingale (probability theory), martingales. Measure-theoretic definition Uniform integrability is an extension to the notion of a family of functions being dominated in L_1 which is central in Dominated convergence theorem, dominated convergence. Several textbooks on real analysis and measure theory use the following definition: Definition A: Let (X,\mathfrak, \mu) be a positive measure space. A set \Phi\subset L^1(\mu) is called uniformly integrable if \sup_\, f\, _0 there corresponds a \delta>0 such that : \int_E , f, \, d\mu 0 such that : \sup_\int_A, f, \, d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Random Walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z which starts at 0, and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion), the search path of a foraging animal, or the price of a fluctuating stock and the financial status of a gambler. Random walks have applications to engineering and many scientific fields including ecology, psychology, computer science, physics, chemistry, biology, economics, and sociology. The term ''random walk'' was first introduced by Karl Pearson in 1905. Lattice random walk A popular random walk model is that of a random walk on a regular lattice, where at each step the location jumps to another site according to some probability distribution. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z which starts at 0, and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion), the search path of a foraging animal, or the price of a fluctuating stock and the financial status of a gambler. Random walks have applications to engineering and many scientific fields including ecology, psychology, computer science, physics, chemistry, biology, economics, and sociology. The term ''random walk'' was first introduced by Karl Pearson in 1905. Lattice random walk A popular random walk model is that of a random walk on a regular lattice, where at each step the location jumps to another site according to some probability distribution. In a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decreasing
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus and analysis In calculus, a function f defined on a subset of the real numbers with real values is called ''monotonic'' if and only if it is either entirely non-increasing, or entirely non-decreasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is called ''monotonically increasing'' (also ''increasing'' or ''non-decreasing'') if for all x and y such that x \leq y one has f\!\left(x\right) \leq f\!\left(y\right), so f preserves the order (see Figure 1). Likewise, a function is called ''monotonically decreasing'' (also ''decreasing'' or ''non-increasing'') if, whenever x \leq y, then f\!\left(x\right) \geq f\!\left(y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
