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23 Cards in this Set
- Front
- Back
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What is stochastic calculus? |
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly. The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion as described by Louis Bachelier in 1900 and by Albert Einstein in 1905 and other physical diffusion processes in space of particles subject to random forces. Since the 1970s, the Wiener process has been widely applied in financial mathematics and economics to model the evolution in time of stock prices and bond interest rates.
(Source: Wikipedia) |
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What is calculus? |
Calculus is the mathematical study of change,[1] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus (concerning rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under and between curves); these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Generally, modern calculus is considered to have been developed in the 17th century by Isaac Newton and Gottfried Leibniz. Today, calculus has widespread uses in science, engineering and economics and can solve many problems that algebra alone cannot.
(Source: Wikipedia) |
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Where does the word calculus come from? |
The word "calculus" comes from Latin (calculus) and refers to a small stone used for counting. More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions.
(Source: Wikipedia) |
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Where was Kioshi Ito born and when? Where did he study? What is he famous for? |
1915, m\e-a prefector in Japan, imperial university of Tokyo, developing stochastic calculus. as a student, he recognized random phenomena was governed by statistical law., was not happy with the state of probability theory since the current papers did not define the random variable. INTUITIVELY knew that a random variable referred to a random experiment but without a formal definition made it difficult to work with such a vague a notion.
Like most mathematicians of the day, he did not take probability theory serious. Upon graduation in 1938, he joined the Cabinet Statistics bureau, and it must have been a "cushy" job because he like Einstein at the Swiss patent office he had time on his hands. The director realize he had more ability than compiling statistics so he let him study the work of Levi and kolMagorov.
Wrote the paper that would make stochastic processes amenable to calculus, published 1942, three years before his doctorate, titled "On Stochastic Processes"
(Source: Pricing The Future: . . ., Chapter 11) |
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What does non-differentiable mean? |
jerky, not smooth. |
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What is brownian motion? |
Brownian motion or pedesis (from Greek: πήδησις /pɛ̌ːdɛːsis/ "leaping") is the random motion of particles suspended in a fluid (a liquid or a gas) resulting from their collision with the quick atoms or molecules in the gas or liquid. The term "Brownian motion" can also refer to the mathematical model used to describe such random movements, which is often called a particle theory.[1] This transport phenomenon is named after the botanist Robert Brown. In 1827, while looking through a microscope at particles found in pollen grains in water, he noted that the particles moved through the water but was not able to determine the mechanisms that caused this motion. Atoms and molecules had long been theorized as the constituents of matter, and many decades later, Albert Einstein published a paper in 1905 that explained in precise detail how the motion that Brown had observed was a result of the pollen being moved by individual water molecules. This explanation of Brownian motion served as definitive confirmation that atoms and molecules actually exist, and was further verified experimentally by Jean Perrin in 1908. Perrin was awarded the Nobel Prize in Physics in 1926 "for his work on the discontinuous structure of matter" (Einstein had received the award five years earlier "for his services to theoretical physics" with specific citation of different research). The direction of the force of atomic bombardment is constantly changing, and at different times the particle is hit more on one side than another, leading to the seemingly random nature of the motion. The mathematical model of Brownian motion has numerous real-world applications. For instance, Stock market fluctuations are often cited, although Benoit Mandelbrot rejected its applicability to stock price movements in part because these are discontinuous.[2] Brownian motion is among the simplest of the continuous-time stochastic (or probabilistic) processes, and it is a limit of both simpler and more complicated stochastic processes (see random walk and Donsker's theorem). This universality is closely related to the universality of the normal distribution. In both cases, it is often mathematical convenience, rather than the accuracy of the models, that motivates their use.
(Source: Wikipedia referred by Pricing The Future: . . . ) |
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What is the variable time called in a differential equation? |
The drift or trend
(Source: Pricing The Future . . . ) |
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Who was the first mathematician to bring calculus to bear on problems involving probability? |
Bu-shell-yay, most recent kolMagorov
(Source: Pricing The Future . . ., Chapter 11) |
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What does differentiable mean? |
smooth mathematical functions |
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What did the chemist Fee Spedburg have trouble with? |
In trying to calculate the speed of a particle he could not determine the length of the path it was traveling because of the jerky, non-smooth motion.` |
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What is the Taylor Expansion? |
This was Ito's way out of the problem that Spedburg was experiencing when he look closely and closely at an atom's path, he saw more and more jags making the problem of computing the distance traveled more difficult. Therefore, the future value could lie anyway between plus infinity or minus infinity.
The Taylor Expansion |
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Who was Brook Taylor? |
An english mathematician who lived in the early 18th century and developed the Taylor Expansion to approximate the value of mathematical functions. With a Taylor expansion there is no need to manipulate unwieldy mathematical expressions to analyze them but can used more simple approximations instead.
For example, the number 2 can be approximated by adding 1 + 1/2 + 1/4 + ... and so on to as close to the number 2 as one would like. |
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What did Ito notice when using the Taylor Expansion? |
That only 3 terms were needed to describe the variables behaviors, additional terms could be ignored without hurting the approximation. |
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What does Ito's lemma? |
In mathematics, Itō's lemma is an identity used in Itō calculus to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule. Typically, it is memorized by forming the Taylor series expansion of the function up to its second derivatives and identifying the square of an increment in the Wiener process with an increment in time. The lemma is widely employed in mathematical finance, and its best known application is in the derivation of the Black–Scholes equation for option values. Itō's Lemma, which is named after Kiyoshi Itō, is occasionally referred to as the Itō–Doeblin Theorem in recognition of the recently discovered work of Wolfgang Doeblin.[1] |
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Why is there a third term in Ito's lemma? |
Typically, to describe a canon ball or movement of stars on required two terms but with the random movement a third term is required.
(Source: Pricing The Future . . . ) |
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Why can customary Brownian Motion not serve as a model for the price of a stock? |
Because in Brownian Motion an molecules path can cross to the left of a line in that path which represents a negative number but stock prices cannot be zero. |
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What is geometric Brownian Motion? |
This is not representing prices in absolute dollar terms but in percentages which prevent negative stocks and implying that an investor can lose more than an original investment when not trading on borrowed capital. |
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Why was Ito's paper initially ignored? |
He wrote it during WWII when Japan was being ignored during and immediately following the war. The paper was virtually unknown for nearly a decade.
Eventually, word filtered out. In 1954, when he was invited to give a lecture at the Institute of for Advanced Studies at Princeton, home of Einstein, Norman and Girdle, his work was already well known by the West.
(Source: Pricing The Future . . ., Chapter 11 ) |
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What is rarely done in Math that Ito achieved? |
For a lemma or theorem to named after a mathematician. It is even more rare for it used in naming a whole toolbox but the method of handling stochastic differential equations is known as Ito's Calculus.
(Source: Pricing The Future . . . ) |
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What does Ito call the basic element of probability theory? |
The random variable. |
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What are Kolmagorov and Levy's nationalities according to Ito in speech in recognizing his achievements in stochastic calculus? |
Russian and French.
(Source: Chapter 11) |
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In 2006, what prize did Ito received and why? |
The Gauss prize for laying the foundation for stochastic calculus.
(Source: Chapter 11) |
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When did Ito pass? |
2008. |
