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We’ve heard it: people that invest on the stock market or that gamble in lotteries, casinos, etc usually say “I’m going through a bad patch” (or bad spell). That is, they have been losing money for a while, but hey! better times are ahead and there’s no reason to quit. Are they sure? Are better times ahead? How close is “ahead” to today? Let’s work through a specific example to see how far is “ahead”.
In 1828, Robert Brown published the manuscript entitled “A brief account of microscopical observations made in the __months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies" in the Edinburgh new Philosophical Journal [download it in pdf format here]. He suspended some of the pollen grains of the species Clarkia pulchella in water and examined them closely, only to see them “filled with particles” of around 5 µm diameter that were “very evidently in motion”.
Stochastic differential equations (SDEs) are basically inhomogenous ordinary differential equations that depend on an external stochastic process.
Typically, that stochastic process is white noise, which is the mathematical idealization of the noise found in nature. This idealization is handy, because it simplifies the mathematical description. However, this idealization comes at some cost: traditional calculus is no longer valid and you have to use the so-call Itô calculus. This introduces some non intuitive changes.
Take a coin and toss it a number \(N\) of times in a time interval of duration \(T\). Suppose that every time you get head you win \(a\) euros and that you lose the same amount of money when you get tail. Then your capital is a random process with ups and dows like this:
This process is a stochastic process usually called “Random Walk” and its properties depend on the parameters $N, a $ and \(T\).
When tea is poured in a cup of hot water, we observe a phenomenon called diffusion: in the end particles of tea spread evenly throughout the mass of water and we enjoy our cup of tea. Diffusion occurs as a result of the second law of thermodynamics (increase of entropy) and can be modeled quantitatively using the diffusion equation (or heat equation). This is a funny equation, since it establishes that the velocity of spreading is infinite while the mean root square fluctuations of the position of the particles grows in time as