Lecture #7, 8: Brownian Motion as a Model of a Fair Game

Ito's Lemma - YouTube

2.5 Brownian motion Brownian motion, which is both Markov and Martingale, is arguably the most important stochastic process. It is commonly used in physical science and also several branches of social sciences. Its history speaks for the diversity of the process. Brownian motion was first • Brownian motion with drift. Now consider a Brownian motion with drift µ and standard deviation σ. That is consider B µ(t) = µt + σB(t), where B is the standard Brownian motion. It is straightforward to show that B µ(t)−µt is a martingale. Also it is simple to see that (B µ(t)−µt)2 −σ2t is also a martingale. • Wald’s ... Stopped Brownian motion is an example of a martingale. It can model an even coin-toss betting game with the possibility of bankruptcy. In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence, given all prior values, is equal to the present value. I feel it's best to interpret the Ito Integral via relating it to a gambling game: the integrator (i.e. the Brownian motion with respect to which we are integrating) is the (random) outcome of the gambling game, whilst the integrand (the function we are integrating) is the betting strategy. The betting strategy can be deterministic or random. From the known completeness of Brownian motion, the Poisson martingale N(t) - At, and the one-dimensional random walk in discrete time, we then conjecture that this model is complete. 7. Concluding remarks This section presents a list of unresolved questions which, we think, merit further study by probabilists and/or economists.

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Ito's Lemma - YouTube

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