Before embarking on this assignment, you should read this pdf file:
SDEs.pdf
Then work on the problems in this Jupyter notebook:
StochasticDiffEqs-1.ipynb

We saw how Brownian motion paths can be constructed by using cumulative sums of innitesimal stochastic jumps in each innitesimal time step. At rst glance, this might seem vaguely similar to what we do when we solve dierential equations, except that the jumps, while innitesimal, are stochastic.
Before talking about stochastic dierential equations, it is helpful to review some ordinary dierential equation (ODE) concepts.

Brownian Motion
In order to understand what happens on the stochastic side, with stochastic dierential equations, we start with the intuitive construction to an approximation for standard Brownian motion based on a random walk that has been re-scaled in time and space.

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If we transform Brownian motion with drift by exponentiation i.e. by defining $G_t = \\exp(S_t)$ (so that the updates are *multiplicative* rather than *additive*) we get what is referred to as \n”,
“**geometric Brownian motion** (with drift) a process often used in finance to model asset prices since it cannot become negative.

用到的数据文件

SP500Daily.csv
SP500Weekly.csv

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