Example of application of stochastic calculus to discrete probability

Question

Integrals extend the concept of sum in the continuous domain, and the usefulness of such extension with respect to the generalized concepts (i.e. discrete mathematics) can be easily illustrated with some examples, such as providing bounds on the value of a series by upper and lower bounding it with appropriate integrals.

Stochastic calculus extends the concept of sum of random variables in a similar fashion. Can we also provide a corresponding motivating example of an application of stochastic calculus in a discrete-probability setting? In other words, can we prove something about a discrete process by the means of stochastic calculus, that we could not easily prove otherwise?

Answer 1

Let me have a go, although this might not be exactly what you were after.

Stochastic integral can be intuitively interpreted as an outcome of a gambling strategy, where the integrand is the strategy and the integrator is the (random) outcome of the game being played.

Example: A gambling game involves a coin flip. If the outcome is heads the bettor wins, if it's tails, the bettor loses. In this case the "integrator" could be modeled via $X_t=\left\{-1,1\right\}$ with probability $\frac{1}{2}$. My gambling strategy as a function of the game outcomes is to bet 2 dollars every time (so that the "integrand" is a constant $2$). The outcome of this game, that I denote $I(t)$ after 10 rounds will be:

$$I(t=10):=\sum_{i=1}^{i=t}2X_i$$

Now what if I am interested in computing the outcome of investing 100 dollars into a specific stock (say the stock costs 100 dollars, so that I purchase one unit of it). In discrete space, suppose the stock can go up or down every unit of time by a certain percentage amount (where this percentage amount is represented by $\sigma$). After $t$ units of time, the stock value would be:

$$S_t=S_0+\sum_{i=1}^{i=t}\sigma S_{i-1} X_i$$

What if the percentage amount by which the stock can go up or down isn't fixed, but rather dependent on all the incoming news and events that affect the wider economy? The sum of all these incoming random events could be modeled as a normally-distributed random shock with a Brownian motion increment, improving the model:

$$S_t=S_0+\sum_{i=1}^{i=t}\sigma S_{i-1}(W_{i}-W_{i-1})$$

However, time is (arguably) continuous, and when the stock market is open, people can buy & sell stocks at any time (prices can move at tiny increments called ticks that make the prices (almost) continuous). How to model this "continuity" to get a more realistic stock price model?

How about via Ito integral:

$$S_t=S_0+\int_{h=0}^{h=t}\sigma S_h dW_h=S_0+\lim_{n\to\infty}\sum_{i=t_1}^{i=t_n}\sigma S_{i-1}( W_{i}-W_{i-1})$$

Above, the stock is now exposed to "continuous shocks" driven by the Brownian motion integrator (reflecting the continuously incoming random news and events affecting the economy). The above equation has the well-known solution (using Ito's lemma):

$$S_t=S_0e^{-0.5 \sigma^2 t + \sigma W_t}$$

The continuous model is "more realistic" and it allowed us reach the conclusion that the stock price at a specific future point in time converges to a log-normally distributed random variable, with the expected value centered on the "starting value" $S_0$. Furthermore, log-normal distribution is nice also because it has a positive domain, and stock prices cannot go negative, since zero price means bankruptcy. Also, the log-normal solution above is a Markov process, meaning that future stock prices only depend on last observed value, rather than the entire path taken to reach that value.

It is nice to get all these properties as a continuous limit of the discrete model.

Answer 2

I guess one of the mightiest tools in order to infer from the contiuous setting to the discrete setting is Skorokhod's second embedding theorem, which will tell you that given a random walk $S_{n}$ and a Brownian motion $B_{t}$ there exist stopping times $\tau_{n}$ such that $S_{n}$ and $B_{\tau_{n}}$ are identically distributed. This can be used for instance to prove the law of the iterated logarithm for random walks using the corresponding statement for Brownian motion. See e.g. https://www.math.upenn.edu/~pemantle/papers/Student-theses/Masters/Choi120424.pdf . I would expect that other asymtotic statements can be proved in a similar fashion, such as arc-sine laws.