I am currently in the process of deciphering the notes written by an instructor. I missed the class.

He writes:

Continuous time trading: $S(T) = W(T)$, "Bachelier Model". $\phi(t)$ denotes number of shares held at time t. Approximation is $\phi^{(n)}(t) = \phi(t_i), t_i < t < t_{i+1}, t_i = \frac{i}{n}T$. Gain from strategy $\phi^{(n)}$ is $\sum_{i=1}^n \phi(t_{i-1})[W_{t_i} > - W_{t_i -1}] = \int_0^T \phi^{(n)}(s) dW(s).$

I don't understand what $n$ or $\phi^{(n)}$ is. Is it a derivative?


n is the number of time steps used in the discrete time solution $\phi^{(n)}$ to approximate the continuous time solution $\phi$ (the time period $T$ is being broken up into n steps). The higher the n the better the approximation. In the expression $\phi^{(n)}$ n is a super-script, not an exponent nor a derivative.

I have a feeling that later on you are going to be told that $n \rightarrow \infty $ ...


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