For Ito Integrals with respect to a Brownian motion, why would the amount of stock held be a stochastic process?

Suppose that $$B$$ is a Wiener process and suppose $$H$$ is a right-continuous, adapted, and locally bounded process. Suppose

$$\int_0^t H dB$$ is the Ito integral of $$H$$ with respect to the Wiener process.

Now, suppose $$B$$ represents a stock price and $$H$$ represents the amount of the stock held.

Intuitively, if the stock follows a Wiener process, it makes sense to treat it using a stochastic process

However, I don't understand why $$H$$ is a stochastic process. Suppose I'm a trading firm. Why would the amount I hold be random?

It could depend on the brownian - e.g., could be a function of B, $$H(B)$$. What it means is you can change your holding over time depending on how the Brownian/randomness evolves, but for Ito's definition, H is supposed to be kinda non-anticipating, roughly speaking H cannot depend on the next move as you cannot predict the next change in the Brownian when choosing how much to invest.
• I can imagine some function $H(B)$ which transforms the Wiener process $B_t$ in such a way to select maximize the expected value of my portfolio. But then shouldn't the fact $H$ is a function of $B$ be explicit? Or are we just assuming the trader's possible holdings be an adapted process that uses any probability distribution? And that might include some pretty terrible choices in how much to hold? – Stan Shunpike Jul 16 '19 at 22:04
• It should certainly be explicit. H can be a deterministic function as well, say $H(t)$, or function of both t and B $H(t,B_t)$ so always good to be clear but you know...! – Magic is in the chain Jul 16 '19 at 22:09
$$H$$ is in general random. The position of a trading firm into a stock is clearly random in terms of it being dependent of the realisation of the stock price. If a firm is not invested in a stock but changes its mind because it keeps increasing, then they may alter their opinion and begin investing in the asset. So, a trading strategy depends on the random nature of the traded stocks.
However, and this is key, the process $$(H_t)$$ needs to be adapted, i.e. at time $$t$$ you need to know how much stocks you hold based on the information available at time $$t$$, this is denoted by $$\mathcal{F}_t$$. Thus, only $$(\mathcal{F}_t)$$-adapted processes qualify as trading strategies. If $$H_t$$ would depend on, for instance, $$\mathcal{F}_{t+1}$$, then you would use future information to make today's decision and this is surely not a reasonable model setup.
In a discrete time model, $$(H_t)$$ needs to be previsible (aka predictable), i.e. you need to know in advance you much you want to invest in an asset. This means you need to know $$H_t$$ based on the information available at the previous time step, $$\mathcal{F}_{t-1}$$.
For a trading strategy (aka portfolio) to be admissible, one needs it to be adapted, self-financing and $$\int_t^T \mathbb{E}[H_uS_u]^2\mathrm{d}u<\infty$$.