# 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? Jul 16, 2019 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...! Jul 16, 2019 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 $$\int_t^T \mathbb{E}[H_uS_u]^2\mathrm{d}u<\infty$$.