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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?

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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.

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  • $\begingroup$ 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? $\endgroup$ – Stan Shunpike Jul 16 at 22:04
  • $\begingroup$ 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...! $\endgroup$ – Magic is in the chain Jul 16 at 22:09
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$H$ 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 $\int_t^T \mathbb{E}[H_uS_u]^2\mathrm{d}u<\infty$.

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