Why Trading Holdings Are Stochastic in Itô Portfolio Models
Summary
The document explains why the amount of stock held in an Itô integral can itself be modeled as a stochastic process. A trading position may change in response to observed price movements or other information, so its future path is uncertain even though the trader chooses the position. The holding process represents a trading strategy that reacts to the evolving market rather than a fixed quantity held throughout.
The key admissibility condition described is that holdings must be adapted to the available information: at each time, the position can depend on what has already been observed, but not on future Brownian movements. In discrete time, the cited response describes the position as predictable, meaning the decision must be based on information available before the relevant step. The exchange also mentions self-financing and an integrability condition for admissibility, without deriving them. It gives conceptual intuition, but does not develop a formal proof or specify all assumptions required for a mathematically valid strategy.
Key ideas
- A trading position can be random because it responds to random market information over time.
- An admissible holding process must be adapted to information available when the position is chosen.
- A strategy that uses future price movements would rely on information unavailable at decision time.
- Discrete-time strategies are commonly expressed using predictable holdings based on prior information.
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# For Ito Integrals with respect to a Brownian motion, why would the amount of stock held be a stochastic process?
# 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?
## Answer by Magic is in the chain (score 3)
https://quant.stackexchange.com/a/46667
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.
## Answer by Kevin (score 1)
https://quant.stackexchange.com/a/46668
$H$ 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 $\int_t^T \mathbb{E}[H_uS_u]^2\mathrm{d}u<\infty$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.