I like to interpret Ito Integral as the outcome of a gambling strategy (which fits in well with the fundamental property of the Integral: i.e. that it is forward-looking). In general, a stochastic Integral can be written as:
$$I_t:=\int_{h=0}^{h=t}Y_hdX_h=\lim_{n \to\infty}\sum_{h=0}^{n-1}Y_h\left(X_{h+1}-X_h\right)$$
Above, the limit is in probability, $X_t$ is some stochastic process (doesn't necessarily need to be a Standard Wiener Process) and $Y_t$ is a square-integrable process (obviously, doesn't need to be stochastic).
I interpret the integrator $X_t$ as the outcome of the gambling game, whilst the integrand $Y_t$ is the betting strategy (that is why the integrator is forward-looking by design: i.e. the bettor who places his bet at time $t$ is unable to see the outcome of the gambling game yet, which only gets realized at the next instance in time).
Simple illustrative example: let's suppose $H_t$ represents a coin-flip for each t (i.e. $H_t\in\left\{−1,1\right\}$ with probability 0.5, $H_0:=0$, $X_t:=\sum_{i=0}^{i=t}H_i$) and $Y_t=1$. Then a "discrete stochastic integral" could be defined as: $$I_{t=10}=\sum_{h=0}^{9}1\left(X_{h+1}-X_h\right)$$
This quantity computes the outcome of a gambling game after 10 rounds of betting, where each round the bettor bets consistently 1 unit of currency, and can either win or lose the amount he / she bets (obviously the above is a finite sum, it's just for illustrative purpose to build up the intuition).
Moving on, taking $X_t=W_t$ and $Y_t=W_t$, I interpret the Ito integral:
$$I_t:=\int_{h=0}^{h=t}W_hdW_h=\lim_{n \to\infty}\sum_{h=0}^{n-1}W_h\left(W_{h+1}-W_h\right)$$
as the outcome of a betting game, where initially the bettor bets $W_0:=0$, but each subsequent moment in time, the bettor bets the realized sum (up to that point in time) of Brownian increments $W_{h+1}−W_h$. These Brownian increments are at the same time the gambling game pay-off (so the game pays the bettor's last bet multiplied by the next Brownian increment realization).
In continuous time, the bettor constantly adjusts his or her bet to the "current" level of the Brownian motion $W_t$, which acts as the integrator: i.e. the betting game pays the realized Brownian $W_t$ at each moment in time multiplied by the bettor's bet corresponding to the last observed realization of $W_t$.
Finally, if the integrator is some stock price process $S_t$ instead of $W_t$, and $Y_t$ is the number of stocks held (could be simply a constant, deterministic quantity), then I interpret the corresponding Stochastic Integral $I_t:=\int_{h=0}^{h=t}ydS_h$ as the profit or loss of that stock portfolio over time.
(the answer above is taken from a similar answer I gave a while ago to a different question in Quant SE).
The concepts in the provided article revolve around stochastic calculus, particularly the interpretation of Ito integrals in terms of gambling strategies and their connection to stochastic processes. Here’s an overview:
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Stochastic Integral: It's represented by (It = \int{h=0}^{h=t} Y_h dX_h), which can be expressed as a limit of a sum. The limit, in probability, reflects the outcome of a betting game. The integrator, (X_t), symbolizes the game's outcome, while the integrand, (Y_t), stands for the betting strategy.
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Illustrative Example: The explanation uses a simple example involving coin flips ((H_t)) and a constant integrand ((Yt = 1)) to compute the outcome of a gambling game after a certain number of rounds ((I{t=10})). This showcases how the integral represents the result of repeated betting based on a strategy.
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Interpretation with Brownian Motion: Transitioning to Brownian motion ((W_t)), the Ito integral ((It = \int{h=0}^{h=t} W_hdW_h)) represents a betting game where bets are continually adjusted based on the realized sums of Brownian increments. Here, the game pays the bettor's last bet multiplied by the next Brownian increment.
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Continuous Betting with Brownian Motion: In continuous time, the bettor adapts bets based on the current level of Brownian motion ((W_t)), which acts as the integrator. The betting game pays the realized Brownian value at each moment multiplied by the bet placed according to the last observed (W_t).
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Stock Price as the Integrator: If the integrator becomes a stock price process ((S_t)), and (Y_t) represents the number of stocks held (could be a constant quantity), the corresponding Stochastic Integral (It = \int{h=0}^{h=t} ydS_h) reflects the profit or loss of that stock portfolio over time.
The interpretation of the Ito integral in terms of gambling strategies provides an intuitive understanding of stochastic integrals, relating them to dynamic processes and their outcomes. This approach helps in grasping the forward-looking nature of integrals and their applicability in various stochastic scenarios.
