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If $W_t$ is standard Brownian motion, what is meant by $\int_0^T W_t dW_t$ in finance?

Furthermore, what then is the meaning of $\int_0^T W_t \ln(W_t) dW_t$?

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    $\begingroup$ Are you sure you mean $\int_0^T W_TdW_t$ and not $\int_0^T W_tdW_t$? Also, what do you mean with ''what do these integrals mean in finance?'' These integrals have a very clear math definition and sometimes appear in option pricing. In general, $\int \alpha_t dS_t$ is the value of a portfolio (trading strategy) with holds $\alpha_t$ many shares of $S_t$. $\endgroup$
    – Alex
    Oct 25, 2020 at 22:46
  • $\begingroup$ perhaps you can tell the other poster to correct $T$ to $t$ as well quant.stackexchange.com/questions/58923/… $\endgroup$
    – develarist
    Oct 25, 2020 at 22:52
  • $\begingroup$ Hi @develarist: It's a good question, I think intuition is always important in finance. I had a crack at the intuition behind Ito Integrals of the type $\int_0^TW_t\,dW_t$ in this answer here. Also a good answer to your question is here: MagicIsInTheChain shows that the integral of the type $\int_0^T W_TdW_t$ can be rewritten as $2\int_0^TW_t\,dW_t$, which then links back to my answer. $\endgroup$ Oct 26, 2020 at 10:38
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    $\begingroup$ The reason for asking is $W_T$ can take negative values, so wanted to see the context. Have seen $\int_0^T{W_T f(W_T) dW_t}$ but not for f=ln. $\endgroup$ Oct 28, 2020 at 17:46
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    $\begingroup$ @Magicisinthechain: out of curiosity, how would one interpret the integral $\int_0^T{W_T f(W_T) dW_t}$ (or $\int_0^T{W_t f(W_t) dW_t}$) ? Where would it arise and what would it represent? $\endgroup$ Nov 8, 2020 at 10:51

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I am sure there will be more thorough answers provided by others, but let me have a quick go at the first part: "what is meant by $\int_0^T W_t dW_t$ in finance?".

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: $$I_{t=10}=\sum_{h=0}^{t}1\left(X_{h+1}-X_h\right)$$

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 betted (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.

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  • $\begingroup$ Any take on the logarithmic part? $\endgroup$
    – develarist
    Nov 8, 2020 at 8:23
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    $\begingroup$ @develarist: as per some of the comments above, log of a negative number is undefined. Brownian motion can take on negative values, so log of a Brownian motion is not a well defined quantity, I am not sure how to interpret it. $\endgroup$ Nov 8, 2020 at 8:35

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