If $W_t$ is standard Brownian motion, what is $\int_0^T W_t \ln(W_t) dW_t$?

Question

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 $\int_0^T W_t \ln(W_t) dW_t$?

Answer 1

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. In general, Ito Integral can be written as:

$$I_t:=\int_{h=0}^{h=t}f(Y_h)dX_h=\lim_{n \to\infty}\sum_{h=0}^{n-1}f(Y_h)\left(X_{h+1}-X_h\right)$$

Above, $X_t$ is a generic stochastic process (doesn't necessarily have to be $W_t$), whilst $Y_t$ is a square-integrable process (doesn't have to be stochastic). $Y_t$ has to be adapted to the filtration generated by $X_t$. $f()$ is some well-behaved function that still makes $f(Y_t)$ square integrable.

I interpret the integrator $X_t$ as the outcome of the gambling game, whilst the integrand $f(Y_t)$ is the betting strategy.

Illustrative example: let's suppose $X_h$ represents a coinflip for each $h$ (i.e. $X_h\epsilon ${$-1,1$} with probability $0.5$), $Y_h=1$ and $f()=2$. Then a discrete Stochastic integral (finite sum, strictly speaking not an Ito integral) could be defined as: $I_{t=10}=\sum_{h=0}^{9}2\left(X_{h+1}-X_h\right)$. 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 twice what he or she bets.

Moving on, taking $X_t=W_t$, $Y_t=W_t$ and $f()=1$, 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 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 $f(Y_t)$ is the number of stocks held (could be simply a constant, deterministic quantity), then I interpret the Ito Integral as the profit or loss of that stock portfolio over time.