Interpreting Itô Integrals as Self-Financing Trading Gains

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


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












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 by Jan Stuller (score 3)

https://quant.stackexchange.com/a/58988

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.