I have a question regarding the îto integral.
In the definition of the integral we basically take the limit in probability of the sum $\Sigma H(t_i)\cdot(B(t_{i+1})-B(t_i))$ for suitable $H$ and a Brownian motion $B$ and call that the îto integral. The reason for us not demanding pathwise convergence being that such a limit might not exist.
Now one of the main uses of this integral is to model the payoff of some continous-time trading strategy consisting of holding an amount $H_t$ of some stock at time t, the stock price being modeled by $B_t$.
Now the question is: If the above is one of its main uses, how does this definition(only p-convergence) of the integral make sense? I mean it feels like we really would want a.s. convergence, it being hard interpreting a trading strategy which does not converge a.s. but in probability.
Much appreciated if anyone can help me.