money we have in total including what our stock is worth, at any given moment. The prices of stocks and other traded financial assets can be modeled by stochastic processes such as Brownian motion or, more often, geometric Brownian motion (see Black–Scholes). Then, the Itô stochastic integral represents the payoff of a continuous-time trading strategy consisting of holding an amount Ht of the stock at time t. In this situation, the condition that H is adapted corresponds to the necessary restriction that the trading strategy can only make use of the available information at any time. This prevents the possibility of unlimited gains through clairvoyance: buying the stock just before each uptick in the market and selling before each downtick. Similarly, the condition that H is adapted implies that the stochastic integral will not diverge when calculated as a limit of Riemann sums . Notation The process Y defined before as is itself a stochastic process with time parameter t, which is also sometimes written as Y = H · X . Alternatively, the integral is often written in differential form dY = H dX, which is equivalent to Y − Y0 = H · X. As Itô calculus is concerned with continuous-time stochastic processes, it is assumed that an underlying filtered probability space is given The σ-algebra Ft represents the information available up until time t, and a process X is adapted if Xt is Ft-measurable. A Brownian motion B is understood to be an Ft-Brownian motion, which is just a standard Brownian motion with the properties that Bt is Ft-measurable and that Bt+s − Bt is independent of Ft for all s,t ≥ 0 . Integration with respect to Brownian motion The Itô integral can be defined in a manner similar to the Riemann–Stieltjes integral, that is as a limit in probability of Riemann sums; such a limit does not necessarily exist pathwise. Suppose that B is a Wiener process (Brownian motion) and that H is a right-continuous (càdlàg), adapted and locally bounded process. If is a sequence of partitions of [0, t] with mesh going to zero, then the Itô integral of H with respect to B up to time t is a random variable It can be shown that this limit converges in probability. For some applications, such as martingale representation theorems and local times, the integral is needed for processes that are not continuous. The