Alexander Cherny, Albert Shiryaev
In the mathematical finance the capital of a self-financing strategy is defined as a stochastic integral. If the time horizon is infinite, the capital of a self-financing strategy is specified as the limit almost surely of the capital at finite times.
We introduce the notion of the improper stochastic integral and propose to use it for defining the terminal capital of a self-finencing strategy. (The existence of an improper stochastic integral is more restrictive than the existence of the limit almost surely of the stochastic integrals up to finite times.)
It has been found that the First and the Second Fundamental Theorems of Asset Pricing remain true if one accepts this new definition of the terminal capital.