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On fluctuations of a multivariate random walk with some applications to stock options trading and hedging. (English) Zbl 1133.91416

Summary: We continue our studies of multivariate marked recurrent processes (initiated in the 1990s) which we refer to as a multivariate random walk. It is formed with a delayed renewal process \(\mathcal T = \{\tau _0 , \tau_1, \cdots\}\), along with marks representing a multidimensional recurrent process \(\mathcal S\). In our interpretation, this recurrent process \(\mathcal S = \{\mathcal S _0 , \mathcal S_1, \cdots\}\) is observed by \(\mathcal T\). Of the entire multitude of components of \(\mathcal S\), some are referred to as active and the rest as passive components. The first passage time (or exit time) \(\tau _\rho \) of \(\mathcal S\) is defined as the first epoch of \(\mathcal T\) when at least one active entry of \(\mathcal S\) exits its preassigned critical set. At this moment, the value of an active entry that exits the set (called excess value) will be “registered” as well as the rest of the active and passive components \(\mathcal S_\rho \) at \(\tau _\rho \). We will form a joint transformation of these values, along with \(\tau _{\rho - 1}\) (pre-exit time) and \(\mathcal S _{\rho -1}\) and find its explicit form. The analysis of the functional is restricted to only four active entries of \(\mathcal S\), which is an upgrade form of a past accomplishment for a maximum three active entries. The latter made the authors conjecture that a very compact formula for the above functional, holding for one, two, and three components and being identical in terms of some previously introduced index operators, must hold for arbitrary many active components. The proof of this conjecture is still out of reach, but the present case of four (using quite tedious calculations) is a confirmation that the conjecture must be true.
The problem in essence stems from a stock option trading with multiple portfolios, and this is where most of our examples are collected from. The results can be applied to many other instances of stock market as well as computer network security and other areas of science and technology mentioned in the paper.

MSC:

91B28 Finance etc. (MSC2000)
60K05 Renewal theory
60K10 Applications of renewal theory (reliability, demand theory, etc.)
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