Self-Financing Portfolios and Lebesgue–Stieltjes Integration

Summary

The document raises a mathematical question about expressing the self-financing condition for a continuously traded portfolio when gains are defined through Lebesgue–Stieltjes integration. It contrasts this setting with discrete trading, where portfolio value changes can be tracked directly, and asks whether taking a continuous-time limit preserves an intuitive interpretation of self-financing. The question is framed around portfolio replication and the formal treatment of trading gains.

As a specific case, it considers integration against fractional Brownian motion in a pathwise framework. For sufficiently regular, Hölder-continuous integrands, the integral can be defined in the Riemann–Stieltjes sense; for less regular integrands, the question asks how to formulate the integral and impose self-financing. The document offers no answer, derivation, or empirical evidence, so it serves as a statement of an unresolved conceptual issue rather than a usable portfolio construction method. It also sets aside arbitrage considerations, which limits the financial conclusions that can be drawn.

Key ideas

• The self-financing condition links portfolio value changes to gains from trading over time.
• Discrete trading offers a direct accounting interpretation that can become less intuitive in continuous time.
• Pathwise integration against fractional Brownian motion depends on regularity conditions for the integrand.
• The document poses, but does not resolve, how to impose self-financing when a Lebesgue–Stieltjes integral is needed.

Tags

# Financial Interpretation of the Lebesgue-Stieltjes Stochastic Integral


# Financial Interpretation of the Lebesgue-Stieltjes Stochastic Integral












I have a perhaps philosophical question about the interpretation of the Lebesgue-Stieltjes integral in a financial context. In particular, the requirements of a self-financing portfolio are trivially verified with the Riemann-Stieltjes integral, but I am having a difficult time envisioning how the self-financing requirement is expressed in a continuous time Lebesgue-Stieltjes framework. A continuous model with discrete trading times is easy enough to reckon with Lebesgue-Stieltjes, but taking the limit seems to break that interpretation. Am I simply overthinking this, or are replicating portfolios rarely defined in the formal Lebesgue-Stieltjes case?

As a concrete example, suppose I am integrating with respect to a Fractional Brownian Motion $B^H_t$ as in the pathewise sense developed by Martina Zähle. For $f$ Holder continuous of degree $\lambda > 1-H$ the integral $$ \int_0^T f dB^H_t $$ can be formulated as a Riemann-Stieltjes integral. However, if $f$ does not meet this condition then the integral, if it it exists at all, must be addressed as Lesbegue-Stieltjes. How then can the self-financing condition be imposed on such a framework (ignoring arbitrage considerations)?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.