Pathwise Interpretation of Stochastic Integrals in Continuous-Time Trading
Summary
This discussion examines a claim that stochastic integrals lack a pathwise meaning, making portfolio gains hard to interpret for a realized asset-price trajectory. The questioner suggests that knowing an integral’s probability distribution might make its value definable for each state of the world, and asks where that reasoning fails.
The accepted response distinguishes a distributional description from the value of an integral along a particular path. It reads the cited work as developing functional Itô calculus for non-anticipative functionals of an observed path, such as a running average. This framework extends classical Itô calculus to those path-dependent quantities; the response suggests the passage may be describing that extension rather than identifying a general defect in classical stochastic integration.
The exchange is conceptual rather than a derivation. It gives no detailed conditions for constructing pathwise integrals, and its interpretation of the cited paper is explicitly tentative.
Key ideas
- Knowing the distribution of a stochastic integral does not by itself specify its value along a realized price path.
- Functional Itô calculus handles non-anticipative functionals that depend on the history of a process.
- A running average of a process is an example of a path-dependent functional.
- The response interprets the cited passage as describing an extension of classical Itô calculus, not a general failure of it.
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# Advantages of pathwise calculus over stochastic calculus in continuous self-financing trading models
# Advantages of pathwise calculus over stochastic calculus in continuous self-financing trading models
I am new to stochastic calculus but the statement below confuses me:
> Beside the issue of the impossible consensus on a probability measure, the representation of the gain from trading lacks a pathwise meaning: while being a limit in probability of approximating Riemann sums, the stochastic integral does not have a well-defined value on a given ‘state of the world’. This causes a gap in the use of probabilistic models, in the sense that it is not possible to compute the gain of a trading portfolio given the realized trajectory of the underlying asset price, which constitutes a drawback in terms of interpretation.
Source: Riga C., (2015), Pathwise functional calculus and applications to continuous-time finance, Page 3
since the computation of a stochastic integral entails essentially computing a distribution, why can we then not define it pathwise? What am I missing? e.g. $\int_{0}^{t} X_{s} dB_{s} \sim \mathcal{N}(0, \sigma^{2})$ surely it is then simply pathwise defined on any state $\omega \in \Omega$ as $\mathcal{N}(0,\sigma^{2})(\omega)$.
Can anyone describe my misunderstanding, and help me?
## Answer by ir7 (score 2, accepted)
https://quant.stackexchange.com/a/51672
Breezing through the referenced paper, the point of it seems to be to develop Ito calculus for non-anticipative functionals $$ F(t, X_t),$$ where $X_t := \left\{X(u)\mid 0\leq u\leq t \right\}$ and $\left(X(u)\right)_{u\geq 0}$ is a stochastic process. For example, for $$ F(t,X_t)=t^{-1}\int_0^t X(u)du.$$ I think that, in that introductive paragraph, the author simply mentions that classical Ito calculus (formula) didn't have 'pathwise interpretability' until the machinery behind functional Ito calculus arrived, which essentially generalizes it, not that classical Ito calculus had any shortcomings.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.