Moment Generating Functions for Brownian Integrals with Time-Dependent Kernels
Summary
The document examines the moment generating function of a stochastic process formed by integrating Brownian increments against a response kernel whose argument depends on both integration time and the current time. Applying Itô’s lemma to the exponential of the process yields an expectation equation with a mixed expectation involving the process and its exponential. Under a simple exponential-kernel assumption, this becomes an equation involving a higher moment, creating a closure problem.
The author suggests expanding the exponential as a power series as a direct route to the MGF. The addendum also states that closure follows when the kernel obeys a first-order linear homogeneous equation, and proposes a normalized-kernel construction for more general response functions. The text is exploratory: it presents a derivation and proposed resolution but no worked verification, assumptions on integrability, or broader proof. Its main relevance is stochastic-process analysis rather than a direct trading strategy.
Key ideas
- The process integrates Brownian increments against a time-dependent response kernel.
- Applying Itô’s lemma produces an expectation equation involving a mixed moment.
- A simple kernel assumption can leave the equation unclosed.
- A power-series expansion of the exponential is proposed for calculating the MGF.
- A normalized response kernel is suggested to obtain closure for more general functions.
Tags
Full text
# MGF of Generalised Itô Integral
# MGF of Generalised Itô Integral
The following derivation produces a moment closure problem - I would appreciate any insight. It may seem trivial at first glance, but the key aspect is the integrand dependence on $t$.
Consider $W_t$ to be a typical Brownian motion. Introduce the process
$$ X_t = \int\limits_0^t dW_s G_{s-t} $$
where $G_0=1$, and $\mathbb{E}[G_t]=G_t$, which implies $dG_t \sim dt$. The moments of $X_t$ (up to a reasonable order) can be found explicitly (EDIT this is only true under a linear homogeneous response $G_t$). To determine the MGF, define
$$ Y_t = e^{\lambda X_t} $$
such that the MGF of $X_t$ is $M_t^{\lambda} = \mathbb{E}[Y_t]$. Using Itô's lemma
$$ dY_t = \left[ \lambda dW_t+\lambda\int\limits_0^t dW_s dG_{s-t}+\frac{\lambda^2}{2}dt\right] Y_t $$
Taking the expectation gives
$$ d\mathbb{E}[Y_t] = \lambda\int\limits_0^t \mathbb{E}[dW_s Y_t] dG_{s-t}+\frac{\lambda^2}{2}dt\mathbb{E}[Y_t] $$
Independence of increment does not appear to admit further simplification. Under the simplifying assumption, $dG_t = G_t dt$, this becomes
$$ d\mathbb{E}[Y_t] = -\lambda d t\mathbb{E}[X_t Y_t]+\frac{\lambda^2}{2}dt\mathbb{E}[Y_t] $$
which presents a moment closure problem. I realise that the MGF of $e^{\lambda G_t X_t}$ can be found explicitly, and that arguments of the $X_t$ distribution can be made accordingly, but is it possible to calculate $M_t^{\lambda}$ directly using only Itô's lemma?
Addendum: Having examined the process again, it appears that applying the power series of the exponential is the most direct way of calculating the MGF.
Furthermore, the assertion $dG_t = \alpha dt G_t$ is required in order to obtain closure. For more general response (Green's) functions, the process must be constructed as
$$ X_t = \int\limits_0^{t} dW_s \frac{G_s}{G_t} $$
where closure can be obtained even if $G_t$ does not satisfy a first-order linear homogeneous differential.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.