The Malliavin derivative of a stochastic integral with an $F$ -predictable integrand is an important topic in Malliavin calculus. A stochastic integral of the form

$Y(t) = \int_0^t \phi(s, X) \, dW(s)$

involves an integrand $\phi(s, X)$ that is predictable with respect to the filtration $\mathcal{F}_s$ (i.e., \( F \)-predictable), and $W(t)$ is a Brownian motion or a general Wiener process.

1. Setting the Stage

Consider the stochastic integral $Y(t)$ defined as:

$Y(t) = \int_0^t \phi(s, X) \, dW(s),$

where:

• $\phi(s, X)$ is an adapted, $F$ -predictable process, which could depend on the time $s$ and a random process $X$ (e.g., another stochastic process).
• $W(t)$ is a Wiener process (or Brownian motion) adapted to the filtration $\mathcal{F}_t$ .

We want to compute the Malliavin derivative of $Y(t)$ , which involves differentiating the stochastic integral with respect to the underlying Brownian motion $W(t)$ .

2. Malliavin Derivative of a Stochastic Integral

In the classical case of a stochastic integral where the integrand is a real-valued adapted process, the Malliavin derivative of $Y(t)$ is given by:

$D_u Y(t) = \int_0^t \phi(s, X) \mathbf{1}_{[0,u]}(s) \, ds.$

Here:

• $\mathbf{1}_{[0,u]}(s)$ is the indicator function, which equals 1 if $s \leq u$ and 0 otherwise.
• $D_u Y(t)$ represents the Malliavin derivative of $Y(t)$ at time $u$ , which quantifies how the integral $Y(t)$ responds to changes in the path of the Brownian motion at time $u$ .

Thus, the Malliavin derivative of the stochastic integral is itself an integral with the integrand modulated by the indicator function $\mathbf{1}_{[0,u]}(s)$ , which captures the impact of the Brownian motion's path up to time $u$ .

3. Interpretation of the Malliavin Derivative

To understand the result, we can think of the stochastic integral $Y(t)$ as representing the cumulative effect of the process $\phi(s, X)$ driven by the Brownian motion up to time $t$ . The Malliavin derivative $D_u Y(t)$ captures how sensitive $Y(t)$ is to changes in the Brownian path at time $u$ . The indicator function $\mathbf{1}_{[0,u]}(s)$ ensures that only the part of the integral up to time $u$ influences the derivative.

4. Extension to Hilbert Space-Valued Processes

If $X$ is Hilbert space-valued, the Malliavin derivative can be generalized to handle vector-valued integrands. The computation becomes more complex but follows a similar structure. If $X(t)$ is a random variable in a Hilbert space $H$ , then the Malliavin derivative involves Fréchet derivatives in the Hilbert space, which may require the use of duality arguments and more sophisticated integration techniques.

For a Hilbert space-valued process $X(t)$ , the derivative would be:

$D_u Y(t) = \int_0^t \phi(s, X) \otimes \mathbf{1}_{[0,u]}(s) \, ds.$

Here, the integral is taken over a tensor product space, reflecting the more complex structure of the Hilbert space-valued processes.

5. Technical Details

In practice, the predictability of $\phi(s, X)$ with respect to the filtration $F$ (or $\mathcal{F}_s$ ) ensures that the integrand is well-behaved for the stochastic integral, which is essential for applying Malliavin calculus. The Malliavin derivative works in conjunction with Itô isometry and integration by parts techniques in stochastic calculus to handle these types of integrals.