Differentiating an Exponentially Weighted Brownian Integral

Summary

The document asks how to find the differential and variance of a stochastic integral with an exponentially decaying kernel. The proposed method factors out the time-dependent exponential, expressing the process as a deterministic multiplier of a simpler Brownian integral.

The answer invokes the variance rule for an integral of a deterministic function against Brownian motion: integrate the squared integrand over time. It then applies standard stochastic calculus to the simpler process, whose differential is the exponential factor times the Brownian increment. These steps provide a route to the requested differential and variance, though the response does not carry out the final algebra or explicitly state the resulting expressions. The method assumes constant parameters and Brownian motion; it is a focused stochastic-calculus derivation rather than an empirical result or a complete treatment of more general kernels.

Key ideas

• Factor the exponential kernel to express the process as a deterministic multiplier of a simpler stochastic integral.
• The variance of a Brownian integral with deterministic integrand is the time integral of its squared integrand.
• The transformed stochastic integral has a differential given by its integrand times the Brownian increment.
• The response outlines the calculation but leaves the final expressions implicit.

Tags

# How to take the differential of a stochastic integral?


# How to take the differential of a stochastic integral?












Denote $$X_t = \int^t_0\sigma e^{-k(t-s)}dW_s$$ here $W_s$ is the Brownian motion, $k,\sigma$ are constants.

I want to calculate $d X_t$ and the variance $Var[X_t].$ I know how to take the derivatives of a integral with parameters, but don't know how to deal with this stochastic integral.

## Answer by Sanjay (score 6, accepted)

https://quant.stackexchange.com/a/38617

You can rewrite $X_t = e^{-kt}Z_t$ and define $Z_t:=\int_{0}^{t}e^{ks}dW_s$. There is a theory (Lemma 4.15 in Björk if you use his book) which states that $$\text{Var}\left[\int_{0}^{t}f(u)dW_s\right]=\int_{0}^{t}(f(u))^2ds$$ You can use that. furthermore, You can use Ito to compute $dX_t$. By standard stochastic calculus theory the dynamics of $Z_t $ is $dZ_t=e^{kt}dW_t$

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