Finding the Dynamics of an Exponential Stochastic Integral

Summary

The document presents a question about deriving the dynamics of a process defined as the exponential of a stochastic integral and a quadratic-variation adjustment. The process is parameterized by an integrand and a driving process, in the setting of an exercise from a mathematical finance text.

The questioner proposes defining an auxiliary process as the stochastic integral, then expressing the original process as a product of exponentials before applying Itô’s lemma. Their proposed function includes an ordinary time integral of the integrand, while the original exponent contains an integral of its square. The document gives no accepted answer or derivation, so it does not establish that this proposed setup is correct.

The useful lesson is the general approach of identifying the exponent as a process and applying Itô’s lemma, while carefully preserving each term in that exponent. The precise dynamics depend on assumptions about the driving process and integrand, which the question does not spell out.

Key ideas

• The exercise asks for the dynamics of an exponential built from a stochastic integral and a quadratic adjustment.
• One possible starting point is to name the stochastic integral as an auxiliary process.
• Applying Itô’s lemma requires accounting for the full exponent, including its time-integral term.
• The proposed auxiliary function differs from the exponent shown in the question, and the document supplies no derivation to resolve that issue.
• The dynamics depend on properties of the driving process and integrand that are not fully specified.

Tags

# Munk (2011) exercise 3.6


# Munk (2011) exercise 3.6












I'm trying to solve the exercise in Munk (2011). The exercise reads: "Find the dynamics of the process: $\xi^{\lambda}_{t} = \exp\left\{-\int^{t}_{0} \lambda_{s} dz_{s} - \frac{1}{2}\int^{t}_{0} \lambda_{s}^{2} ds\right\}$". My issue is how do I start finding the dynamics of such a process. Usually, we have a process defined by fx $y_{t} = x_{t}w_{t}$ and then we have to find the dynamics of $y$. A hint would be very much appreciated on how to get started with such a process.

My idea was to define a process $y_{t} = \int^{t}_{0} \lambda_{s} dz_{s}$. Then we have $$\xi^{\lambda}_{t} = \exp\left\{-y_{t} - \frac{1}{2}\int^{t}_{0} \lambda_{s}^{2} ds\right\} = \exp\{-y_{t}\}\exp\left\{-\frac{1}{2}\int^{t}_{0} \lambda_{s}^{2} ds\right\}.$$

Then we can define $g(y,t) = \mathrm{e}^{y}\mathrm{e}^{\int^{t}_{0}\lambda_{s}ds}$. From here I would apply It's Lemma to find the dynamics.

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