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.

• Define $X_t = -\int_{0}^t \lambda_s \: dz_s$ and $Y_t = -\frac{1}{2} \int_0^t \lambda^2_s \: ds$ such that the original process can be rewritten as $\varepsilon^\lambda_t = e^{X_t + Y_t}$. Try to use bivariate Ito's lemma on this process wrt. X and Y.
– Pleb
Jul 23, 2022 at 19:20
• @Pleb Thank you so much. I'll try this approach. Jul 23, 2022 at 19:22