Deriving Present Values with Stochastic Growth and Markov Switching
Summary
The document poses a continuous-time valuation problem for a cash flow tied to two stochastic state variables and a pricing kernel. It specifies correlated Brownian drivers, state-dependent growth, and a two-state continuous-time Markov process for the cash-flow multiplier. A proposed infinite-horizon present-value expression reduces the valuation to the current state-variable product multiplied by a coefficient involving discounting and Markov transition rates.
The question asks how to derive that coefficient and how to extend the calculation to finite horizons, including an exponential decay term. However, the source provides the model setup and proposed formulas only; it contains no derivation, worked steps, numerical evidence, or resolution of the questions. The expressions therefore cannot be independently checked from the document alone, and their applicability depends on the model assumptions and parameter conventions used.
Key ideas
- The valuation integrates discounted cash flows driven by stochastic state variables and a pricing kernel.
- The cash-flow multiplier follows a two-state continuous-time Markov process.
- The proposed infinite-horizon value is proportional to the current state-variable product.
- The source asks for finite-horizon extensions but does not supply their derivation or answers.
Tags
Full text
# How to derive the the following integral in continuous time?
# How to derive the the following integral in continuous time?
Here is a standard exogenous system in continuous time, where $\pi_t$ is the pricing kernel \begin{align*} \frac{dx_t}{x_t} &= \mu_x dt + \sigma_x dB_{xt}\\ \frac{dz_t}{z_t} &= \mu_z dt + \sigma_z dB_{zt}\\ \frac{d\pi_t}{\pi_t} &= -r_f dt - \gamma_x dB_{xt}- \gamma_z dB_{zt} \end{align*}
Assume $\lambda_{ft}=\lambda_f \lambda_t =\lambda_f \lambda_H $ at time $t$. I know the following solution. \begin{align*} PVGO_{ft} &= \mathbb{E}_t \bigg[ \int_{t}^{\infty} \frac{\pi_s}{\pi_t} \:\lambda_{fs} \:x_s z_s^{\frac{\alpha}{1-\alpha}} \: ds\bigg] = x_t z_t^{\frac{\alpha}{1-\alpha}} G \\ G &= \lambda_f \bigg[ \rho^{-1} + \frac{\mu_L}{\mu_L+\mu_H} (\rho+\mu_H+\mu_L)^{-1} \bigg] \\ \rho &= r+ \gamma_x \sigma_x - \mu_x - \frac{\alpha}{1-\alpha} \bigg(\mu_x-\gamma_z \sigma_z - \frac{1}{2}\sigma^2_z\bigg) - \frac{1}{2}\bigg( \frac{\alpha}{1-\alpha}\bigg)^2\sigma^2_z \end{align*} where $\lambda_t$ follows a two-state continuous-time Markov process $\lambda_t \in \{\lambda_L, \lambda_H\}$ with instant transition probabilities $\mu_L$ and $\mu_H$ and $\mathbb{E}[\lambda_t]=1$.
How is the solution derived?
How to derive the following integrals? Again assume $\lambda_{ft}=\lambda_f \lambda_H $ at time $t$ \begin{align*} \mathbb{E}_t \bigg[ \int_{t}^{T} \frac{\pi_s}{\pi_t} \:\lambda_{fs} \:x_s z_s^{\frac{\alpha}{1-\alpha}}\: ds\bigg] \end{align*} and \begin{align*} \mathbb{E}_t \bigg[ \int_{t}^{T} \frac{\pi_s}{\pi_t} \:\lambda_{fs} \:x_s z_s^{\frac{\alpha}{1-\alpha}}\: e^{-\kappa(T-s)} \: ds\bigg] \end{align*}
I'd appreciate if you can provide a detailed derivation!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.