Setting Up a Squared-Return Calculation for a Jump Price Model

Summary

The document poses a stochastic-calculus problem for a price process whose relative changes are driven by a compensated jump measure. It also specifies a state variable and a jump intensity that depends on that state. The requested quantity is the conditional expectation of the squared relative price jump over an infinitesimal time interval.

The text provides the model equations and identifies the jump measure, its compensator, and the state-dependent intensity, but it does not include a derivation or answer. A solution would need to account for the squared jump sizes under the relevant conditional intensity, while distinguishing the jump contribution from the compensated integral’s continuous compensator term. The notation and intensity specification may need clarification before calculation, including how the two coordinates of the jump measure contribute to price changes and whether the expectation is intended as an infinitesimal conditional second moment or a finite-interval quantity. It is therefore a useful problem setup, rather than a worked method or result.

Key ideas

• The price process is modeled with a compensated random measure that counts jumps.
• The jump intensity depends on a state variable that also evolves through jumps and mean reversion.
• The target is a conditional second moment of relative price changes.
• The document supplies no solution, and the role of each jump coordinate may need clarification.

Tags

# Computing squared returns given differential equation for prices


# Computing squared returns given differential equation for prices












I am looking for general advice on how to start tackling the problem below. My background in math is fairly bad when it comes to stochastic differential equations, but if you have any recommendations that will guide me to the solution please let me know.

Consider the following process for prices: \begin{align} \frac{dX_t}{X_{t-}}&=\int_{\mathbb{R}^2}(e^x-1)\tilde{\mu}(dt, dx, dy)\\ dU_t&=-\alpha U_tdt+\mu\int_{\mathbb{R}^2}[(1-\rho)x^2\mathbb{1}_{\{x<0\}}+\rho y^2]\mu(dt, dx, dy)\\ \tilde{\mu}(dt, dx, dy)&=\mu(dt, dx, dy)-dt\otimes v_t(dx,dy)\\ \frac{v_t(dx,dy)}{dx dy}&=\begin{cases} c_te^{\lambda x}\mathbb{1}{\{x<0\}}+e^{-\lambda x}\mathbb{1}{\{x>0\}}, &\text{if }y=0\\ c_te^{\lambda y},&\text{if }x=0,y<0 \end{cases}\\ c_t&=U_{t-} \end{align} where $\mu$ is an integer-valued measure counting the jumps in the price $X$ and the state variable $U$, the corresponding jump intensity is $dt\otimes v_t(dx,dy)$, and $\tilde{\mu}(dt, dx, dy)=\mu(dt,dx,dy)-dtv_t(dx,dy)$ is the associated martingale jump measure.

Question: How should I approach computing $\mathbb{E}_t\left[\left(\frac{dX_t}{X_{t-}}\right)^2\right]$?

Thank you for your help! :D

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