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Modeling Diffusions That Reset After State-Dependent Jumps

Article Quant Q&A · Author: sayWhat
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Summary

The document formulates a stochastic model that begins in one diffusion regime and switches to another after a jump. The jump intensity depends on the current modeled state. After the first jump, a second process evolves with drift and volatility depending on its value and on elapsed time since the most recent jump; each subsequent jump resets that elapsed time and restarts the second process from an initial value. The setup also tracks whether a jump has occurred and assumes independent Brownian drivers for the two diffusion components.

The author asks how to solve the system to obtain an expected value and a European payoff, and whether Feynman–Kac or a stochastic-clock approach applies. The document supplies a model specification and a numerical-analysis question, but no answer, derivation, boundary conditions, or valuation results. It is useful as an example of a state-dependent jump process with regenerative reset dynamics, while the appropriate solution method and any existence or regularity requirements remain unresolved.

Key ideas

  • The model switches from an initial diffusion to a replacement process when the first jump occurs.
  • Jump intensity depends on the current state of the modeled variable.
  • Each jump resets the replacement process and the clock measuring time since the last jump.
  • The document raises Feynman–Kac and stochastic-clock methods as possible tools but does not resolve which method applies.

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Full text
# jump-resetted diffusion process


# jump-resetted diffusion process












I'm working on a model in which there are two processes, $H$ and $L$, and the final variable to model starts as $H$ and then whenever a jump occurs, an instance of the $L$ processes starts and replaces the previous process. The $L$ processes can be reset themselves by the jumps to become a new $L$ process. Both $H$ and $L$ have processes with local drift and coefficients, which we can assume to be $\mu_{H/L}(x,t) \in \mathcal{C}^{\infty}(\mathbb{R}^2)$ and $\sigma_{H/L}(x,t) \in \mathcal{C}^{\infty}(\mathbb{R}^2)$ . I wrote down the set of SDE's for the model. I need advice on how to solve them to evaluate the expected value and an european payoff.

Is Feynman Kac applicable here? Notice that $\tau$ is reset to zero whenever a jump occurs, so a subordinator / stochastic clock method might be useful.

The final variable is $x_{B,t}$ . $N_t$ is a counting process, whose jump intensity $\lambda$ is a local function of $x_{B,t}$ $Y_t$ is an indicator variable denoting whether the first jump has occurred or not. $\tau$ is the time that has passed since the last jump.

$E\left[dN_t\right] = \lambda(x_{B,t}) \,\,\, ; \,\,\, \lambda \in \mathcal{C}^{\infty}(\mathbb{R}) $ $dY_t= (1-Y_t) \cdot dN_t \,\,\, ; \,\,\, Y_0= 0$ $dx_{H,t} = \mu_{H}(x_{H,t},t)\cdot dt + \sigma_{H}(x_{H,t},t) \cdot dW_{1,t}$ $d\tau = dt + dN_{t} \cdot (-\tau) \,\,\, ; \,\,\, \tau(0) = 0 $ $dx_{LR,t} = Y_t \cdot \left[ dN_{t} \cdot (x_{L,0}-x_{LR,t}) + (1-dN_{t}) \cdot \mu_{L}(x_{LR,t},\tau) \cdot dt +\sigma_{L}(x_{LR,t},\tau) \cdot dW_{2,t} \right]\\ dx_{B,t} = (1-Y_t) \cdot dx_{H,t} +dY_t \cdot (x_{L,0}-x_{B,t}) +d x_{LR,t}$ $E\left[dW_{1,t} \cdot dW_{2,t}\right]=0$

Thanks

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