Stochastic Differential Equation after a Change in the Time Parameter

Question

How does the stochastic differential equation for a stochastic process change under a change in the time parameter? For example, consider the Bessel Squared Process $$ dR_t = m \, dt + 2 \sqrt{R_t} \, d\beta_t$$ where $\beta_t$ is some standard Brownian motion. How would one find the stochastic differential equation to, for example, the new process $Z_t := R_{1 - e^{-t}}$.

Answer 1

You have the following lemma from Kei Kobayashi's paper (which is in fact due to Jacod (1979)) :

Lemma 2.3 : Let $Z$ be an $(\mathcal F_t)$-semimartingale which is in synchronization with an $(\mathcal F_t)$-time change $(T_t)$. If $H\in L(Z,\mathcal F_t)$, then $H_{T(t-)}\in L(Z\circ T,\mathcal G_t)$, where $\mathcal G_t :=\mathcal F_{T_t}$. Moreover, with probability one, for all $t\ge 0$ $$\int_0^{T_t} H_s dZ_s = \int_0^t H_{T(s-)}dZ_{T_s} $$

You can check pages 3 and 4 of the paper to get the precise definitions of all the terms introduced, but in practice, this formula will hold if $Z$ is an $\mathcal F_t$-Brownian Motion, $T_t$ is a deterministic and continuous function of $t$ and $H$ is an $\mathcal F_t$-predictable process for which $\int_0^t H_sdZ_s $ is well defined.

Applying the above lemma to the Bessel Squared Process $R_t$ and time change $T(t):=1-e^{-t}$ yields $$\begin{align}Z_t := R_{T_t}&= Z_0 + mT_t + \int_0^{T_t} 2\sqrt{R_s}d\beta_s\\ &=Z_0 + mT_t + \int_0^{t} 2\sqrt{R_{T(s)}}d\beta_{T(s)}\\ &=Z_0 + mT_t + \int_0^{t} 2\sqrt{Z_s}d\beta_{T(s)}\\ &=Z_0 + \int_0^{t}m T'(s)ds + \int_0^{t} 2\sqrt{Z_s}d\beta_{T(s)}\end{align} $$ Which, you could write in differential form as $$dZ_t = m\cdot T'(t)dt + 2\sqrt{Z_t}d\beta_{T(t)}\tag1$$

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Addendum : If, like me, it bothers you to integrate against $\beta_{T(t)}$, I found in this post the following theorem from Oksendal's book. Here I give a version limited to the 1 dimensional case, and where the time change $T_t$ is deterministic. I also skip the technical conditions (measurability, adaptedness...) :

Theorem 8.5.7 (simplified version) : If the time change $T_t$ is a continuously differentiable and non-decreasing function of $t$ such that $T_0=0$, and if the process $H$ is continuous and bounded then $$\int_0^{T_t} H_s\,\mathbb{d}\beta_s = \int_0^{t} H_{T(s)}\sqrt{T'(s)}\,\mathbb{d}\beta_s $$

Applying this theorem yields, by proceeding similarly as above, the following (arguably nicer) SDE for the time-changed Bessel Squared Process : $$dZ_t = m\cdot T'(t)dt + 2\sqrt{T'(t)\cdot Z_t}\ d\beta_t\tag2 $$