Suppose $S_t$ satisfies the equation $$dS_t=S_t(a(X_t)dt+b(X_t)dW_t)$$
Where $X_t$ is a Markov process.Then is the joint process $(S_t,X_t)$ Markov?
2026-02-23 15:12:31.1771859551
Showing a joint process is markov
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The joint process $\boldsymbol{Y}_t=(S_t,X_t)$ is Markov because it satisfies the two dimensional SDE $d\boldsymbol{Y}_t=\boldsymbol{\alpha}(\boldsymbol{Y}_t)\,dt+\boldsymbol{\beta}(\boldsymbol{Y}_t)\,d\boldsymbol{W}_t$ with $\boldsymbol{\alpha}(S_t,X_t)=(S_ta(X_t),\alpha_2(X_t))$ and $\boldsymbol{\beta}(S_t,X_t)=(S_tb(X_t),\beta_2(X_t))\,.$ The extra functions $\alpha_2,\beta_2$ exist because $X_t$ was Markov.