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I get curious after reading this post on maximum of two Brownian motion: Process properties of the maximum of two independent linear Brownian motions.

It occurs that for two stochastic processes $A=(A_t)_{t\ge0},B=(B_t)_{t\ge0}.$ If we define $$X=(X_t)=\max\{A_t,B_t\},$$ for $A,B$ being independent linear Brownian motions, then $X$ is a submartingale, and also a semimartingale. By Bitchteler-Dellacherie theorem, an adapted, càdlàg process $X$ is semimartingale if it can be written as $X=M+Y,$ where $M$ is a local martingale and $Y$ is a finite variation process.

My question is that $A,B$ are semimartingale, will $X_t=\max\{A_t,B_t\}$ also be semimartingale? (My intuition tells that it is unlikely, but I don't know where to start.)

John He
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    $\max(A_t,B_t)=\frac{A_t+B_t+\vert A_t-B_t\vert}{2}$. Is is true that the absolute value preserves the semimartingale property? – Will Jul 06 '24 at 10:12
  • Thank you for your hint, I find that by Meyer-Tanaka formula, we have for $X_t$ being a semimartingale with continuous path, $|X_t|=|X_0|+\int_{0^+}^{t}sign(X_s)dX_s+L_t^0,$ where $L$ is the local time at $0$ of $X.$ So absolute value doesn't perserve semimartingale property. – John He Jul 06 '24 at 19:35

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