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Let $0\leq \underline{\sigma} \leq \overline{\sigma}$ be two constant matrices in $\mathbb{S}^d$. Let $W$ be a Brownian motion under the measure $P_0$ and define

$$ \mathcal{P} := \{P^\sigma \colon \sigma \in L^0(F; \mathbb{S}^d) \text{ such that } 0\leq \underline{\sigma} \leq \sigma \leq \overline{\sigma} \} $$

where $P^\sigma = P_0 \circ (X^\sigma)^{-1}$ and $X^{\sigma}_t = \int_0^t \sigma(s) \mathrm{d}W_s$ $\quad$ $P_0$ almost surely and $L^0(F; \mathbb{S}^d)$ denotes the set of F measurable processes with values in $\mathbb{S}^d$. This construction is from "Backward Stochastic Differential Equations" by Jianfeng Zhang, Section 9.2.2.

Is this set $\mathcal{P}$ of non-equivalent martingale measures convex?

EDIT:

Suppose that the probability space is the classical Wiener space, i.e.,

$$ \Omega = \{\omega \in C([0,T],\mathbb{R}^d), \omega_0=0 \} $$

and $P_0$ is the Wiener measure under which $W$ is a Brownian motion.

T-at-R
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    Do you want $\mathcal{P}$ to be the set of all possible distributions over any space that carries a Brownian Motion or do you want to stick to one fixed probability space, for which you a priori don't know anything else? If you are okay with extending, then replace $\Omega$ by $\Omega\times [0,1]$, $F$ by $F\otimes \mathcal{B}([0,1])$ and replace $P_0$ by $P_0\otimes m$ with $m$being the Lesbegue measure. Associate to each $u\in [0,1]$ and $\sigma,\sigma'$ the process $u\sigma+(1-u)\sigma'$, which will still satisfy your positivity bounds and clearly, its integral has the desired distribution. – WoolierThanThou Aug 05 '20 at 12:56
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    In sum total, you of course want to replace $\Omega$ by $\Omega\times [0,1]^{\mathbb{N}}$ so that you can do the same trick with convex combinations of the new matrix distributions that arise. – WoolierThanThou Aug 05 '20 at 12:59
  • @WoolierThanThou Thank you for this construction. However, I like to stay on the classical Wiener space and edited the question accordingly. Am I right in thinking that in this case my set will not be convex? – T-at-R Aug 06 '20 at 08:32

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