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Suppose that there are multiple martingale measures $Q_1$ and $Q_2$ that attain the minimal variance. Then the convex combination $Q_* := \frac{1}{2}Q_1 + \frac{1}{2}Q_2$ is also a martingale measure. Due to the strict convexity of $f(x) = x^2$, it can be shown that  E_P \left[\frac{dQ_*}{dP}^2 \right] < \frac{1}{2} \left[ \frac{dQ_1}{dP}^2 \right] + ...

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Any of a wide variety of local vol models, where (from your equation) $b(\cdot,\cdot)$ is some fitted surface, are unlikely to have closed-form solutions for the terminal distribution. Indeed it's well-known that these models tend to have very unusual forward term structures of volatility. As a specific example, take $b(\cdot,\cdot)$ to be an approximation ...

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I haven't read all the paper, just the section you mentioned. The previsible/predictable strategy $\pi_t$ represents the number of shares of the asset $S$ held at time $t$. The paper looks to use power utility in some way, and as is common in those types of problems, generally you want to think of $\tilde{\pi}_t$, which is the percentage of wealth at time ...

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If you allow $X_t$ to be two dimensional then a model with a stock price $X_t^1$ and its variance process $X_t^2$ (stochastic volatility) would fit your definition. In such cases to my knowledge we often don't have a closed form of the density of $X_T^1$ but in some cases we have a closed form of the Laplace transform. An example is the Heston model.

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