48 problems

Let $P$ be the price path, let $T$ be the terminal time, and let $K$ be the strike price. Define … When $P(T)=K$, let $(\alpha,\beta)$ be the strategy constructed from the function…

The signature framework is formulated in a continuous setting, with the transform theorem established under a weight class that controls the relevant continuous dynamics. Jump-acti…

The superhedging sets are subsets of $\mathbb{L}^2$ spaces indexed by time, and their approximate versions satisfy a set-valued Bellman's principle relating approximate superhedgin…

Canonical-filtration conjecture. It should be enough to restrict the search for an optimal model to models equipped with the canonical filtration of the price process.

Let $Y_t$ denote the risky position, let $y_\star$ be the perfect target, and let $y_-<y_\star<y_+$ be boundary levels. Trading incurs transaction costs, and admissible strategies…

Let $\widetilde\pi_-$ and $\widetilde\pi_+$ be the lower and upper shadow-market trading boundaries, let $\Lambda$ be the target proportion, let $\gamma$ be the risk-aversion param…

Let $\widetilde\pi_-$ and $\widetilde\pi_+$ denote the lower and upper trading boundaries in the shadow market, and let $\varepsilon$ be the transaction-cost parameter. The compari…

Let $V(t,x)$ denote the value function of the robust investment-consumption problem, and let $Y(t)$ be an auxiliary process. Suppose that $G$ is a function continuously differentia…

Consider the multi-agent relative investment game in Section … can only be constant. The numerical experiments suggest convergence to a constant Nash equilibrium even when the suff…

Gaussian equilibrium potential conjecture. The pricing rule should satisfy $H^*(t,0)=\mu$, and the potential should satisfy

Bernoulli fixed-point conjecture.

Conjecture. The proposed method can be adapted, with additional technicalities, to compute super-hedging prices for general payoff functions.

Let ${\cal B}$ denote the constrained region and ${\cal U}$ the unconstrained region for the optimal consumption-investment problem, and let $x^*$ be a positive real number. Interv…

Let $D$ denote the dividend process, and let $(a,Z_a)$, $(Y_1,Z_1)$, and $(Y_2,Z_2)$ be an ${\mathcal S}^\infty\times\operatorname{bmo}$ solution to the stated BSDE system. Set…

Kreps' conjecture. Under suitable conditions, the optimal expected utility for the sequence of discrete-time economies converges to the optimal expected utility of the Black–Schole…

Let $\nu$ be a Gaussian distribution, and let $\phi$ be the convex function constructed in the paper's fixed-point theorem. In the Gaussian case, transport maps between Gaussian di…

The paper studies the multidimensional extension of a Kyle–Back model, where the construction involves a convex potential and the associated Monge–Ampère equation. Monge–Ampère con…

The paper considers a multidimensional Kyle–Back model in which the price pressure is constructed through optimal transport maps between probability laws. Given the strong regulari…

A no-fee CFMM has a path-independent payoff, whereas an unconstrained, continually rebalanced delta-hedging strategy can be path-dependent and may benefit from positive theta. Cons…

A constant-mean market maker is a particular CFMM, and the paper considers whether fees can mitigate arbitrage losses sufficiently to allow replication without additional initial c…

A covered-call position is replicated by a no-fee CFMM whose trading function must be updated as maturity approaches, requiring additional capital because the CFMM does not capture…

In the Kyle–Back model, let $\xi$ be a state process whose dynamics are to be determined and which is observable by all market participants. State-process pricing conjecture. The m…

Let $\zeta$ denote the equilibrium stock-volatility process in the one-source-of-randomness economy described above. Nonvanishing equilibrium volatility conjecture. We conjecture t…

Optimal exercise-time conjecture. The sequence of times

Let $X(k)$ be the return of the bet at time $k$, let $\mathcal{X}_n=\prod_{k=0}^{n-1}(1+X(k))-1$ be the total return over $n$ steps, and let $g_n^*$ denote the optimal expected log…