In a pseudo-Euclidean space with scalar product $S(\cdot, \cdot)$, we show that the singularities of projections on $S$-monotone sets and of the associated Fitzpatrick functions are covered by countable $c-c$ surfaces having positive normal vectors with respect to the $S$-product. By Zaj\'{\i}\v{c}ek [24], the singularities of a convex function $f$ can be covered by a countable collection of $c-c$ surfaces. We show that the normal vectors to these surfaces are restricted to the cone generated by $F-F$, where $F:=\text{cl range } \nabla f$, the closure of the range of the gradient of $f$.

We study an optimal transport problem with a backward martingale constraint in a pseudo-Euclidean space $S$. We show that the dual problem consists in the minimization of the expected values of the Fitzpatrick functions associated with maximal $S$-monotone sets. An optimal plan $\gamma$ and an optimal maximal $S$-monotone set $G$ are characterized by the condition that the support of $\gamma$ is contained in the graph of the $S$-projection on $G$. For a Gaussian random variable $Y$, we get a unique decomposition: $Y = X+Z$, where $X$ and $Z$ are independent Gaussian random variables taking values, respectively, in complementary positive and negative linear subspaces of the $S$-space.

We consider an optimal transport problem with backward martingale constraint. The objective function is given by the scalar product of a pseudo-Euclidean space S. We show that the supremums over maps and plans coincide, provided that the law ν of the input random variable Y is atomless. An optimal map X exists if ν does not charge any c-c surface (the graph of a difference of convex functions) with strictly positive normal vectors in the sense of the S-space. The optimal map X is unique if ν does not charge c-c surfaces with nonnegative normal vectors in the S-space. As an application, we derive sharp conditions for the existence and uniqueness of equilibrium in a multi-asset version of the model with insider from Rochet and Vila [10]. In the linear-Gaussian case, we characterize Kyle's lambda, the sensitivity of price to trading volume, as the unique positive solution of a non-symmetric algebraic Riccati equation.

We study a single-period optimal transport problem on $\mathbb{R}^2$ with a covariance-type cost function $c(x,y) = (x_1-y_1)(x_2-y_2)$ and a backward martingale constraint. We show that a transport plan $\gamma$ is optimal if and only if there is a maximal monotone set $G$ that supports the $x$-marginal of $\gamma$ and such that $c(x,y) = \min_{z\in G}c(z,y)$ for every $(x,y)$ in the support of $\gamma$. We obtain sharp regularity conditions for the uniqueness of an optimal plan and for its representation in terms of a map. Our study is motivated by a variant of the classical Kyle model of insider trading from Rochet and Vila (1994).

Let $\psi$ be a multi-dimensional random variable. We show that the set of probability measures $\mathbb{Q}$ such that the $\mathbb{Q}$-martingale $S^{\mathbb{Q}}_t=\mathbb{E}^{\mathbb{Q}}\left[\psi\lvert\mathcal{F}_{t}\right]$ has the Martingale Representation Property (MRP) is either empty or dense in $\mathcal{L}_\infty$-norm. The proof is based on a related result involving analytic fields of terminal conditions $(\psi(x))_{x\in U}$ and probability measures $(\mathbb{Q}(x))_{x\in U}$ over an open set $U$. Namely, we show that the set of points $x\in U$ such that $S_t(x) = \mathbb{E}^{\mathbb{Q}(x)}\left[\psi(x)\lvert\mathcal{F}_{t}\right]$ does not have the MRP, either coincides with $U$ or has Lebesgue measure zero. Our study is motivated by the problem of endogenous completeness in financial economics.

We consider a financial model where the prices of risky assets are quoted by a representative market maker who takes into account an exogenous demand. We characterize these prices in terms of a system of BSDEs with quadratic growth. We show that this system admits a unique solution for every bounded demand if and only if the market maker's risk-aversion is sufficiently small. The uniqueness is established in the natural class of solutions, without any additional norm restrictions. To the best of our knowledge, this is the first study that proves such (global) uniqueness result for a system of fully coupled quadratic BSDEs.

We consider an economy where agents' consumption sets are given by the cone ${\bf L}^0_+$ of nonnegative measurable functions and whose preferences are defined by additive utilities satisfying the Inada conditions. We extend to this setting the results of Dana [J. Math. Econom., 22 (1993), pp. 563--579; Econometrica, 61 (1993), pp. 953--957] on the existence and uniqueness of Arrow--Debreu equilibria. In the case of existence, our conditions are necessary and sufficient.

In the problem of optimal investment with a utility function defined on (0,∞), we formulate sufficient conditions for the dual optimizer to be a uniformly integrable martingale. Our key requirement consists of the existence of a martingale measure whose density process satisfies the probabilistic Muckenhoupt (Ap) condition for the power p=1/(1−a), where a∈(0,1) is a lower bound on the relative risk-aversion of the utility function. We construct a counterexample showing that this (Ap) condition is sharp.

We obtain stability estimates and derive analytic expansions for local solutions of multi-dimensional quadratic BSDEs. We apply these results to a financial model where the prices of risky assets are quoted by a representative dealer in such a way that it is optimal to meet an exogenous demand. We show that the prices are stable under the demand process and derive their analytic expansions for small risk aversion coefficients of the dealer.

We develop a single-period model for a large economic agent who trades with market makers at their utility indifference prices. We compute the sensitivities of these market indifference prices with respect to the size of the investor’s order. It turns out that the price impact of an order is determined both by the market makers’ joint risk tolerance and by the variation of individual risk tolerances. On a technical level, a key role in our analysis is played by a pair of conjugate saddle functions associated with the description of Pareto optimal allocations in terms of the aggregate utility function.

The existence of complete Radner equilibria is established in an economy whose parameters are driven by a diffusion process. Our results complement those in the literature. In particular, we work under essentially minimal regularity conditions and treat the time-inhomogeneous case.

We develop from basic economic principles a continuous-time model for a large investor who trades with a finite number of market makers at their utility indifference prices. In this model, the market makers compete with their quotes for the investor's orders and trade among themselves to attain Pareto optimal allocations. We first consider the case of simple strategies and then, in analogy to the construction of stochastic integrals, investigate the transition to general continuous dynamics. As a result, we show that the model's evolution can be described by a nonlinear stochastic differential equation for the market makers' expected utilities.

Let Q and P be equivalent probability measures and let ψ be a J-dimensional vector of random variables such that dQdP and ψ are defined in terms of a weak solution X to a d-dimensional stochastic differential equation. Motivated by the problem of endogenous completeness in financial economics we present conditions which guarantee that every local martingale under Q is a stochastic integral with respect to the J-dimensional martingale St≜EQ[ψ|Ft]. While the drift b=b(t,x) and the volatility σ=σ(t,x) coefficients for X need to have only minimal regularity properties with respect to x, they are assumed to be analytic functions with respect to t. We provide a counter-example showing that this t-analyticity assumption for σ cannot be removed.

We describe the sample paths of the stochastic field $F = F_t(v,x,q)$ of aggregate utilities parameterized by Pareto weights $v$ and total cash amounts $x$ and stocks' quantities $q$ in an economy. We also describe the sample paths of the stochastic field $G = G_t(u,y,q)$, which is conjugate to $F$ with respect to the saddle arguments $(v,x)$, and obtain various conjugacy relations between these stochastic fields. The results of this paper play a key role in our study of a continuous-time price impact model.

We provide sufficient conditions for the existence and uniqueness of solutions to a stochastic differential equation which arises in a price impact model. These conditions are stated as smoothness and boundedness requirements on utility functions or Malliavin differentiability of payoffs and endowments.

Let Q and P be equivalent probability measures and let be a Jdimensional vector of random variables such that dQ dP and are dened in terms of a weak solution X to a d-dimensional stochastic dierential equation. Motivated by the problem of endogenous completeness in nancial economics we present conditions which guarantee that any local martingale under Q is a stochastic integral with respect to the J-dimensional martingale St , E Q [ jFt]. While the drift b = b(t;x) and the volatility = (t;x) coecients for X need to have only minimal regularity properties with respect to x, they are assumed to be analytic functions with respect to t. We provide a counter-example showing that this t-analyticity assumption for cannot be removed.

We study the linear approximation of utility-based hedging strategies for small number of contingent claims. We show that this approximation is actually a mean-variance hedging strategy under an appropriate choice of a numéraire and a risk-neutral probability. In contrast to previous studies, we work in the general framework of a semimartingale financial model and a utility function defined on the positive real line.

In the general framework of a semimartingale financial model and a utility function $U$ defined on the positive real line, we compute the first-order expansion of marginal utility-based prices with respect to a ``small'' number of random endowments. We show that this linear approximation has some important qualitative properties if and only if there is a risk-tolerance wealth process. In particular, they hold true in the following polar cases: \begin{tabular}@p97mm@ for any utility function $U$, if and only if the set of state price densities has a greatest element from the point of view of second-order stochastic dominance;for any financial model, if and only if $U$ is a power utility function ($U$ is an exponential utility function if it is defined on the whole real line). \end{tabular}

We study the two-times differentiability of the value functions of the primal and dual optimization problems that appear in the setting of expected utility maximization in incomplete markets. We also study the differentiability of the solutions to these problems with respect to their initial values. We show that the key conditions for the results to hold true are that the relative risk aversion coefficient of the utility function is uniformly bounded away from zero and infinity, and that the prices of traded securities are sigma-bounded under the numéraire given by the optimal wealth process.

We study the uniqueness of the marginal utility‐based price of contingent claims in a semimartingale model of an incomplete financial market. In particular, we obtain that a necessary and sufficient condition for all bounded contingent claims to admit a unique marginal utility‐based price is that the solution to the dual problem defines an equivalent local martingale measure.