arXiv subjects

Mihai Sîrbu

Publications and source records attributed to Mihai Sîrbu.

11 recordsLinked to original sources

arXiv2024

Backward martingale transport maps and equilibrium with insider

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.

math.PR↗

arXiv2023

Backward martingale transport and Fitzpatrick functions in pseudo-Euclidean spaces

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 $γ$ and an optimal maximal $S$-monotone set $G$ are characterized by the condition that the support of $γ$ 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.

math.PR↗

arXiv2023

Singularities of Fitzpatrick and convex functions

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\'ıček [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 $f$.

math.CA↗

arXiv2021

On the analyticity of the value function in optimal investment and stochastically dominant markets

We study the analyticity of the value function in optimal investment with expected utility from terminal wealth and the relation to stochastically dominant financial models. We identify both a class of utilities and a class of semi-martingale models for which we establish analyticity. Specifically, these utilities have completely monotonic inverse marginals, while the market models have a maximal element in the sense of infinite-order stochastic dominance. We construct two counterexamples, themselves of independent interest, which show that analyticity fails if either the utility or the market model does not belong to the respective special class. We also provide explicit formulas for the derivatives, of all orders, of the value functions as well as their optimizers. Finally, we show that for the set of supermartingale deflators, stochastic dominance of infinite order is equivalent to the apparently stronger dominance of second order.

math.PR↗

arXiv2019

Optimal investment and consumption with labor income in incomplete markets

We consider the problem of optimal consumption from labor income and investment in a general incomplete semimartingale market. The economic agent cannot borrow against future income, so the total wealth is required to be positive at (all or some) previous times. Under very general conditions, we show that an optimal consumption and investment plan exists and is unique, and provide a dual characterization in terms of martingale deflators and decreasing parts, which allow for a limit that charges only the times when the no-borrowing constraint is binding. The analysis relies on the infinite-dimensional parametrization of the income/liability streams and, therefore, provides the first-order dependence of the optimal investment and consumption plans on future income/liabilities (as well as a pricing rule). An emphasis is placed on mathematical generality.

math.PR↗

arXiv2017

Sensitivity analysis of the utility maximization problem with respect to model perturbations

We study the sensitivity of the expected utility maximization problem in a continuous semi-martingale market with respect to small changes in the market price of risk. Assuming that the preferences of a rational economic agent are modeled with a general utility function, we obtain a second-order expansion of the value function, a first-order approximation of the terminal wealth, and construct trading strategies that match the indirect utility function up to the second order. If a risk-tolerance wealth process exists, using it as a numéraire and under an appropriate change of measure, we reduce the approximation problem to a Kunita-Watanabe decomposition.

q-fin.PM↗

arXiv2016

Zero-sum stochastic differential games without the Isaacs condition: random rules of priority and intermediate Hamiltonians

For a zero-sum stochastic game which does not satisfy the Isaacs condition, we provide a value function representation for an Isaacs-type equation whose Hamiltonian lies in between the lower and upper Hamiltonians, as a convex combination of the two. For the general case (i.e. the convex combination is time and state dependent) our representation amounts to a random change of the rules of the game, to allow each player at any moment to see the other player's action or not, according to a coin toss with probabilities of heads and tails given by the convex combination appearing in the PDE. If the combination is state independent, then the rules can be set all in advance, in a deterministic way. This means that tossing the coin along the game, or tossing it repeatedly right at the beginning leads to the same value. The representations are asymptotic, over time discretizations. Space discretization is possible as well, leading to similar results.

math.PR↗

arXiv2015

Asymptotic Perron's method and simple Markov Strategies in stochastic games and control

We introduce a modification of Perron's method, where semi-solutions are considered in a carefully defined asymptotic sense. With this definition, we can show, in a rather elementary way, that in a zero-sum game or a control problem (with or without model uncertainty), the value function over all strategies coincides with the value function over Markov strategies discretized in time. Therefore, there are always discretized Markov $\varepsilon$-optimal strategies, (uniform with respect to the bounded initial condition). With a minor modification, the method produces a value and approximate saddle points for an asymmetric game of feedback strategies vs. counter-strategies.

math.OC↗

arXiv2014

On martingale problems with continuous-time mixing and values of zero-sum games without Isaacs condition

We consider a zero-sum stochastic differential game over elementary mixed feed-back strategies. These are strategies based only on the knowledge of the past state, randomized continuously in time from a sampling distribution which is kept constant in between some stopping rules. Once both players choose such strategies, the state equation admits a unique solution in the sense of the martingale problem of Stroock and Varadhan. We show that the game defined over martingale solutions has a value, which is the unique continuous viscosity solution of the randomized Isaacs equation.

math.OC↗

arXiv2014

A note on the strong formulation of stochastic control problems with model uncertainty

We consider a Markovian stochastic control problem with model uncertainty. The controller (intelligent player) observes only the state, and, therefore, uses feed-back (closed-loop) strategies. The adverse player (nature) who does not have a direct interest in the pay-off, chooses open-loop controls that parametrize Knightian uncertainty. This creates a two-step optimization problem (like half of a game) over feed-back strategies and open-loop controls. The main result is to show that, under some assumptions, this provides the same value as the (half of) the zero-sum symmetric game where the adverse player also plays feed-back strategies and actively tries to minimize the pay-off. The value function is independent of the filtration accessible to the adverse player. Aside from the modeling issue, the present note is a technical companion to [SÎ3b].

math.OC↗

arXiv2014

Stochastic Perron's method and elementary strategies for zero-sum differential games

We develop here the Stochastic Perron Method in the framework of two-player zero-sum differential games. We consider the formulation of the game where both players play, symmetrically, feed-back strategies (as in [CR09] or [PZ12]) as opposed to the Elliott-Kalton formulation prevalent in the literature. The class of feed-back strategies we use is carefully chosen so that the state equation admits strong solutions and the technicalities involved in the Stochastic Perron Method carry through in a rather simple way. More precisely, we define the game over elementary strategies, which are well motivated by intuition. Within this framework, the Stochastic Perron Method produces a viscosity sub-solution of the upper Isaacs equation dominating the upper value of the game, and a viscosity super-solution of the upper Isaacs equation lying below the upper value of the game. Using a viscosity comparison result we obtain that the upper value is the unique and continuous viscosity solution of the upper Isaacs equation. An identical statement holds for the lower value and lower Isaacs equation. A version of the Dynamic Programming Principle is obtained as a by-product. If the Isaacs condition holds, the game has a value over elementary (pure) strategies.

math.OC↗