Schedule for: 23w5011 - Applications of Stochastic Control to Finance and Economics

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

In this work, we study the mean field Schrödinger problem from a purely probabilistic point of view by exploiting its connection to stochastic control theory for McKean–Vlasov diffusions. Our main result shows that the mean field Schrödinger problem arises as the limit of "standard" Schrödinger problems over interacting particles. Due to the stochastic maximum principle and a suitable penalization procedure, the result follows as a consequence of a novel quantitative propagation of chaos result for forward–backward particle systems. Our stochastic control technique further allows us to solve the mean field Schödinger problem and characterize its solution, the mean field Schrödinger bridge, by a forward–backward planning equation. The approach described in the paper seems flexible enough to address other questions in the theory. For instance, it allows us to study zero noise limits for the Schrödinger problem.

In recent times, there has been a growing interest in the study of Mean-field Langevin (MFL) dynamics, primarily due to its natural application in training two-layer neural networks. To simulate the MFL dynamics' invariant distribution, one relies on the corresponding N-particle system, hoping that the error between the particle system and the mean-field dynamics remains small over an extended period. Our recent research focuses on the uniform-in-time propagation of chaos for the MFL dynamics with convex mean-field potential, motivated by this observation. We establish that this holds true for the L_2-Wasserstein distance and relative entropy under mild conditions.

We consider the setting described by $(\Omega, \mathcal{G}, \mathbb{F}, \tau, \mathbb{P} )$. Here $(\Omega, \mathcal{G}, \mathbb{F}, \mathbb{P} )$ is a complete filtered probability space, called initial model hereafter, in which $\mathbb{F}$ represents the flow of information available to all agents throughout time. $\tau$ is an arbitrary random time, which might not be observable via $\mathbb{F}$, and represents default time of a firm in credit risk theory, or death time of an insured in life insurance, and/or the occurrence time of any event that might impact the market somehow. In this framework of informational model, various valuation approaches and various hedging methods —which we are developing— point to the following family of reflected backward stochastic differential equations (RBSDE hereafter) \[ \begin{cases| \displaystyle \mathrm{d}Y_t = f(t,Y_t,Z_t)\mathrm{d}(t\wedge\tau) + g(t,Y_t,Z_t)\mathrm{d}U_t^\tau +Z_t\mathrm{d}W_t^\tau +\mathrm{d}M_t - \mathrm{d}K_t, \; Y_\tau =\xi,\\ \displaystyle Y\geq S\; \text{\rm on}\; \llbracket[0,\tau\llbracket,\; \int_0^\tau\big(Y_{s-}-S_{s-}\big)\mathrm{d}K_s=0, \mathbb{P}\text{\rm--a.s.} \end{cases} \] Here $W$ is an $\\mathbb{F}$--Brownian motion, and $(f,g,\xi,S,U)$ is the data-quintuplet in which $U$ is an RCLL nondecreasing and $\mathbb{F}$-adapted process, $f$ and $g$ are the drivers functionals, $\xi$ is the terminal condition and the barrier process $S$ is RCLL $\mathbb{F}$-adapted. We allow the initial model $(\Omega, \mathcal{G}, \mathbb{F}, \mathbb{P})$ to be chosen as "nice enough" as possible if needed, while $\tau$ is let to be as arbitrary general as possible. This will allow to measure adequately the impact of $\tau$ on the initial model in various aspects without interferences. In my talk, I will single out the challenges arising from allowing $\tau$ to be arbitrary general, and I will detail our approaches for these challenges and the intermediate problems arising from them as well. Among these, we address the impact of $\tau$ on the mathematical structures of these RBSDEs, which yields to the mathematical structures in the optimal stopping problem under random horizon $\tau$. Besides being intimately related to RBSDEs and vital in addressing the mathematical structures, the optimal stopping problem is interesting on its own and is useful in solving other financial problems. This talk is based on joint works with Safa Alsheyab.

In this talk I will present quantitative convergence results for a class of mean field games with common noise and controlled volatility. The basic strategy we employ is the one introduced recently by Laurière and myself–roughly speaking, we use a synchronous coupling argument to prove a “forward–backward propagation of chaos” result for the FBSDEs which characterize the (open-loop) equilibria of the N-player and mean field games. Unlike in earlier works which have adopted this strategy, we do not require smallness conditions, and instead rely on monotonicity. In particular, (displacement) monotonicity of the Hamiltonian and the terminal cost allow us to establish a (uniform in N) stability estimate for the N-player FBSDEs, which implies the convergence result. The arguments are relatively simple, and flexible enough to yield similar results in the setting of mean field control and infinite horizon (discounted) mean field games.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet at the Vistas Dining Room for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

We study the optimal execution problem in a principal-agent setting. A client (for example, a pension fund, endowment, or other institution) contracts to purchase a large position from a dealer at a future point in time. In the interim, the dealer acquires the position from the market, choosing how to divide his trading across time. Price impact may have temporary and permanent components. There is hidden action in that the client cannot directly dictate the dealer's trades. Rather, she chooses a contract with the goal of minimizing her expected payment, given the price process and an understanding of the dealer's incentives. Many contracts used in practice prescribe a payment equal to some weighted average of the market prices within the execution window. We explicitly characterize the optimal such weights: they are symmetric and generally U-shaped over time. The talk is based on joint work with Markus Baldauf (University of British Columbia) and Joshua Mollner (Northwestern University).

We formulate a general optimal stopping problem that can represent a wide variety of non-stationary environments, e.g., where the decision maker’s patience, time pressure, and learning speed can change gradually and abruptly over time. We show that, under some mild regularity conditions, this problem has a well-defined solution. Furthermore, we characterize the shape of the stopping region in a large class of “monotone” environments. As a result, we obtain comparative statics on the timing and quality of decisions for many sequential sampling problems à la Wald. For example, we show that accuracy is increasing (decreasing) over time when (i) learning speed increases (decreases) in time, or (ii) the discount rate decreases (increases) over time (i.e., the decision maker values the future more over time), or (iii) time pressure decreases (increases) over time. Since our main comparative static results hold locally, we can also capture non-monotone relations between time and accuracy that consistently arise in both perceptual and cognitive testing.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

McKean Vlasov or mean-field control problems are very closely related to the mean-field games. Dynamic programming for these problems results in a nonlinear partial differential equation on the space of probability measures. These equations require the value function to be not only differentiable but also its derivatives (which are in the dual of the set of measures, hence continuous functions) to be twice differentiable. Despite these difficulties, several approaches to characterize the value function as the unique appropriate weak solutions have been developed. We employ the classical viscosity solutions using the intrinsic linear derivative. We obtain a comparison result between the Lipschitz viscosity sub and super solutions under a structural assumption on the control set together with the given functions. Proof follows the classical variable doubling argument of Crandall & Lions by using the Fourier characterization of the dual Sobolev norm. The value function is also shown to be Lipschitz continuous with respect to this metric. This is joint work with Qinxin Yan of Princeton University.

This paper develops a methodology to solve dynamic principal-agent problems in which the agent features present-biased time preferences and naive beliefs. There are three insights. First, the problem has a recursive representation using the agent's perceived continuation value as a state variable (i.e., the remaining value the agent (wrongly) anticipates getting from the contract). Second, incentive compatibility corresponds to a volatility constraint on the agent's perceived continuation value. Finally, due to the agent's naivete, a perceived action constraint needs to be satisfied. This constraint is accommodated by linking the agent's perceived effort policy and the volatility of his perceived continuation value. Novel economic insights regarding optimal time-varying incentives and the term-structure of compensation are also explored.

We study how decentralized utility transfer promises affect collective decision-making by voting. Committee members with varying levels of support and opposition for an efficient reform can make enforceable promises before voting. An equilibrium requires stability and minimal promises. Equilibrium promises exist and are indeterminate, but do share several key characteristics. Equilibria require transfer promises from high to low intensity members and result in enacting the reform. When reform supporters lack sufficient voting power, promises must reach across the aisle. Even if the coalition of reform supporters is decisive, promises must preclude the least enthusiastic supporters of the reform from being enticed to overturn the decision. In that case, equilibrium promises do not need to reach across the aisle. We also discuss a finite sequence of promises that achieve an equilibrium.

I will present a connection between solutions to equilibrium problems in markets with asymmetric information, and solutions of Schroedinger problems. In particular, we will see how in some cases the former ones can be found as limit of the latter ones. Based on joint ongoing work with Umut Çetin.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider the problem of choosing an investment strategy that will maximise utility over distributions, under capital gains tax and constraints on the expected liquidation date. We show that the problem can be decomposed in two separate ones. The first involves choosing an optimal target distribution, while the second involves optimally realising this distribution via an investment strategy and stopping time. The latter step may be regarded as a variant of the Skorokhod embedding problem. A solution is given very precisely in terms of the first time that the wealth of the growth optimal portfolio, properly taxed, crosses a moving stochastic (depending on its minimum-to-date) level. The suggested solution has the additional optimality property of stochastically minimising maximal losses over the investment period.

We study the well-posedness of BSDEs and reflected BSDEs on random time horizons with stochastic Lipschitz constants and driven by general (possibly pure-jump) martingales. We employ a fixed-point theorem and provide various conditions, depending on the complexity of the generator, that ensure the existence and uniqueness within a class of processes.

In this talk, we will study Nash equilibria in two player continuous time stochastic differential games with diffusion control, and where the Brownian motions driving the state processes are correlated. We consider zero-sum ranking games, in the sense that the criteria to optimize only depends on the difference of the two players' state processes. We explicitly compute the players' equilibrium strategies, depending on the correlation of the Brownian motions driving the two state equations: in particular, if the correlation coefficient is smaller than some explicit threshold, then the equilibrium strategies consist of strong controls, whereas if the correlation exceeds the threshold, then the equilibrium controls are mixed strategies. To characterize these equilibria, we rely on a relaxed formulation of the game based on solutions to martingale problems, allowing the players to randomize their actions. The talk is based on a joint work with Stefan Ankirchner (University of Jena) and Julian Wendt (University of Jena).

We propose a novel generative model for time series based on Schrödinger bridge (SB) approach. This consists in the entropic interpolation via optimal transport between a reference probability measure on path space and a target measure consistent with the joint data distribution of the time series. The solution is characterized by a stochastic differential equation on finite horizon with a path-dependent drift function, hence capturing the temporal dynamics of the time series distribution. We can estimate the drift function from data samples either by kernel regression methods or with LSTM neural networks, and the simulation of the SB diffusion yields new synthetic data samples of the time series. The performance of our generative model is evaluated through a series of numerical experiments. First, we test with a toy autoregressive model, a GARCH Model, and the example of fractional Brownian motion, and measure the accuracy of our algorithm with marginal and temporal dependencies metrics. Next, we use our SB generated synthetic samples for the application to deep hedging on real-data sets. Finally, we illustrate the SB approach for generating sequence of images. Based on joint work with M. Hamdouche and P. Henry-Labordère

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider an economy with two agents. Each of the two agents receives a random endowment flow. We model this cumulative flow as the stochastic integral of a deterministic function of the economy's state, which we model by means of a general Ito diffusion. Each of the two agents has mean-variance preferences with different risk-aversion coefficients. The two agents can also trade a risky asset. We determine the agents' optimal equilibrium trading strategies in the presence of proportional transaction costs. In particular, we derive a new free-boundary problem that provides the solution to the agents' optimal equilibrium problem. Furthermore, we derive the explicit solution to this free-boundary problem when the problem data is such that the frictionless optimiser is a strictly increasing or a strictly increasing and then strictly decreasing function of the economy's state.

We study a mean-field principal-agent model where interactions take place through the allocation of capital to agents. We analyse the impact of information asymmetry on risk-sharing and investment. The optimal contract is explicitely characterized and implemented with money and taxes.

The practice of retail internalization has been a controversial topic since the late 1990s. The crux of this debate is whether this practice benefits, via the price improvement relative to exchange, or disadvantages, via the reduced liquidity on exchange, retail traders. To answer this question we set two models of market design that differ in their mode of liquidity provision: in the model capturing retail order internalization the liquidity is provided by market makers (representing wholesalers) competing for the retail order flow in a Bertrand fashion, whereas in the model characterizing the open exchange the price-taking competitive agents act as liquidity providers. We discover that, when liquidity providers in both market designs are risk averse, routing of the marketable orders to the wholesalers is preferred by retail traders: informed, uninformed and noisy. In addition to addressing optimal order routing problem, we identify a universal parameter that allows comparison of market liquidity, profit and value of information across different markets and demonstrate that the risk aversion of liquidity providers fundamentally changes market outcomes. In particular, we observe mean reverting inventories, price reversal, and lower market depth as the result of retail investor (informed or not) absorbing large shocks in their inventory to compensate for the unwillingness of liquidity providers to bear risk.

We study the optimal execution problem for the Central Risk Book (CRB), a centralized trading unit recently established in many large banks and trading companies. The CRB aggregates orders from the other business units within the organization in real time, netting opposite orders and executing outstanding orders such as to minimize transaction costs. Thus, the in-flow orders of the CRB are a stochastic process. We introduce a tractable model for the price impact and spread cost paid by the out-flow orders and find the optimal execution strategy for a general class of in-flow processes. The strategy highlights how future in-flows are taken into account to determine the optimal trade-off between trading speed and transaction costs. (Joint work with Kevin Webster and Long Zhao.)

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

This paper revisits Williams' (2011) (henceforth PPI’s) continuous-time principal-agent model of optimal dynamic insurance with persistent private information. We identify three independent issues in PPI that implicate its characterizations of incentive compatible and optimal contracts: (i) the agent cannot over-report increments of his type, a constraint that does not follow from the common assumption that the agent cannot over-report his type; (ii) the agent's feasible set of reporting strategies does not include standard "no Ponzi" constraints, without which PPI's main analysis of infinite-horizon incentive compatibility is incomplete; and (iii) most importantly, in PPI's main application, which concerns hidden endowments, the contract identified as optimal is generically strictly suboptimal. For this application, we address the first two issues and elucidate the third one by analyzing a class of "self-insurance contracts" that can be implemented as consumption-saving problems for the agent, and which includes the contract derived in PPI as a particular case. We characterize the optimal self-insurance contract and show that, generically, it strictly dominates PPI's. Our analysis does not support PPI’s main economic finding that immiseration generally fails or its attribution of this failure to continuous time and persistence. (Joint work zith R. Vijay Krishna and Bruno Strulovici).

We develop a dynamic principal-agent model that examines the intricate balance between monitoring and rewards in executive compensation. In the optimal contract, monitoring and success reward dynamically substitute each other, exhibiting two main characteristics: (i) a reduction in contract sensitivity to monitoring and a rise in the success reward when positive evidence of the agent's effort accumulates, and (ii) a larger success reward when the signal about the agent's effort is more noisy, particularly when negative evidence accumulates. Using the changes in the availability of direct flights for board directors to the firm's headquarters as an exogenous proxy of signal quality, we present empirical evidence that supports our model's predictions. This is a joint work with Kerry Back, Clint Hamilton, and Ali Kakhbod.

We study the properties of value functions in joint optimal control and stopping problems where (i) the state variable may be multi-dimensional, (ii) the domain may be unbounded, and (iii) the primitives may be time-inhomogeneous. We show that the value function is (i) the unique Lp-solution of the Hamilton–Jacobi–Bellman equation, (ii) twice parabolically differentiable a.e., and (iii) continuously differentiable in the non-time variables, under general conditions relevant for most economic applications. In particular, we show that the smooth-pasting property holds everywhere with respect to all space variables. We also derive sufficient conditions under which smooth pasting also holds with respect to time. Our results imply that numerical solutions obtained by standard methods converge uniformly to the value function.

Based on recent work by Possamaï and Touzi, we consider an extension of Sannikov's principal-agent problem by letting the agent control the drift and jump intensity of the output process. We investigate whether the problem exhibits a golden parachute, that is, whether there is a scenario in which the agent retires and receives a continuous stream of payments or the agent receives a lump-sum payment as compensation for the termination of the contract by the principal. With the introduction of the face-lifted utility, we can study the two cases simultaneously in a different principal-agent problem that we reduce to a standard mixed control-stopping problem.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We develop a stationary mean-field model of singular control irreversible investment where production capacity is subject to random gaussian fluctuations while prices are affected by macroeconomic shocks following a two-state Markov chain (boom/burst episodes). We prove existence and uniqueness of the mean-field stationary equilibrium and we characterize it through a system of nonlinear equations. We provide several insights on the effects of crisis on the equilibrium, in particular on the firms size distribution and on the firms value at equilibrium. [Joint work with Giorgio Ferrari and Matteo Basei].

In an economy with random growth, several long-lived agents with heterogeneous risk aversions, time-preferences, and personal income streams make consumption and investment decisions, trading stocks and a consol bond, and borrowing from and lending to each other. We find in closed form equilibrium stock prices, interest rates, consumption, and trading policies. Agents do not trade stocks, although their returns are time-varying and predictable. Agents dynamically trade the consol bond in response to growth shocks, as to hedge their effect on interest rates, dividends, and personal incomes. Static fund separation holds if agents have also access to a linear bond and two additional hedges for dividend and growth shocks. Such additional assets can be dynamically replicated with the stock and the consol bond. No representative agent exists.

We present an optimal linear insurance demand model, where the protection buyer can also exert a time-dynamic costly prevention effort to reduce her risk exposure. This is expressed as a stochastic control problem that consists in maximizing an exponential utility of a terminal wealth. We assume that the effort reduces the intensity of the jump arrival process and interpret this as dynamic self-protection. We solve the problem by using a dynamic programming approach, and we provide a representation of the certainty equivalent of the buyer as the solution to a backward stochastic differential equation (BSDE). Using this representation, we prove that an exponential utility maximizer has an incentive to modify her effort dynamically only in the presence of a terminal reimbursement in the contract. Otherwise, the dynamic effort is actually constant, for a class of compound Poisson loss processes. If there is no terminal reimbursement, we solve the problem explicitly and identify the dynamic certainty equivalent of the protection buyer. This shows in particular that the Lévy property of the loss process is preserved under exponential utility maximization. We also characterize the constant effort as the unique minimizer of an explicit Hamiltonian, from which we can determine the optimal effort in particular cases. Finally, after studying the dependence of the BSDE associated to the insurance buyer on the linear insurance contract parameter, we prove the existence of an optimal linear cover that is not necessarily zero or full insurance.

We study a dynamic general equilibrium model with costly-to-short stocks and heterogeneous beliefs. The closed-form solution to the model shows that costly short sales drive a wedge between the valuation of assets that promise identical cash flows but are subject to different trading arrangements. Specifically, we show that the price of an asset is given by the risk-adjusted present value of future cash flows which include both dividends and an endogenous lending yield. This formula implies that returns satisfy a modified CAPM and sheds light on recent findings about the explanatory power of lending fees in the cross-section of returns. In particular, we show that once returns are appropriately adjusted for lending fees, stocks with low and high shorting costs offer similar risk-return tradeoffs.

We study a principal–agent problem with a lump-sum payment on a finite-time horizon. Extending the dynamic programming approach in Cvitanić, Possamaï and Touzi (2017), we consider problems involving constraints on the optimal contract. We introduce a framework, in which the agent is allowed to randomize his actions. This in turn gives more freedom to the principal, when she chooses the contract, and allows us to show existence of an optimal contract in problems with fairly general constraints, for which the standard PDE approach ceased to be tractable.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk we investigate path dependent mean field optimal control problems with both drift and volatility controls. The value function is characterized by a fully nonlinear path dependent HJB equation on the Wasserstein space of probability measures. By lifting to the space of processes, we introduce a new notion of viscosity solutions and establish both existence and comparison principle. The main feature of our notion is that the test function consists of an extra component, besides the standard smooth part, which helps to get around of some major difficulty arising from the volatility control. We shall use the doubling variable arguments, combined with the Borwein-Preiss Variational Principle in order to overcome the noncompactness of the state space. A smooth Gauge-type function on the path space is crucial for our estimates. The talk is based on an ongoing joint work with Nizar Touzi and Jianjun Zhou.

We are interested in the study of stochastic games for which each player faces an optimal stopping problem. In our setting, the players may interact through the criterion to optimize as well as through their dynamics. After briefly discussing the N-players game, we formulate the corresponding mean field problem. In particular, we introduce a weak formulation of the game for which we are able to prove existence of Nash equilibria for a large class of criteria. We also prove that equilibria for the mean field problem provide approximated Nash equilibria for the N-players game, and we formally derive the master equation associated with our mean field game. Joint work with Dylan Possamaï.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.