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Register a new userMinimal Quantile Functions Subject to Stochastic Dominance Constraints
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We consider a problem of finding an SSD-minimal quantile function subject to the mixture of multiple first-order stochastic dominance (FSD) and second-order stochastic dominance (SSD) constraints. The solution is explicitly worked out and has a closed relation to the Skorokhod problem. We then apply this result to solve an expenditure minimization problem with the mixture of an FSD constraint and an SSD constraint in financial economics.
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Optimized certainty equivalents (OCEs) is a family of risk measures widely used by both practitioners and academics. This is mostly due to its tractability and the fact that it encompasses important examples, including entropic risk measures and average value at risk. In this work we consider stochastic optimal control problems where the objective criterion is given by an OCE risk measure, or put in other words, a risk minimization problem for controlled diffusions. A major difficulty arises since OCEs are often time inconsistent. Nevertheless, via an enlargement of state space we achieve a substitute of sorts for time consistency in fair generality. This allows us to derive a dynamic programming principle and thus recover central results of (risk-neutral) stochastic control theory. In particular, we show that the value of our risk minimization problem can be characterized via the viscosity solution of a Hamilton--Jacobi--Bellman--Issacs equation. We further establish the uniqueness of the latter under suitable technical conditions.
We study existence and uniqueness of continuous-time stochastic Radner equilibria in an incomplete market model among a group of agents whose preference is characterized by cash invariant time-consistent monetary utilities. An assumption of smallness type is shown to be sufficient for existence and uniqueness. In particular, this assumption encapsulates settings with small endowments, small time-horizon, or a large population of weakly heterogeneous agents. Central role in our analysis is played by a fully-coupled nonlinear system of quadratic BSDEs.
The t copula is often used in risk management as it allows for modelling tail dependence between risks and it is simple to simulate and calibrate. However, the use of a standard t copula is often criticized due to its restriction of having a single parameter for the degrees of freedom (dof) that may limit its capability to model the tail dependence structure in a multivariate case. To overcome this problem, grouped t copula was proposed recently, where risks are grouped a priori in such a way that each group has a standard t copula with its specific dof parameter. In this paper we propose the use of a grouped t copula, where each group consists of one risk factor only, so that a priori grouping is not required. The copula characteristics in the bivariate case are studied. We explain simulation and calibration procedures, including a simulation study on finite sample properties of the maximum likelihood estimators and Kendalls tau approximation. This new copula can be significantly different from the standard t copula in terms of risk measures such as tail dependence, value at risk and expected shortfall. Keywords: grouped t copula, tail dependence, risk management.
This paper investigates optimal consumption in the stochastic Ramsey problem with the Cobb-Douglas production function. Contrary to prior studies, we allow for general consumption processes, without any a priori boundedness constraint. A non-standard stochastic differential equation, with neither Lipschitz continuity nor linear growth, specifies the dynamics of the controlled state process. A mixture of probabilistic arguments are used to construct the state process, and establish its non-explosiveness and strict positivity. This leads to the optimality of a feedback consumption process, defined in terms of the value function and the state process. Based on additional viscosity solutions techniques, we characterize the value function as the unique classical solution to a nonlinear elliptic equation, among an appropriate class of functions. This characterization involves a condition on the limiting behavior of the value function at the origin, which is the key to dealing with unbounded consumptions. Finally, relaxing the boundedness constraint is shown to increase, strictly, the expected utility at all wealth levels.
Latency (i.e., time delay) in electronic markets affects the efficacy of liquidity taking strategies. During the time liquidity takers process information and send marketable limit orders (MLOs) to the exchange, the limit order book (LOB) might undergo updates, so there is no guarantee that MLOs are filled. We develop a latency-optimal trading strategy that improves the marksmanship of liquidity takers. The interaction between the LOB and MLOs is modelled as a marked point process. Each MLO specifies a price limit so the order can receive worse prices and quantities than those the liquidity taker targets if the updates in the LOB are against the interest of the trader. In our model, the liquidity taker balances the tradeoff between missing trades and the costs of walking the book. We employ techniques of variational analysis to obtain the optimal price limit of each MLO the agent sends. The price limit of a MLO is characterized as the solution to a new class of forward-backward stochastic differential equations (FBSDEs) driven by random measures. We prove the existence and uniqueness of the solution to the FBSDE and numerically solve it to illustrate the performance of the latency-optimal strategies.
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