arXiv subjects

Virginia R. Young

Publications and source records attributed to Virginia R. Young.

At least 19 recordsLinked to original sources

arXiv2025

Equilibrium Strategies for Singular Dividend Control Problems under the Mean-Variance Criterion

We revisit the optimal dividend problem of de Finetti by adding a variance term to the usual criterion of maximizing the expected discounted dividends paid until ruin, in a singular control framework. Investors do not like variability in their dividend distribution, and the mean-variance (MV) criterion balances the desire for large expected dividend payments with small variability in those payments. The resulting MV singular dividend control problem is time-inconsistent, and we follow a game-theoretic approach to find a time-consistent equilibrium strategy. Our main contribution is a new verification theorem for the novel dividend problem, in which the MV criterion is applied to an integral of the control until ruin, a random time that is endogenous to the problem. We demonstrate the use of the verification theorem in two cases for which we obtain the equilibrium dividend strategy (semi-)explicitly, and we provide a numerical example to illustrate our results.

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arXiv2025

Equilibrium Mean-Variance Dividend Rate Strategies

This paper studies an optimal dividend problem for a company that aims to maximize the mean-variance (MV) objective of the accumulated discounted dividend payments up to its ruin time. The MV objective involves an integral form over a random horizon that depends endogenously on the company's dividend strategy, and these features lead to a novel time-inconsistent control problem. To address the time inconsistency, we seek a time-consistent equilibrium dividend rate strategy. We first develop and prove a new verification lemma that characterizes the value function and equilibrium strategy by an extended Hamilton-Jacobi-Bellman system. Next, we apply the verification lemma to obtain the equilibrium strategy and show that it is a barrier strategy for small levels of risk aversion.

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arXiv2024

Optimal Insurance to Maximize Exponential Utility when Premium is Computed by a Convex Functional

We find the optimal indemnity to maximize the expected utility of terminal wealth of a buyer of insurance whose preferences are modeled by an exponential utility. The insurance premium is computed by a convex functional. We obtain a necessary condition for the optimal indemnity; then, because the candidate optimal indemnity is given implicitly, we use that necessary condition to develop a numerical algorithm to compute it. We prove that the numerical algorithm converges to a unique indemnity that, indeed, equals the optimal policy. We also illustrate our results with numerical examples.

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arXiv2022

Optimal Consumption under a Habit-Formation Constraint: the Deterministic Case

We formulate and solve a deterministic optimal consumption problem to maximize the discounted CRRA utility of an individual's consumption-to-habit process assuming she only invests in a riskless market and that she is unwilling to consume at a rate below a certain proportion $α\in(0,1]$ of her consumption habit. Increasing $α$, increases the degree of addictiveness of habit formation, with $α=0$ (respectively, $α=1$) corresponding to non-addictive (respectively, completely addictive) model. We derive the optimal consumption policies explicitly in terms of the solution of a nonlinear free-boundary problem, which we analyze in detail. Impatient individuals (or, equivalently, those with more addictive habits) always consume above the minimum rate; thus, they eventually attain the minimum wealth-to-habit ratio. Patient individuals (or, equivalently, those with less addictive habits) consume at the minimum rate if their wealth-to-habit ratio is below a threshold, and above it otherwise. By consuming patiently, these individuals maintain a wealth-to-habit ratio that is greater than the minimum acceptable level. Additionally, we prove that the optimal consumption path is hump-shaped if the initial wealth-to-habit ratio is either: (1) larger than a high threshold; or (2) below a low threshold and the agent is more risk seeking (that is, less risk averse). Thus, we provide a simple explanation for the consumption hump observed by various empirical studies.

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arXiv2022

Optimal Reinsurance to Minimize the Probability of Drawdown under the Mean-Variance Premium Principle: Asymptotic Analysis

In this paper, we consider an optimal reinsurance problem to minimize the probability of drawdown for the scaled Cramér-Lundberg risk model when the reinsurance premium is computed according to the mean-variance premium principle. We extend the work of Liang et al. [16] to the case of minimizing the probability of drawdown. By using the comparison method and the tool of adjustment coefficients, we show that the minimum probability of drawdown for the scaled classical risk model converges to the minimum probability for its diffusion approximation, and the rate of convergence is of order $O(n^{-1/2})$. We further show that using the optimal strategy from the diffusion approximation in the scaled classical risk model is $O(n^{-1/2})$-optimal.

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arXiv2021

Optimal Investment and Consumption under a Habit-Formation Constraint

We formulate an infinite-horizon optimal investment and consumption problem, in which an individual forms a habit based on the exponentially weighted average of her past consumption rate, and in which she invests in a Black-Scholes market. The individual is constrained to consume at a rate higher than a certain proportion $α$ of her consumption habit. Our habit-formation model allows for both addictive ($α=1$) and nonaddictive ($0<α<1$) habits. The optimal investment and consumption policies are derived explicitly in terms of the solution of a system of differential equations with free boundaries, which is analyzed in detail. If the wealth-to-habit ratio is below (resp. above) a critical level $x^*$, the individual consumes at (resp. above) the minimum rate and invests more (resp. less) aggressively in the risky asset. Numerical results show that the addictive habit formation requires significantly more wealth to support the same consumption rate compared to a moderately nonaddictive habit. Furthermore, an individual with a more addictive habit invests less in the risky asset compared to an individual with a less addictive habit but with the same wealth-to-habit ratio and risk aversion, which provides an explanation for the equity-premium puzzle.

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arXiv2021

Optimal Dividends under Model Uncertainty

We consider a diffusive model for optimally distributing dividends, while allowing for Knightian model ambiguity concerning the drift of the surplus process. We show that the value function is the unique solution of a non-linear Hamilton-Jacobi-Bellman variational inequality. In addition, this value function embodies a unique optimal threshold strategy for the insurer's surplus, thereby making it the smooth pasting of a non-linear and linear part at the location of the threshold. Furthermore, we obtain continuity and monotonicity of the value function and the threshold strategy with respect to the parameter that measures ambiguity of our model.

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arXiv2021

Minimizing the Probability of Lifetime Exponential Parisian Ruin

We find the optimal investment strategy in a Black-Scholes market to minimize the probability of so-called {\it lifetime exponential Parisian ruin}, that is, the probability that wealth exhibits an excursion below zero of an exponentially distributed time before the individual dies. We find that leveraging the risky asset is worse for negative wealth when minimizing the probability of lifetime exponential Parisian ruin than when minimizing the probability of lifetime ruin. Moreover, when wealth is negative, the optimal amount invested in the risky asset increases as the hazard rate of the exponential ``excursion clock'' increases. In view of the heavy leveraging when wealth is negative, we also compute the minimum probability of lifetime exponential Parisian ruin under a constraint on investment. Finally, we derive an asymptotic expansion of the minimum probability of lifetime exponential Parisian ruin for small values of the hazard rate of the excursion clock. It is interesting to find that, for small values of this hazard rate, the minimum probability of lifetime exponential Parisian ruin is proportional to the minimum occupation time studied in Bayraktar and Young, and the proportion equals the hazard rate. To the best of our knowledge, our work is the first to {\it control} the probability of Parisian ruin.

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arXiv2020

Optimal Insurance to Minimize the Probability of Ruin: Inverse Survival Function Formulation

We find the optimal indemnity to minimize the probability of ruin when premium is calculated according to the distortion premium principle with a proportional risk load, and admissible indemnities are such that both the indemnity and retention are non-decreasing functions of the underlying loss. We reformulate the problem with the inverse survival function as the control variable and show that deductible insurance with maximum limit is optimal. Our main contribution is in solving this problem via the inverse survival function.

q-fin.RM↗

arXiv2020

Optimal Dividend Problem: Asymptotic Analysis

We re-visit the classical problem of optimal payment of dividends and determine the degree to which the diffusion approximation serves as a valid approximation of the classical risk model for this problem. Our results parallel some of those in Bäuerle (2004), but we obtain sharper results because we use a different technique for obtaining them. Specifically, Bäuerle (2004) uses probabilistic techniques and relies on convergence in distribution of the underlying processes. By contrast, we use comparison results from the theory of differential equations, and these methods allow us to determine the rate of convergence of the value functions in question.

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arXiv2020

Minimizing the Discounted Probability of Exponential Parisian Ruin via Reinsurance

We study the problem of minimizing the discounted probability of exponential Parisian ruin, that is, the discounted probability that an insurer's surplus exhibits an excursion below zero in excess of an exponentially distributed clock. The insurer controls its surplus via reinsurance priced according to the mean-variance premium principle, as in Liang, Liang, and Young (2019). We consider the classical risk model and apply stochastic Perron's method, as introduced by Bayraktar and Sirbu (2012,2013,2014), to show that the minimum discounted probability of exponential Parisian ruin is the unique viscosity solution of its Hamilton-Jacobi-Bellman equation with boundary conditions at $\pm \infty$. A major difficulty in proving the comparison principle arises from the discontinuity of the Hamiltonian.

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arXiv2020

Optimal Reinsurance under the Mean-Variance Premium Principle to Minimize the Probability of Ruin

We consider the problem of minimizing the probability of ruin by purchasing reinsurance whose premium is computed according to the mean-variance premium principle, a combination of the expected-value and variance premium principles. We derive closed-form expressions of the optimal reinsurance strategy and the corresponding minimum probability of ruin under the diffusion approximation of the classical Cramér-Lundberg risk process perturbed by a diffusion. We find an explicit expression for the reinsurance strategy that maximizes the adjustment coefficient for the classical risk process perturbed by a diffusion. Also, for this risk process, we use stochastic Perron's method to prove that the minimum probability of ruin is the unique viscosity solution of its Hamilton-Jacobi-Bellman equation with appropriate boundary conditions. Finally, we prove that, under an appropriate scaling of the classical risk process, the minimum probability of ruin converges to the minimum probability of ruin under the diffusion approximation.

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arXiv2020

Rate of Convergence of the Probability of Ruin in the Cramér-Lundberg Model to its Diffusion Approximation

We analyze the probability of ruin for the {\it scaled} classical Cramér-Lundberg (CL) risk process and the corresponding diffusion approximation. The scaling, introduced by Iglehart \cite{I1969} to the actuarial literature, amounts to multiplying the Poisson rate $\la$ by $n$, dividing the claim severity by $\sqrtn$, and adjusting the premium rate so that net premium income remains constant. %Therefore, we think of the associated diffusion approximation as being "asymptotic for large values of $\la$." We are the first to use a comparison method to prove convergence of the probability of ruin for the scaled CL process and to derive the rate of convergence. Specifically, we prove a comparison lemma for the corresponding integro-differential equation and use this comparison lemma to prove that the probability of ruin for the scaled CL process converges to the probability of ruin for the limiting diffusion process. Moreover, we show that the rate of convergence for the ruin probability is of order $\mO\big(n^{-1/2}\big)$, and we show that the convergence is {\it uniform} with respect to the surplus. To the best of our knowledge, this is the first rate of convergence achieved for these ruin probabilities, and we show that it is the tightest one in the general case. For the case of exponentially-distributed claims, we are able to improve the approximation arising from the diffusion, attaining a uniform $\mO\big(n^{-k/2}\big)$ rate of convergence for arbitrary $k \in \N$. We also include two examples that illustrate our results.

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arXiv2019

Optimal Dividend Distribution Under Drawdown and Ratcheting Constraints on Dividend Rates

We consider the optimal dividend problem under a habit formation constraint that prevents the dividend rate to fall below a certain proportion of its historical maximum, the so-called drawdown constraint. This is an extension of the optimal Duesenberry's ratcheting consumption problem, studied by Dybvig (1995) [Review of Economic Studies 62(2), 287-313], in which consumption is assumed to be nondecreasing. Our problem differs from Dybvig's also in that the time of ruin could be finite in our setting, whereas ruin was impossible in Dybvig's work. We formulate our problem as a stochastic control problem with the objective of maximizing the expected discounted utility of the dividend stream until bankruptcy, in which risk preferences are embodied by power utility. We semi-explicitly solve the corresponding Hamilton-Jacobi-Bellman variational inequality, which is a nonlinear free-boundary problem. The optimal (excess) dividend rate $c^*_t$ - as a function of the company's current surplus $X_t$ and its historical running maximum of the (excess) dividend rate $z_t$ - is as follows: There are constants $0 < w_α < w_0 < w^*$ such that (1) for $0 < X_t \le w_α z_t$, it is optimal to pay dividends at the lowest rate $αz_t$, (2) for $w_α z_t < X_t < w_0 z_t$, it is optimal to distribute dividends at an intermediate rate $c^*_t \in (αz_t, z_t)$, (3) for $w_0 z_t < X_t < w^* z_t$, it is optimal to distribute dividends at the historical peak rate $z_t$, (4) for $X_t > w^* z_t$, it is optimal to increase the dividend rate above $z_t$, and (5) it is optimal to increase $z_t$ via singular control as needed to keep $X_t \le w^* z_t$. Because, the maximum (excess) dividend rate will eventually be proportional to the running maximum of the surplus, "mountains will have to move" before we increase the dividend rate beyond its historical maximum.

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arXiv2017

Optimality of Excess-Loss Reinsurance under a Mean-Variance Criterion

In this paper, we study an insurer's reinsurance-investment problem under a mean-variance criterion. We show that excess-loss is the unique equilibrium reinsurance strategy under a spectrally negative Lévy insurance model when the reinsurance premium is computed according to the expected value premium principle. Furthermore, we obtain the explicit equilibrium reinsurance-investment strategy by solving the extended Hamilton-Jacobi-Bellman equation.

q-fin.RM↗

arXiv2016

Optimally Investing to Reach a Bequest Goal

We determine the optimal strategy for investing in a Black-Scholes market in order to maximize the probability that wealth at death meets a bequest goal $b$, a type of goal-seeking problem, as pioneered by Dubins and Savage (1965, 1976). The individual consumes at a constant rate $c$, so the level of wealth required for risklessly meeting consumption equals $c/r$, in which $r$ is the rate of return of the riskless asset. Our problem is related to, but different from, the goal-reaching problems of Browne (1997). First, Browne (1997, Section 3.1) maximizes the probability that wealth reaches $b < c/r$ before it reaches $a < b$. Browne's game ends when wealth reaches $b$. By contrast, for the problem we consider, the game continues until the individual dies or until wealth reaches 0; reaching $b$ and then falling below it before death does not count. Second, Browne (1997, Section 4.2) maximizes the expected discounted reward of reaching $b > c/r$ before wealth reaches $c/r$. If one interprets his discount rate as a hazard rate, then our two problems are {\it mathematically} equivalent for the special case for which $b > c/r$, with ruin level $c/r$. However, we obtain different results because we set the ruin level at 0, thereby allowing the game to continue when wealth falls below $c/r$.

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arXiv2016

Minimizing the Probability of Lifetime Drawdown under Constant Consumption

We assume that an individual invests in a financial market with one riskless and one risky asset, with the latter's price following geometric Brownian motion as in the Black-Scholes model. Under a constant rate of consumption, we find the optimal investment strategy for the individual who wishes to minimize the probability that her wealth drops below some fixed proportion of her maximum wealth to date, the so-called probability of {\it lifetime drawdown}. If maximum wealth is less than a particular value, $m^*$, then the individual optimally invests in such a way that maximum wealth never increases above its current value. By contrast, if maximum wealth is greater than $m^*$ but less than the safe level, then the individual optimally allows the maximum to increase to the safe level.

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arXiv2016

Optimal Investment to Minimize the Probability of Drawdown

We determine the optimal investment strategy in a Black-Scholes financial market to minimize the so-called {\it probability of drawdown}, namely, the probability that the value of an investment portfolio reaches some fixed proportion of its maximum value to date. We assume that the portfolio is subject to a payout that is a deterministic function of its value, as might be the case for an endowment fund paying at a specified rate, for example, at a constant rate or at a rate that is proportional to the fund's value.

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