arXiv subjects

Andreea Minca

Publications and source records attributed to Andreea Minca.

16 recordsLinked to original sources

arXiv2023

Approximations of semi-Markov processes and insurance policy valuation

Inspired by a duration-dependent life insurance model, we consider continuous-time semi-Markov jump processes, initially assumed to have a finite state-space. We develop approximations using jump processes that are time-homogeneous Markov, conditioned on a high-intensity Poissonian grid (grid-conditional). Our results are based on a recent adaptation of the uniformization principle, which yields a strongly pathwise convergent sequence of jump processes. In contrast to traditional methods that use classical approximations to integro-differential equation solutions to compute value functions, our approximations result in easily implementable expressions, making them valuable in situations where evaluating pathwise distributional functionals for the original semi-Markov process is challenging. Our homogeneous approximation, initially grid-conditional, evolves into an unconditional version that remains effective under reasonable regularity assumptions. We then relax the finite state-space assumption and show how our results can be extended to a general measurable state-space. We illustrate the practicality of our approach with a disability insurance model, using realistic underlying semi-Markov process parameters.

math.PR↗

arXiv2023

Reinforcement Learning for SBM Graphon Games with Re-Sampling

The Mean-Field approximation is a tractable approach for studying large population dynamics. However, its assumption on homogeneity and universal connections among all agents limits its applicability in many real-world scenarios. Multi-Population Mean-Field Game (MP-MFG) models have been introduced in the literature to address these limitations. When the underlying Stochastic Block Model is known, we show that a Policy Mirror Ascent algorithm finds the MP-MFG Nash Equilibrium. In more realistic scenarios where the block model is unknown, we propose a re-sampling scheme from a graphon integrated with the finite N-player MP-MFG model. We develop a novel learning framework based on a Graphon Game with Re-Sampling (GGR-S) model, which captures the complex network structures of agents' connections. We analyze GGR-S dynamics and establish the convergence to dynamics of MP-MFG. Leveraging this result, we propose an efficient sample-based N-player Reinforcement Learning algorithm for GGR-S without population manipulation, and provide a rigorous convergence analysis with finite sample guarantee.

cs.GT↗

arXiv2023

Ruin-dependent bivariate stochastic fluid processes

This paper presents a novel model for bivariate stochastic fluid processes that incorporate a ruin-dependent behavioral switch. Unlike typical models that assume a shared underlying process, our model allows each process to operate independently until a ruin event in one triggers a change in the other. We develop a mathematical framework for our model, exploring its properties and providing closed-form expressions for approximations of key performance metrics, particularly the joint law of the ruin times. Our approach introduces a class of compatible pathwise approximations to analyze ruin probabilities, which we subsequently study through a matrix-analytic framework.

math.PR↗

arXiv2023

Duration-dependent stochastic fluid processes and solar energy revenue modeling

We endow the classical stochastic fluid process with a duration-dependent Markovian arrival process (DMArP). We show that this provides a flexible model for the revenue of a solar energy generator. In particular, it allows for heavy-tailed interarrival times and for seasonality embedded into the state-space. It generalizes the calendar-time inhomogeneous stochastic fluid process. We provide descriptors of the first return of the revenue process. Our main contribution is based on the uniformization approach, by which we reduce the problem of computing the Laplace transform to the analysis of the process on a stochastic Poissonian grid. Since our process is duration dependent, our construction relies on translating duration form its natural grid to the Poissonian grid. We obtain the Laplace transfrom of the project value based on a novel concept of $n$-bridge and provide an efficient algorithm for computing the duration-level density of the $n$-bridge. Other descriptors such as the Laplace transform of the ruin process are further provided.

math.PR↗

arXiv2023

Optimal Intervention in Economic Networks using Influence Maximization Methods

We consider optimal intervention in the Elliott-Golub-Jackson network model \cite{jackson14} and we show that it can be transformed into an influence maximization-like form, interpreted as the reverse of a default cascade. Our analysis of the optimal intervention problem extends well-established targeting results to the economic network setting, which requires additional theoretical steps. We prove several results about optimal intervention: it is NP-hard and cannot be approximated to a constant factor in polynomial time. In turn, we show that randomizing failure thresholds leads to a version of the problem which is monotone submodular, for which existing powerful approximations in polynomial time can be applied. In addition to optimal intervention, we also show practical consequences of our analysis to other economic network problems: (1) it is computationally hard to calculate expected values in the economic network, and (2) influence maximization algorithms can enable efficient importance sampling and stress testing of large failure scenarios. We illustrate our results on a network of firms connected through input-output linkages inferred from the World Input Output Database.

cs.GT↗

arXiv2023

Ruin Probabilities for Risk Processes in Stochastic Networks

We study multidimensional Cramér-Lundberg risk processes where agents, located on a large sparse network, receive losses form their neighbors. To reduce the dimensionality of the problem, we introduce classification of agents according to an arbitrary countable set of types. The ruin of any agent triggers losses for all of its neighbours. We consider the case when the loss arrival process induced by the ensemble of ruined agents follows a Poisson process with general intensity function that scales with the network size. When the size of the network goes to infinity, we provide explicit ruin probabilities at the end of the loss propagation process for agents of any type. These limiting probabilities depend, in addition to the agents' types and the network structure, on the loss distribution and the loss arrival process. For a more complex risk processes on open networks, when in addition to the internal networked risk processes the agents receive losses from external users, we provide bounds on ruin probabilities.

math.PR↗

arXiv2022

While Stability Lasts: A Stochastic Model of Non-Custodial Stablecoins

The `Black Thursday' crisis in cryptocurrency markets demonstrated deleveraging risks in over-collateralized non-custodial stablecoins. We develop a stochastic model that helps explain deleveraging crises in these over-collateralized systems. In our model, the stablecoin supply is decided by speculators who optimize the profitability of a leveraged position while incorporating the forward-looking cost of collateral liquidations, which involves the endogenous price of the stablecoin. We formally characterize regimes that are interpreted as stable and unstable for the stablecoin. We prove bounds on quadratic variation and the probability of large deviations in the stable domain and we demonstrate distinctly greater price variance in the unstable domain. We identify a deflationary deleveraging spiral by means of a submartingale. These deleveraging spirals, which resemble short squeezes, lead to faster collateral drawdown (and potential shortfalls) and are accompanied by higher price variance, as experienced on Black Thursday. We conclude by discussing non-custodial ways in which the issues raised in this paper can be mitigated.

q-fin.TR↗

arXiv2022

Decentralized Governance of Stablecoins with Closed Form Valuation

We model incentive security in non-custodial stablecoins and derive conditions for participation in a stablecoin system across risk absorbers (vaults/CDPs) and holders of governance tokens. We apply option pricing theory to derive closed form solutions to the stakeholders' problems, and to value their positions within the capital structure of the stablecoin. We derive the optimal interest rate that is incentive compatible, as well as conditions for the existence of equilibria without governance attacks, and discuss implications for designing secure protocols.

econ.GN↗

arXiv2021

Epidemic Spreading and Equilibrium Social Distancing in Heterogeneous Networks

We study a multi-type SIR epidemic process among a heterogeneous population that interacts through a network. When we base social contact on a random graph with given vertex degrees, we give limit theorems on the fraction of infected individuals. For a given social distancing individual strategies, we establish the epidemic reproduction number $R_0$ which can be used to identify network vulnerability and inform vaccination policies. In the second part of the paper we study the equilibrium of the social distancing game, in which individuals choose their social distancing level according to an anticipated global infection rate, which then must equal the actual infection rate following their choices. We give conditions for the existence and uniqueness of equilibrium. For the case of random regular graphs, we show that voluntary social distancing will always be socially sub-optimal.

physics.soc-ph↗

arXiv2021

(In)Stability for the Blockchain: Deleveraging Spirals and Stablecoin Attacks

We develop a model of stable assets, including non-custodial stablecoins backed by cryptocurrencies. Such stablecoins are popular methods for bootstrapping price stability within public blockchain settings. We derive fundamental results about dynamics and liquidity in stablecoin markets, demonstrate that these markets face deleveraging feedback effects that cause illiquidity during crises and exacerbate collateral drawdown, and characterize stable dynamics of the system under particular conditions. The possibility of such `deleveraging spirals' was first predicted in the initial release of our paper in 2019 and later directly observed during the `Black Thursday' crisis in Dai in 2020. From these insights, we suggest design improvements that aim to improve long-term stability. We also introduce new attacks that exploit arbitrage-like opportunities around stablecoin liquidations. Using our model, we demonstrate that these can be profitable. These attacks may induce volatility in the `stable' asset and cause perverse incentives for miners, posing risks to blockchain consensus. A variant of such attacks also later occurred during Black Thursday, taking the form of mempool manipulation to clear Dai liquidation auctions at near zero prices, costing $8m.

q-fin.TR↗

arXiv2020

Stablecoins 2.0: Economic Foundations and Risk-based Models

Stablecoins are one of the most widely capitalized type of cryptocurrency. However, their risks vary significantly according to their design and are often poorly understood. We seek to provide a sound foundation for stablecoin theory, with a risk-based functional characterization of the economic structure of stablecoins. First, we match existing economic models to the disparate set of custodial systems. Next, we characterize the unique risks that emerge in non-custodial stablecoins and develop a model framework that unifies existing models from economics and computer science. We further discuss how this modeling framework is applicable to a wide array of cryptoeconomic systems, including cross-chain protocols, collateralized lending, and decentralized exchanges. These unique risks yield unanswered research questions that will form the crux of research in decentralized finance going forward.

econ.GN↗

arXiv2020

Provable Fictitious Play for General Mean-Field Games

We propose a reinforcement learning algorithm for stationary mean-field games, where the goal is to learn a pair of mean-field state and stationary policy that constitutes the Nash equilibrium. When viewing the mean-field state and the policy as two players, we propose a fictitious play algorithm which alternatively updates the mean-field state and the policy via gradient-descent and proximal policy optimization, respectively. Our algorithm is in stark contrast with previous literature which solves each single-agent reinforcement learning problem induced by the iterates mean-field states to the optimum. Furthermore, we prove that our fictitious play algorithm converges to the Nash equilibrium at a sublinear rate. To the best of our knowledge, this seems the first provably convergent single-loop reinforcement learning algorithm for mean-field games based on iterative updates of both mean-field state and policy.

cs.LG↗

arXiv2020

Cascading Losses in Reinsurance Networks

We develop a model for contagion in reinsurance networks by which primary insurers' losses are spread through the network. Our model handles general reinsurance contracts, such as typical excess of loss contracts. We show that simpler models existing in the literature--namely proportional reinsurance--greatly underestimate contagion risk. We characterize the fixed points of our model and develop efficient algorithms to compute contagion with guarantees on convergence and speed under conditions on network structure. We characterize exotic cases of problematic graph structure and nonlinearities, which cause network effects to dominate the overall payments in the system. We lastly apply our model to data on real world reinsurance networks. Our simulations demonstrate the following: (1) Reinsurance networks face extreme sensitivity to parameters. A firm can be wildly uncertain about its losses even under small network uncertainty. (2) Our sensitivity results reveal a new incentive for firms to cooperate to prevent fraud, as even small cases of fraud can have outsized effect on the losses across the network. (3) Nonlinearities from excess of loss contracts obfuscate risks and can cause excess costs in a real world system.

q-fin.RM↗

arXiv2019

Clustering Degree-Corrected Stochastic Block Model with Outliers

For the degree corrected stochastic block model in the presence of arbitrary or even adversarial outliers, we develop a convex-optimization-based clustering algorithm that includes a penalization term depending on the positive deviation of a node from the expected number of edges to other inliers. We prove that under mild conditions, this method achieves exact recovery of the underlying clusters. Our synthetic experiments show that our algorithm performs well on heterogeneous networks, and in particular those with Pareto degree distributions, for which outliers have a broad range of possible degrees that may enhance their adversarial power. We also demonstrate that our method allows for recovery with significantly lower error rates compared to existing algorithms.

cs.LG↗

arXiv2015

On the central management of risk networks

We introduce a family of risk networks composed from a) several subsidiary branches $U_i(t), i=1,...,I$ necessary for coping with different types of risks, which must all be kept above $0$, and b) a central branch (CB) which bails out the subsidiaries whenever necessary. Ruin occurs when the central branch is ruined. We find out that with one subsidiary ($I=1$), the finite time ruin probability of the central branch may be explicitly written out in terms of the finite time ruin probability of the subsidiary, provided that the CB in the absence of subsidiary bailouts is a deterministic drift. To study other problems, like for example the optimization of dividends to the CB with one subsidiary over a barrier, it is convenient to restrict to the case of phase-type claims to the subsidiary, and study the Markovian phase process at the moments when the CB process reaches new minima. The resulting structure is quite close to that of the phase of a PH/G/1 queue at the moments when it reaches new minima, and this yields in principle numeric approaches to several problems, based on the iterative calculation of the Laplace transforms of the upwards and downwards ladder times (also called excursions, and busy periods). In this work we propose a different approach of further providing matrix exponential approximations for the distributions of the ladder times. The advantage of this approach is that once a SNMAP approximation is obtained, many similar problems may be solved just by applying the recently developed scale matrix methodology. A numeric experiment for the CB barrier optimization problem is provided.

math.PR↗

arXiv2011

Resilience to Contagion in Financial Networks

Propagation of balance-sheet or cash-flow insolvency across financial institutions may be modeled as a cascade process on a network representing their mutual exposures. We derive rigorous asymptotic results for the magnitude of contagion in a large financial network and give an analytical expression for the asymptotic fraction of defaults, in terms of network characteristics. Our results extend previous studies on contagion in random graphs to inhomogeneous directed graphs with a given degree sequence and arbitrary distribution of weights. We introduce a criterion for the resilience of a large financial network to the insolvency of a small group of financial institutions and quantify how contagion amplifies small shocks to the network. Our results emphasize the role played by "contagious links" and show that institutions which contribute most to network instability in case of default have both large connectivity and a large fraction of contagious links. The asymptotic results show good agreement with simulations for networks with realistic sizes.

q-fin.RM↗