CyberSec Research

This paper addresses the problem of determining the optimal time for an individual to convert retirement savings into a lifetime annuity. The individual invests their wealth into a dividend-paying fund that follows the dynamics of a geometric Brownian motion, exposing them to market risk. At the same time, they face an uncertain lifespan influenced by a stochastic mortality force. The latter is modelled as a piecewise deterministic Markov process (PDMP), which captures sudden and unpredictable changes in the individual's mortality force. The individual aims to maximise expected lifetime linear utility from consumption and bequest, balancing market risk and longevity risk under an irreversible, all-or-nothing annuitization decision. The problem is formulated as a three-dimensional optimal stopping problem and, by exploiting the PDMP structure, it is reduced to a sequence of nested one-dimensional problems. We solve the optimal stopping problem and find a rich structure for the optimal annuitization rule, which cover all parameter specifications. Our theoretical analysis is complemented by a numerical example illustrating the impact of a single health shock on annuitization timing, along with a sensitivity analysis of key model parameters.

Strassen's theorem asserts that for given marginal probabilities $\mu,\nu$ there exists a martingale starting in $\mu$ and terminating in $\nu$ if and only if $\mu,\nu$ are in convex order. From a financial perspective, it guarantees the existence of market-consistent martingale pricing measures for arbitrage-free prices of European call options and thus plays a fundamental role in robust finance. Arbitrage-free prices of American options demand a stronger version of martingales which are 'biased' in a specific sense. In this paper, we derive an extension of Strassen's theorem that links them to an appropriate strengthening of the convex order. Moreover, we provide a characterization of this order through integrals with respect to compensated Poisson processes.

In this paper we apply a probabilistic approach to analyze the impact of contagious default among investment group members. A general formula is given to compute the group survival probability with the presence of contagion effect. Special cases of this probability model are examined. In particular, we show that if the investment group is homogeneous, defined in the paper, then including more members into the group will eventually lead to default with probability 1, contrasting with the non-contagious scenario, where the default probability increases monotonically with respect to the group size. Also, we provide an upper bound of the optimal group size under the homogeneous setup; so, one can run a linear search with finite time to locate this optimizer.

We revisit the problem of portfolio selection, where an investor maximizes utility subject to a risk constraint. Our framework is very general and accommodates a wide range of utility and risk functionals, including non-concave utilities such as S-shaped utilities from prospect theory and non-convex risk measures such as Value at Risk. Our main contribution is a novel and complete characterization of well-posedness for utility-risk portfolio selection in one period that takes the interplay between the utility and the risk objectives fully into account. We show that under mild regularity conditions the minimal necessary and sufficient condition for well-posedness is given by a very simple either-or criterion: either the utility functional or the risk functional need to satisfy the axiom of sensitivity to large losses. This allows to easily describe well-posedness or ill-posedness for many utility-risk pairs, which we illustrate by a large number of examples. In the special case of expected utility maximization without a risk constraint (but including non-concave utilities), we show that well-posedness is fully characterised by the asymptotic loss-gain ratio, a simple and interpretable quantity that describes the investor's asymptotic relative weighting of large losses versus large gains.

In this article, we study optimal investment and consumption in an incomplete stochastic factor model for a power utility investor on the infinite horizon. When the state space of the stochastic factor is finite, we give a complete characterisation of the well-posedness of the problem, and provide an efficient numerical algorithm for computing the value function. When the state space is a (possibly infinite) open interval and the stochastic factor is represented by an It\^o diffusion, we develop a general theory of sub- and supersolutions for second-order ordinary differential equations on open domains without boundary values to prove existence of the solution to the Hamilton-Jacobi-Bellman (HJB) equation along with explicit bounds for the solution. By characterising the asymptotic behaviour of the solution, we are also able to provide rigorous verification arguments for various models, including -- for the first time -- the Heston model. Finally, we link the discrete and continuous setting and show that that the value function in the diffusion setting can be approximated very efficiently through a fast discretisation scheme.

This paper investigates the continuous-time limit of score-driven models with long memory. By extending score-driven models to incorporate infinite-lag structures with coefficients exhibiting heavy-tailed decay, we establish their weak convergence, under appropriate scaling, to fractional Ornstein-Uhlenbeck processes with Hurst parameter $H < 1/2$. When score-driven models are used to characterize the dynamics of volatility, they serve as discrete-time approximations for rough volatility. We present several examples, including EGARCH($\infty$) whose limits give rise to a new class of rough volatility models. Building on this framework, we carry out numerical simulations and option pricing analyses, offering new tools for rough volatility modeling and simulation.

Options are contingent claims regarding the value of underlying assets. The Black-Scholes formula provides a road map for pricing these options in a risk-neutral setting, justified by a delta hedging argument in which countervailing positions of appropriate size are taken in the underlying asset. However, what if an underlying asset is expensive to trade? It might be better to hedge with a different, but related asset that is cheaper to trade. This study considers this question in a setting in which the option written on a portfolio containing $\alpha$ shares of one asset $S_{t_1}$ and $(1-\alpha)$ shares of another security $S_{t_2}$ correlated with $S_{t_1}$. We suppose that the asset is hedged against only one of $S_{t_1}$ or $S_{t_2}.$ In the case of $\alpha=0~\text{or}~1$ we can consider this model to cover the case where an option on one asset is hedged against either the ``right" (underlying) asset or the``wrong" (related, different) asset. We hedge our portfolio on simulated data using varying trading intervals, correlation coefficients, $\rho$ and transaction costs. We calculated the risk-adjusted values ($RAV$) as the risk and return measures to make meaningful decisions on when to trade $S_{t_1}$ or $S_{t_2}.$ From the conclusions made based on $RAV,$ the size of the market price of risk and that of transaction costs on both assets are key to making a decision while hedging. From our results, trading the wrong asset can be opted for when $\rho$ is very high for reasonably small transaction costs for either of the assets.

We study the problem of optimal liquidity withdrawal for a representative liquidity provider (LP) in an automated market maker (AMM). LPs earn fees from trading activity but are exposed to impermanent loss (IL) due to price fluctuations. While existing work has focused on static provision and exogenous exit strategies, we characterise the optimal exit time as the solution to a stochastic control problem with an endogenous stopping time. Mathematically, the LP's value function is shown to satisfy a Hamilton-Jacobi-Bellman quasi-variational inequality, for which we establish uniqueness in the viscosity sense. To solve the problem numerically, we develop two complementary approaches: a Euler scheme based on operator splitting and a Longstaff-Schwartz regression method. Calibrated simulations highlight how the LP's optimal exit strategy depends on the oracle price volatility, fee levels, and the behaviour of arbitrageurs and noise traders. Our results show that while arbitrage generates both fees and IL, the LP's optimal decision balances these opposing effects based on the pool state variables and price misalignments. This work contributes to a deeper understanding of dynamic liquidity provision in AMMs and provides insights into the sustainability of passive LP strategies under different market regimes.

We introduce a novel rough Bergomi (rBergomi) model featuring a variance-driven exponentially weighted moving average (EWMA) time-dependent Hurst parameter $H_t$, fundamentally distinct from recent machine learning and wavelet-based approaches in the literature. Our framework pioneers a unified rough differential equation (RDE) formulation grounded in rough path theory, where the Hurst parameter dynamically adapts to evolving volatility regimes through a continuous EWMA mechanism tied to instantaneous variance. Unlike discrete model-switching or computationally intensive forecasting methods, our approach provides mathematical tractability while capturing volatility clustering and roughness bursts. We rigorously establish existence and uniqueness of solutions via rough path theory and derive martingale properties. Empirical validation on diverse asset classes including equities, cryptocurrencies, and commodities demonstrates superior performance in capturing dynamics and out-of-sample pricing accuracy. Our results show significant improvements over traditional constant-Hurst models.

We study the optimal liquidation problem in both lit and dark pools for investors facing execution uncertainty in a continuous-time setting with market impact. First, we design an optimal make--take fee policy for a large investor liquidating her position across both pools, interacting with small investors who pay trading fees. We explicitly characterize the large investor's optimal liquidation strategies in both lit and dark pools using BSDEs under a compensation scheme proposed by an exchange to mitigate market impact in the lit venue. Second, we consider a purely competitive model with major--minor traders in the absence of regulation. We provide explicit solutions to the associated HJB--Fokker--Planck system. Finally, we illustrate our results through numerical experiments, comparing market impact under a regulated market with a strategic large investor to that in a purely competitive market with both small and large investors.

This paper considers two investors who perform mean-variance portfolio selection with asymmetric information: one knows the true stock dynamics, while the other has to infer the true dynamics from observed stock evolution. Their portfolio selection is interconnected through relative performance concerns, i.e., each investor is concerned about not only her terminal wealth, but how it compares to the average terminal wealth of both investors. We model this as Stackelberg competition: the partially-informed investor (the "follower") observes the trading behavior of the fully-informed investor (the "leader") and decides her trading strategy accordingly; the leader, anticipating the follower's response, in turn selects a trading strategy that best suits her objective. To prevent information leakage, the leader adopts a randomized strategy selected under an entropy-regularized mean-variance objective, where the entropy regularizer quantifies the randomness of a chosen strategy. The follower, on the other hand, observes only the actual trading actions of the leader (sampled from the randomized strategy), but not the randomized strategy itself. Her mean-variance objective is thus a random field, in the form of an expectation conditioned on a realized path of the leader's trading actions. In the idealized case of continuous sampling of the leader's trading actions, we derive a Stackelberg equilibrium where the follower's trading strategy depends linearly on the actual trading actions of the leader and the leader samples her trading actions from Gaussian distributions. In the realistic case of discrete sampling of the leader's trading actions, the above becomes an $\epsilon$-Stackelberg equilibrium.

We study mean-field stochastic differential equations (SDEs) driven by G-Brownian motion, extending recent work on existence and uniqueness by developing a full quantitative stability framework. Our main contribution is the construction of an intrinsic stability modulus that provides explicit bounds on the sensitivity of solutions with respect to perturbations in initial data (and, indirectly, coefficients). Using Bihari-Osgood type inequalities under G-expectation, we establish sharp continuity estimates for the data-to-solution map and analyze the asymptotic properties of the stability modulus. In particular, we identify contraction behavior on short horizons, leading to a contraction principle that guarantees uniqueness and global propagation of stability. The results apply under non-Lipschitz, non-deterministic coefficients with square-integrable initial data, thereby significantly broadening the scope of mean-field G-SDEs. Beyond existence and uniqueness, our framework quantifies robustness of solutions under volatility uncertainty, opening new directions for applications in stochastic control, risk management, and mean-field models under ambiguity.

This paper presents a unified exposition of rough path methods applied to optimal control, robust filtering, and optimal stopping, addressing a notable gap in the existing literature where no single treatment covers all three areas. By bringing together key elements from Lyons' theory of rough paths, Gubinelli's controlled rough paths, and related developments, we recast these classical problems within a deterministic, pathwise framework. Particular emphasis is placed on providing detailed proofs and explanations where these have been absent or incomplete, culminating in a proof of the central verification theorem, which is another key contribution of this paper. This result establishes the rigorous connection between candidate solutions to optimal control problems and the Hamilton-Jacobi-Bellman equation in the rough path setting. Alongside these contributions, we identify several theoretical challenges -- most notably, extending the verification theorem and associated results to general p-variation with -- and outline promising directions for future research. The paper is intended as a self-contained reference for researchers seeking to apply rough path theory to decision-making problems in stochastic analysis, mathematical finance, and engineering.

In this paper, we first conduct a study of the portfolio selection problem, incorporating both exogenous (proportional) and endogenous (resulting from liquidity risk, characterized by a stochastic process) transaction costs through the utility-based approach. We also consider the intrinsic relationship between these two types of costs. To address the associated nonlinear two-dimensional Hamilton-Jacobi-Bellman (HJB) equation, we propose an innovative deep learning-driven policy iteration scheme with three key advantages: i) it has the potential to address the curse of dimensionality; ii) it is adaptable to problems involving high-dimensional control spaces; iii) it eliminates truncation errors. The numerical analysis of the proposed scheme, including convergence analysis in a general setting, is also discussed. To illustrate the impact of these two types of transaction costs on portfolio choice, we conduct through numerical experiments using three typical utility functions.

We use classical tools from calculus of variations to formally derive necessary conditions for a Markov control to be optimal in a standard finite time horizon stochastic control problem. As an example, we solve the well-known Merton portfolio optimization problem.

This paper presents a derivation of the explicit price for the perpetual American put option in the Black-Scholes model, time-capped by the first drawdown epoch beyond a predefined level. We demonstrate that the optimal exercise strategy involves executing the option when the asset price first falls below a specified threshold. The proof relies on martingale arguments and the fluctuation theory of L\'evy processes. To complement the theoretical findings, we provide numerical analysis.

We study an American option pricing problem with liquidity risks and transaction fees. As endogenous transaction costs, liquidity risks of the underlying asset are modeled by a mean-reverting process. Transaction fees are exogenous transaction costs and are assumed to be proportional to the trading amount, with the long-run liquidity level depending on the proportional transaction costs rate. Two nonlinear partial differential equations are established to characterize the option values for the holder and the writer, respectively. To illustrate the impact of these transaction costs on option prices and optimal exercise prices, we apply the alternating direction implicit method to solve the linear complementarity problem numerically. Finally, we conduct model calibration from market data via maximum likelihood estimation, and find that our model incorporating liquidity risks outperforms the Leland model significantly.

This paper presents a derivation of the explicit price for the perpetual American put option time-capped by the first drawdown epoch beyond a predefined level. We consider the market in which an asset price is described by geometric L\'evy process with downward exponential jumps. We show that the optimal stopping rule is the first time when the asset price gets below a special value. The proof relies on martingale arguments and the fluctuation theory of L\'evy processes. We also provide a numerical analysis.