Search Results (40)

This paper investigates a robust optimal reinsurance and investment problem for an insurance company operating in a Markov-modulated financial market. The insurer’s surplus process is modeled by a diffusion process with jumps, which is correlated with financial risky assets through a common shock structure. The economic regime switches according to a continuous-time Markov chain. To address model uncertainty concerning both diffusion and jump components, we formulate the problem within a robust optimal control framework. By applying the Girsanov theorem for semimartingales, we derive the dynamics of the wealth process under an equivalent martingale measure. We then establish the associated Hamilton–Jacobi–Bellman (HJB) equation, which constitutes a coupled system of nonlinear second-order integro-differential equations. An explicit form of the relative entropy penalty function is provided to quantify the cost of deviating from the reference model. The theoretical results furnish a foundation for numerical solutions using actor–critic reinforcement learning algorithms. Full article

This paper demonstrates that perfectly calibrating a multi-asset model to observed market prices of all basket call options is insufficient to uniquely determine the price of a best-of call option. Previous research on multi-asset option pricing has primarily focused on complete market settings or assumed specific parametric models, leaving fundamental questions about model risk and pricing uniqueness in incomplete markets inadequately addressed. This limitation has critical practical implications: derivatives practitioners who hedge best-of options using basket-equivalent instruments face fundamental distributional uncertainty that compounds the well-recognized non-linearity challenges. We establish this non-uniqueness using convex analysis (extreme ray characterization demonstrating geometric incompatibility between payoff structures), measure theory (explicit construction of distinct equivalent probability measures), and geometric analysis (payoff structure comparison). Specifically, we prove that the set of equivalent probability measures consistent with observed basket prices contains distinct measures yielding different best-of option prices, with explicit no-arbitrage bounds $[a_K,b_K]$ quantifying this uncertainty. Our theoretical contribution provides the first rigorous mathematical foundation for several empirically observed market phenomena: wide bid-ask spreads on extremal options, practitioners’ preference for over-hedging strategies, and substantial model reserves for exotic derivatives. We demonstrate through concrete examples that substantial model risk persists even with perfect basket calibration and equivalent measure constraints. For risk-neutral pricing applications, equivalent martingale measure constraints can be imposed using optimal transport theory, though this requires additional mathematical complexity via Schrödinger bridge techniques while preserving our fundamental non-uniqueness results. The findings establish that additional market instruments beyond basket options are mathematically necessary for robust exotic derivative pricing. Full article

We focus on solving stochastic differential equations driven by jump processes (SDEJs) with measurable drifts that may exhibit quadratic growth. Our approach leverages a space transformation and Itô-Krylov’s formula to effectively eliminate the singular component of the drift, allowing us to obtain a transformed SDEJ that satisfies classical solvability conditions. By applying the inverse transformation proven to be a one-to-one mapping, we retrieve the solution to the original equation. This methodology offers several key advantages. First, it extends the well-known result of Le Gall (1984) from Brownian-driven SDEs to the jump process setting, broadening the range of applicable stochastic models. Second, it provides a robust framework for handling singular drifts, enabling the resolution of equations that would otherwise be intractable. Third, the approach accommodates drifts with quadratic growth, making it particularly relevant for financial modeling, insurance risk assessment, and other applications where such growth behavior is common. Finally, the inclusion of multiple examples illustrates the practical effectiveness of our method, demonstrating its flexibility and applicability to real-world problems. Full article

In this work, we propose a wavelet-based framework for estimating the derivatives of a density function in the setting of continuous, stationary, and ergodic processes. Our primary focus is the derivation of the integrated mean square error (IMSE) over compact subsets of ${\mathbb{R}}^d$, which provides a quantitative measure of the estimation accuracy. In addition, a uniform convergence rate and normality are established. To establish the asymptotic behavior of the proposed estimators, we adopt a martingale approach that accommodates the ergodic nature of the underlying processes. Importantly, beyond ergodicity, our analysis does not require additional assumptions regarding the data. By demonstrating that the wavelet methodology remains valid under these weaker dependence conditions, we extend earlier results originally developed in the context of independent observations. Full article

In this study, we present a novel mathematical framework for pricing financial derivates and modelling asset behaviour by bringing together fractional Brownian motion (fBm), fuzzy logic, and jump processes, all aligned with no-arbitrage principle. In particular, our mathematical developments include fBm defined through Mandelbrot-Van Ness kernels, and advanced mathematical tools such Molchan martingale and BDG inequalities ensuring rigorous theoretical validity. We bring together these different concepts to model uncertainties like sudden market shocks and investor sentiment, providing a fresh perspective in financial mathematics and derivatives pricing. By using fuzzy logic, we incorporate subject factors such as market optimism or pessimism, adjusting volatility dynamically according to the current market environment. Fractal mathematics with the Hurst exponent close to zero reflecting rough market conditions and fuzzy set theory are combined with jumps, representing sudden market changes to capture more realistic asset price movements. We also bridge the gap between complex stochastic equations and solvable differential equations using tools like Feynman-Kac approach and Girsanov transformation. We present simulations illustrating plausible scenarios ranging from pessimistic to optimistic to demonstrate how this model can behave in practice, highlighting potential advantages over classical models like the Merton jump diffusion and Black-Scholes. Overall, our proposed model represents an advancement in mathematical finance by integrating fractional stochastic processes with fuzzy set theory, thus revealing new perspectives on derivative pricing and risk-free valuation in uncertain environments. Full article

We introduce a new coherent risk measure, the minimal-entropy risk measure, which is built on the minimal-entropy $\sigma$-martingale measure—a concept inspired by the well-known minimal-entropy martingale measure used in option pricing. While the minimal-entropy martingale measure is commonly used for pricing and hedging, the minimal-entropy $\sigma$-martingale measure has not previously been studied, nor has it been analyzed as a traditional risk measure. We address this gap by clearly defining this new risk measure and examining its fundamental properties. In addition, we revisit the entropic risk measure, typically expressed through an exponential formula. We provide an alternative definition using a supremum over Kullback–Leibler divergences, making its connection to entropy clearer. We verify important properties of both risk measures, such as convexity and coherence, and extend these concepts to dynamic situations. We also illustrate their behavior in scenarios involving optimal risk transfer. Our results link entropic concepts with incomplete-market pricing and demonstrate how both risk measures share a unified entropy-based foundation. Full article

The Dupire formula is a very useful tool for pricing financial derivatives. This paper is dedicated to deriving the aforementioned formula for the European call option in the space of distributions by applying a mathematically rigorous approach developed in our previous paper concerning the case of the Margrabe option. We assume that the underlying asset is described by the Merton jump-diffusion model. Using this stochastic process allows us to take into account jumps in the price of the considered asset. Moreover, we assume that the instantaneous interest rate follows the Merton model (1973). Therefore, in contrast to the models combining a constant interest rate and a continuous underlying asset price process, frequently observed in the literature, applying both stochastic processes could accurately reflect financial market behaviour. Moreover, we illustrate the possibility of using the minimal entropy martingale measure as the risk-neutral measure in our approach. Full article

We survey some of the main developments in probability theory during the so-called “heroic age”; that is, the period from the nineteen twenties to the early nineteen fifties. It was during the heroic age that probability finally attained the status of a mathematical discipline in the full sense of the term, with a complete axiomatic basis and incontestable standards of intellectual rigor to complement and support the extraordinarily rich intuitive content that has always been part of probability theory from its very inception. The axiomatic basis and mathematical rigor are themselves rooted in the abstract theory of measure and integration, which now comprises the very bedrock of modern probability, and among the central characters in the measure-theoretic “re-invention” of probability during the heroic age one finds, in particular, Kolmogorov, Doob, Lévy, Khinchin and Feller, each of whom fundamentally shaped the structure of modern probability. In this survey, we attempt a brief sketch of some of the main contributions of each of these pioneers. Full article

This paper focuses on the pricing problem of binary options under stochastic interest rates, stochastic volatility, and a mixed exponential jump diffusion model. Considering the negative interest rates in the market in recent years, this paper assumes that the stochastic interest rate follows the Hull–White (HW) model. In addition, we assume that the stochastic volatility follows the Heston volatility model, and the price of the underlying asset follows the jump diffusion model in which the jumps follow the mixed exponential jump model. Considering these factors comprehensively, the mixed exponential jump diffusion of the Heston–HW (abbreviated as MEJ-Heston–HW) model is established. Using the idea of measure transformation, the pricing formula of binary call options is derived by the martingale method, eigenfunction, and Fourier transform. Finally, the effects of the volatility term and the parameters of the mixed-exponential jump diffusion model on the option price in the O-U process are analyzed. In the numerical simulation, compared with the double exponential jump Heston–HW (abbreviated as DEJ-Heston–HW) model and the Heston–HW model, the mixed exponential jump model is an extension of the double exponential jump model, which can approximate any distribution in the sense of weak convergence, including arbitrary discrete distributions, normal distributions, and various thick-tailed distributions. Therefore, the MEJ-Heston–HW model adopted in this paper can better describe the price of the underlying asset. Full article

In this paper, we explore the connection between a single index model under the real-world probability measure and martingale pricing via minimal relative entropy or Esscher transform, within the context of a one-period market model, possibly incomplete, with multiple risky assets and a single risk-free asset. The minimal relative entropy martingale measure and the Esscher martingale measure coincide in such a market, provided they both exist. From their Radon–Nikodym derivative, we derive a portfolio of risky assets in a natural way, termed portfolio G. Our analysis shows that pricing using the Esscher or minimal relative entropy martingale measure is equivalent to a single index model (SIM) incorporating portfolio G. In the special case of elliptical returns, portfolio G coincides with the classical tangency portfolio. Furthermore, in the case of jointly normal returns, Esscher or minimal relative entropy martingale measure pricing is equivalent to CAPM pricing. Full article

We develop a measure-theoretic framework for dynamic Shapley values in Banach spaces, extending classical cooperative game theory to continuous-time, infinite-dimensional settings. We prove the existence and uniqueness of strong solutions to stochastic differential equations modeling competence evolution in Banach spaces, establishing sample path continuity and moment estimates. The dynamic Shapley value is rigorously defined as a càdlàg stochastic process with an axiomatic characterization. We derive a martingale representation for this process and establish its asymptotic properties, including a strong law of large numbers and a functional central limit theorem under α-mixing conditions. This framework provides a rigorous basis for analyzing dynamic value attribution in abstract spaces, with potential applications to economic and game-theoretic models. Full article

Recognizing the importance of incorporating different risk measures in the portfolio management model, this paper examines the dynamic mean-risk portfolio optimization problem using both variance and value at risk (VaR) as risk measures. By employing the martingale approach and integrating the quantile optimization technique, we provide a solution framework for this problem. We demonstrate that, under a general market setting, the optimal terminal wealth may exhibit different patterns. When the market parameters are deterministic, we derive the closed-form solution for this problem. Examples are provided to illustrate the solution procedure of our method and demonstrate the benefits of our dynamic portfolio model compared to its static counterpart. Full article

The detection of seismic activity precursors as part of an alarm system will provide opportunities for minimization of the social and economic impact caused by earthquakes. It has long been envisaged, and a growing body of empirical evidence suggests that the Earth’s electromagnetic field could contain precursors to seismic events. The ability to capture and monitor electromagnetic field activity has increased in the past years as more sensors and methodologies emerge. Missions such as Swarm have enabled researchers to access near-continuous observations of electromagnetic activity at second intervals, allowing for more detailed studies on weather and earthquakes. In this paper, we present an approach designed to detect anomalies in electromagnetic field data from Swarm satellites. This works towards developing a continuous and effective monitoring system of seismic activities based on SWARM measurements. We develop an enhanced form of a probabilistic model based on the Martingale theories that allow for testing the null hypothesis to indicate abnormal changes in electromagnetic field activity. We evaluate this enhanced approach in two experiments. Firstly, we perform a quantitative comparison on well-understood and popular benchmark datasets alongside the conventional approach. We find that the enhanced version produces more accurate anomaly detection overall. Secondly, we use three case studies of seismic activity (namely, earthquakes in Mexico, Greece, and Croatia) to assess our approach and the results show that our method can detect anomalous phenomena in the electromagnetic data. Full article

A mean-field linear quadratic stochastic (MF-SLQ for short) optimal control problem with hybrid disturbances and cross terms in a finite horizon is concerned. The state equation is a systems driven by the Wiener process and the Poisson random martingale measure disturbed by some stochastic perturbations. The cost functional is also disturbed, which means more general cases could be characterized, especially when extra environment perturbations exist. In this paper, the well-posedness result on the jump diffusion systems is obtained by the fixed point theorem and also the solvability of the MF-SLQ problem. Actually, by virtue of adjoint variables, classic variational calculus, and some dual representation, an optimal condition is derived. Throughout our research, in order to connect the optimal control and the state directly, two Riccati differential equations, a BSDE with random jumps and an ordinary equation (ODE for short) on disturbance terms are obtained by a decoupling technique, which provide an optimal feedback regulator. Meanwhile, the relationship between the two Riccati equations and the so-called mean-field stochastic Hamilton system is established. Consequently, the optimal value is characterized by the initial state, disturbances, and original value of the Riccati equations. Finally, an example is provided to illustrate our theoretic results. Full article

We construct a model where, at each time instance, risky securities can only take a limited number of values and the equity-linked policy sold by the insurer to policyholders pays benefits linked to these securities. Since the number of states in the model exceeds the number of securities in the (incomplete) market, the martingale measure is not unique, posing a problem in pricing insurance instruments. In this framework, we consider how a super-replicating strategy violates the assumption of absence of arbitrage, yet simultaneously allows the insurance company to fully hedge against financial risk. Since the super-replicating strategy, when considered alone, would be too costly for any rational insured person, through the definition of the safety loading, we demonstrate how the insurer can still hedge against financial risk, albeit at the expense of increasing its exposure to demographic risk. This approach does not aim to show how the pricing of the index-linked policy can actually be performed but rather highlights how risk theory-based approaches (via the definition of the profit and loss random variable) enable the management of the trade-off between financial risk and demographic risk. Full article