Search Results (29)

We study a finite-horizon optimal job-switching and portfolio allocation problem where an agent faces a mandatory retirement date. The agent can freely switch between two jobs with differing levels of income and leisure. The financial market consists of a risk-free asset and a risky asset, with the agent making dynamic consumption, investment, and job-switching decisions to maximize lifetime utility. The utility function follows a Cobb–Douglas form, incorporating both consumption and leisure preferences. Using a dual-martingale approach, we derive the optimal policies and establish a verification theorem confirming their optimality. Our results provide insights into the trade-offs between labor income and leisure over a finite career horizon and their implications for retirement planning and investment behavior. Full article

In this paper, we introduce the martingale Hardy spaces and $BMO$ spaces generated by an operator T in continuous time and establish the atomic decomposition theorem of the space $H_p^T$ under the condition that T is predictable. We show that the $BM{O}_q$ spaces generated by the operator T are all equivalent and consider the sharp operator. Using the real interpolation method, we identify the interpolation spaces between the Hardy spaces and the $BMO$ spaces. With the aid of atomic decomposition, we establish some martingale inequalities between the Hardy spaces generated by two different operators. 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

This study introduces a wavelet-based framework for estimating derivatives of a general regression function within discrete-time, stationary ergodic processes. The analysis focuses on deriving the integrated mean squared error (IMSE) over compact subsets of ${\mathbb{R}}^d$, while also establishing rates of uniform convergence and the asymptotic normality of the proposed estimators. To investigate their asymptotic behavior, we adopt a martingale-based approach specifically adapted to the ergodic nature of the data-generating process. Importantly, the framework imposes no structural assumptions beyond ergodicity, thereby circumventing restrictive dependence conditions. By establishing the limiting behavior of the wavelet estimators under these minimal assumptions, the results extend existing findings for independent data and highlight the flexibility of wavelet methods in more general stochastic settings. 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

This article studies the theoretical properties of the numerical scheme for backward stochastic differential equations, extending the relevant results of Briand et al. with more general assumptions. To be more precise, the Brown motion will be approximated using the sum of a sequence of martingale differences or a sequence of i.i.d. Gaussian variables instead of the i.i.d. Bernoulli sequence. We cope with an adaptation problem of $Y^n$ by defining a new process ${\widehat{Y}}^n$; then, we can obtain the Donsker-type theorem for numerical solutions using a similar method to Briand et al. Full article

In this paper, we consider a class of stochastic differential equations driven by multiplicative $\alpha$-stable ($0<\alpha <2$) Lévy noises. Firstly, we show that there exists a unique strong solution under a local one-sided Lipschitz condition and a general non-explosion condition. Next, the weak Feller and stationary properties are derived. Furthermore, a concrete sufficient condition for the coefficients is presented, which is different from the conditions for SDEs driven by Brownian motion or general squared-integrable martingales. Finally, some ergodic and mixing properties are obtained by using the Foster–Lyapunov criteria. Full article

The conditional Gaussian nonlinear system (CGNS) is a broad class of nonlinear stochastic dynamical systems. Given the trajectories for a subset of state variables, the remaining follow a Gaussian distribution. Despite the conditionally linear structure, the CGNS exhibits strong nonlinearity, thus capturing many non-Gaussian characteristics observed in nature through its joint and marginal distributions. Desirably, it enjoys closed analytic formulae for the time evolution of its conditional Gaussian statistics, which facilitate the study of data assimilation and other related topics. In this paper, we develop a martingale-free approach to improve the understanding of CGNSs. This methodology provides a tractable approach to proving the time evolution of the conditional statistics by deriving results through time discretization schemes, with the continuous-time regime obtained via a formal limiting process as the discretization time-step vanishes. This discretized approach further allows for developing analytic formulae for optimal posterior sampling of unobserved state variables with correlated noise. These tools are particularly valuable for studying extreme events and intermittency and apply to high-dimensional systems. Moreover, the approach improves the understanding of different sampling methods in characterizing uncertainty. The effectiveness of the framework is demonstrated through a physics-constrained, triad-interaction climate model with cubic nonlinearity and state-dependent cross-interacting noise. Full article

In this study, we propose a smoothed weighted quantile regression (SWQR), which combines convolution smoothing with a weighted framework to address the limitations. By smoothing the non-differentiable quantile regression loss function, SWQR can improve computational efficiency and allow for more stable model estimation in complex datasets. We construct an efficient optimization process based on gradient-based algorithms by introducing weight refinement and iterative parameter estimation methods to minimize the smoothed weighted quantile regression loss function. In the simulation studies, we compare the proposed method with two existing methods, including martingale-based quantile regression (MartingaleQR) and weighted quantile regression (WeightedQR). The results emphasize the superior computational efficiency of SWQR, outperforming other methods, particularly WeightedQR, by requiring significantly less runtime, especially in settings with large sample sizes. Additionally, SWQR maintains robust performance, achieving competitive accuracy and handling the challenges of right censoring effectively, particularly at higher quantiles. We further illustrate the proposed method using a real dataset on primary biliary cirrhosis, where it exhibits stable coefficient estimates and robust performance across quantile levels with different censoring rates. These findings highlight the potential of SWQR as a flexible and robust method for analyzing censored data in survival analysis, particularly in scenarios where computational efficiency is a key concern. Full article

This study explores market efficiency and behavior by integrating key theories such as the Efficient Market Hypothesis (EMH), Adaptive Market Hypothesis (AMH), Informational Efficiency and Random Walk theory. Using LSTM enhanced by optimizers like Stochastic Gradient Descent (SGD), Adam, AdaGrad, and RMSprop, we analyze market inefficiencies in the Standard and Poor’s (SPX) index over a 22-year period. Our results reveal “pockets in time” that challenge EMH predictions, particularly with the AdaGrad optimizer at a size of the hidden layer (HS) of 64. Beyond forecasting, we apply the Dominguez–Lobato (DL) and General Spectral (GS) tests as part of the Martingale Difference Hypothesis to assess statistical inefficiencies and deviations from the Random Walk model. By emphasizing “informational efficiency”, we examine how quickly new information is reflected in stock prices. We argue that market inefficiencies are transient phenomena influenced by structural shifts and information flow, challenging the notion that forecasting alone can refute EMH. Additionally, we compare LSTM with ARIMA with Exponential Smoothing, and LightGBM to highlight the strengths and limitations of these models in financial forecasting. The LSTM model excels at capturing temporal dependencies, while LightGBM demonstrates its effectiveness in detecting non-linear relationships. Our comprehensive approach offers a nuanced understanding of market dynamics and inefficiencies. 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

We show that three prominent consumption-based asset pricing models—the Bansal–Yaron, Campbell–Cochrane and Cecchetti–Lam–Mark models—cannot explain the dynamic properties of stock market returns. We show this by estimating these models with GMM, deriving ex-ante expected returns from them and then testing whether the difference between realised and expected returns is a martingale difference sequence, which it is not. Mincer–Zarnowitz regressions show that the models’ out-of-sample expected returns are systematically biased. Furthermore, semi-parametric tests of whether the models’ state variables are consistent with the degree of own-history predictability in stock returns suggest that only the Campbell–Cochrane habit variable may be able to explain return predictability, although the evidence on this is mixed. Full article

This paper is concerned with the problem of asymptotic stability for a class of stochastic differential equations with impulsive effects. A sufficient criterion on asymptotic stability is derived for such impulsive stochastic differential equations via Lyapunov stability theory, bounded difference condition and martingale convergence theorem. The results show that the impulses can facilitate the stability of the stochastic differential equations when the original system is not stable. Finally, the feasibility of our results is confirmed by two numerical examples and their simulations. Full article

Next-generation networks are expected to handle a wide variety of internet of everything (IoE) services, notably including virtual reality (VR) for smart industrial-oriented applications. VR for industrial environments subtends strict quality of service constraints, requiring sixth-generation terahertz communications to be satisfied. In such an environment, an additional important issue is trying to get high utilization of network and computing resources. This implies identifying efficient access techniques and methodologies to increase bandwidth utilization and enable flows related to services, with different service requirements, to coexist on the same computation node. Towards this goal, this paper addresses the problem of coexistence of the traffic flows related to different services with given delay requirements, on the same computation node arranged to execute flow processing. In such a context, a theoretical comprehensive performance analysis, to the best of the authors’ knowledge, is still missing in the literature. As a consequence, this lack strongly limits the possibility of fully capturing the performance advantages of computation node sharing among different traffic flows, i.e., services. The proposed analysis aims to give a measure of the ability of the system in accomplishing services before the expiration of corresponding deadlines. The integration of martingale bounds within the stochastic network calculus tool is provided, assuming both the first-in–first-out and the earliest deadline first scheduling policies. Finally, the validity of the analysis proposed is confirmed by the tightness emerging from the comparison between analytical predictions and simulation results. Full article