Search Results (99)

Modeling of impurity diffusion processes in a multiphase randomly inhomogeneous body is performed using the Feynman diagram technique. The impurity diffusion equations are formulated for each of the phases separately. Their random boundaries are subject to non-ideal contact conditions for concentration. The contact mass transfer problem is reduced to a partial differential equation describing diffusion in the body as a whole, which accounts for jump discontinuities in the searched function as well as in its derivative at the stochastic interfaces. The obtained problem is transformed into an integro-differential equation involving a random kernel, whose solution is constructed as a Neumann series. Averaging over the ensemble of phase configurations is performed. The Feynman diagram technique is developed to investigate the processes described by parabolic partial differential equations. The mass operator kernel is constructed as a sum of strongly connected diagrams. An integro-differential Dyson equation is obtained for the concentration field. In the Bourret approximation, the Dyson equation is specified for a multiphase randomly inhomogeneous medium with uniform phase distribution. The problem solution, obtained using Feynman diagrams, is compared with the solutions of diffusion problems for a homogeneous layer, one having the coefficients of the base phase and the other having the characteristics averaged over the body volume. Full article

Let $\left\{X\right(t),t\ge 0\}$ be a one-dimensional jump-diffusion process whose continuous part is either a Wiener, Ornstein–Uhlenbeck, or generalized Bessel process. The process starts at $X\left(0\right)=x\in [-d,d]$. Let $\tau \left(x\right)$ be the first time that $X\left(t\right)=0$ or $\left|X\right(t\left)\right|=d$. The jumps follow a uniform distribution on the interval $(-2x,0)$ when x is positive and on the interval $(0,-2x)$ when x is negative. We are interested in the moment-generating function of $\tau \left(x\right)$, its mean, and the probability that $X\left[\tau \right(x\left)\right]=0$. We must solve integro-differential equations, subject to the appropriate boundary conditions. Analytical and numerical results are presented. Full article

Denoising diffusion probabilistic models (DDPMs) have achieved remarkable success across various research domains. However, their high complexity when processing 3D images remains a limitation. To mitigate this, researchers typically preprocess data into 2D slices, enabling the model to perform segmentation in a reduced 2D space. This paper introduces DiffBTS, an end-to-end, lightweight diffusion model specifically designed for 3D brain tumor segmentation. DiffBTS replaces the conventional self-attention module in the traditional diffusion models by introducing an efficient 3D self-attention mechanism. The mechanism is applied between down-sampling and jump connections in the model, allowing it to capture long-range dependencies and global semantic information more effectively. This design prevents computational complexity from growing in square steps. Prediction accuracy and model stability are crucial in brain tumor segmentation; we propose the Edge-Blurring Guided (EBG) algorithm, which directs the diffusion model to focus more on the accuracy of segmentation boundaries during the iterative sampling process. This approach enhances prediction accuracy and stability. To assess the performance of DiffBTS, we compared it with seven state-of-the-art models on the BraTS 2020 and BraTS 2021 datasets. DiffBTS achieved an average Dice score of 89.99 and an average HD95 value of 1.928 mm on BraTS2021 and 86.44 and 2.466 mm on BraTS2020, respectively. Extensive experimental results demonstrate that DiffBTS achieves state-of-the-art performance in brain tumor segmentation, outperforming all competing models. Full article

First-exit problems are studied for two-dimensional diffusion processes with jumps according to a Poisson process. The size of the jumps is distributed as an exponential random variable. We are interested in the random variable that denotes the first time that the sum of the two components of the process leaves a given interval. The function giving the probability that the process will leave the interval on its left-hand side satisfies a partial differential–integral equation. This equation is solved analytically in particular cases by making use of the method of similarity solutions. The problem of calculating the mean and the moment-generating function of the first-passage time random variable is also considered. The results obtained have applications in various fields, notably, financial mathematics and reliability theory. Full article

In order to realize green and low-carbon transformation, some ports have explored the path of sustainable equipment upgrading by adjusting the energy structure of yard cranes in recent years. However, there are multiple uncertainties in the investment process of hydrogen-powered yard cranes, and the existing valuation methods fail to effectively deal with these dynamic changes and lack scientifically sound decision support tools. To address this problem, this study constructs a multi-factor real options model that integrates the dynamic uncertainties of hydrogen price, carbon price, and technology maturity. In this study, a geometric Brownian motion is used for hydrogen price simulation, a Markov chain model with jump diffusion term and stochastic volatility is used for carbon price simulation, and a learning curve method is used to quantify the evolution of technology maturity. Aiming at the long investment cycle of ports, a hybrid option strategy of “American and European” is designed, and the timing and scale of investment are dynamically optimized by Monte Carlo simulation and least squares regression. Based on the empirical analysis of Qingdao Port, the results show that the optimal investment plan for hydrogen-powered yard cranes project under the framework of a multi-factor option model is to use an American-type option to maintain moderate flexibility in the early stage, and to use a European-type option to lock in the return in the later stage. The study provides decision support for the green development of ports and enhances economic returns and carbon emission reduction benefits. 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

Jump-diffusion algorithms are applied to sampling from Bayesian posterior distributions. We consider a class of random sampling algorithms based on continuous-time jump processes. The semigroup theory of random processes lets us show that limiting cases of certain jump processes acting on discretized spaces converge to diffusion processes as the discretization is refined. One of these processes leads to the familiar Langevin diffusion equation; another leads to an entirely new diffusion equation. 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

Deviation of an asset price from its fundamental value, commonly referred to as a price bubble, is a well-known phenomenon in financial markets. Mathematically, a bubble arises when the deflated price process transitions from a martingale to a strict local martingale. This paper explores price bubbles using the framework of optional semimartingale calculus within nonstandard probability spaces, where the underlying filtration is not necessarily right-continuous or complete. We present two formulations for financial markets with bubbles: one in which asset prices are modeled as càdlàg semimartingales and another where they are modeled as làdlàg semimartingales. In both models, we demonstrate that the formation and re-emergence of price bubbles are intrinsically tied to the lack of right continuity in the underlying filtration. These theoretical findings are illustrated with practical examples, offering novel insights into bubble dynamics that hold significance for both academics and practitioners in the field of mathematical finance. Full article

We develop a deep reinforcement learning (RL) framework for an optimal market-making (MM) trading problem, specifically focusing on price processes with semi-Markov and Hawkes Jump-Diffusion dynamics. We begin by discussing the basics of RL and the deep RL framework used; we deployed the state-of-the-art Soft Actor–Critic (SAC) algorithm for the deep learning part. The SAC algorithm is an off-policy entropy maximization algorithm more suitable for tackling complex, high-dimensional problems with continuous state and action spaces, like those in optimal market-making (MM). We introduce the optimal MM problem considered, where we detail all the deterministic and stochastic processes that go into setting up an environment to simulate this strategy. Here, we also provide an in-depth overview of the jump-diffusion pricing dynamics used and our method for dealing with adverse selection within the limit order book, and we highlight the working parts of our optimization problem. Next, we discuss the training and testing results, where we provide visuals of how important deterministic and stochastic processes such as the bid/ask prices, trade executions, inventory, and the reward function evolved. Our study includes an analysis of simulated and real data. We include a discussion on the limitations of these results, which are important points for most diffusion style models in this setting. Full article

This paper is dedicated to studying the optimal investment proportions of three types of assets with symmetry, namely, risky assets, risk-free assets, and wealth management products, when the stochastic expenditure process follows a jump-diffusion model. The stochastic expenditure process is treated as an exogenous cash flow and is assumed to follow a stochastic differential process with jumps. Under the Cox–Ingersoll–Ross interest rate term structure, it is presumed that the prices of multiple risky assets evolve according to a multi-dimensional geometric Brownian motion. By employing stochastic control theory, the Hamilton–Jacobi–Bellman (HJB) equation for the household portfolio problem is formulated. Considering various risk-preference functions, particularly the Hyperbolic Absolute Risk Aversion (HARA) function, and given the algebraic form of the objective function through the terminal-value maximization condition, an explicit solution for the optimal investment strategy is derived. The findings indicate that when household investment behavior is characterized by random expenditures and symmetry, as the risk-free interest rate rises, the optimal proportion of investment in wealth-management products also increases, whereas the proportion of investment in risky assets continually declines. As the expected future expenditure increases, households will decrease their acquisition of risky assets, and the proportion of risky-asset purchases is sensitive to changes in the expectation of unexpected expenditures. Full article

This paper addresses the portfolio optimisation problem within the jump-diffusion stochastic differential equations (SDEs) framework. We begin by recalling a fundamental theoretical result concerning the existence of solutions to the Black–Scholes–Merton partial differential equation (PDE), which serves as the cornerstone for subsequent analysis. Then, we explore a range of financial applications, spanning scenarios characterised by the absence of jumps, the presence of jumps following a log-normal distribution, and jumps following a distribution of greater generality. Additionally, we delve into optimising more complex portfolios composed of multiple risky assets alongside a risk-free asset, shedding new light on optimal allocation strategies in these settings. Our investigation yields novel insights and potentially groundbreaking results, offering fresh perspectives on portfolio management strategies under jump-diffusion dynamics. Full article

Accurate crude oil price forecasting is essential, considering oil’s critical role in the global economy. However, the crude oil market is significantly influenced by external, transient events, posing challenges in capturing price fluctuations’ complex dynamics and uncertainties. Traditional time series forecasting models, such as ARIMA and LSTM, often rely on assumptions regarding data structure, limiting their flexibility to estimate volatility or account for external shocks effectively. Recent research highlights Generative Adversarial Networks (GANs) as a promising alternative approach for capturing intricate patterns in time series data, leveraging the adversarial learning framework. This paper introduces a Crude Oil-Driven Conditional GAN (CO-CGAN), a hybrid model for enhancing crude oil price forecasting by combining advanced AI frameworks (GANs), oil market sentiment analysis, and stochastic jump-diffusion models. By employing conditional supervised training, the inherent structure of the data distribution is preserved, thereby enabling more accurate and reliable probabilistic price forecasts. Additionally, the CO-CGAN integrates a Lévy process and sentiment features to better account for uncertainties and price shocks in the crude oil market. Experimental evaluations on two real-world oil price datasets demonstrate the superior performance of the proposed model, achieving a Mean Squared Error (MSE) of 0.000054 and outperforming benchmark models. 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

Sodium borohydride-closo-hydroborate Na₃(BH₄)(B₁₂H₁₂) exhibits high room-temperature ionic conductivity and high electrochemical stability. To study the dynamical properties of this mixed-anion compound at the microscopic level, we have measured the ¹H, ¹¹B, and ²³Na nuclear magnetic resonance spectra and nuclear spin-lattice relaxation rates over the temperature range of 8–573 K. Our ¹H and ¹¹B spin-lattice relaxation measurements have revealed two types of reorientational jump motion. The faster motional process attributed to reorientations of the [BH₄]⁻ anions is characterized by an activation energy of 159 meV, and the corresponding reorientational jump rate reaches ~10⁸ s⁻¹ near 130 K. The slower process ascribed to reorientations of the larger [B₁₂H₁₂]⁻ anions is characterized by an activation energy of 319 meV, and the corresponding reorientational jump rate reaches ~10⁸ s⁻¹ near 240 K. The results of the ²³Na nuclear magnetic resonance measurements are consistent with the fast long-range diffusion of Na⁺ ions in Na₃(BH₄)(B₁₂H₁₂). The diffusive jump rate of Na⁺ is found to reach ~10⁴ s⁻¹ at 300 K and ~8 × 10⁸ s⁻¹ at 530 K. A comparison of these jump rates with the ionic conductivity data suggests the importance of correlations between diffusing ions. Full article