Search Results (29)

We establish a Bernstein–von Mises (BvM) theorem for parametric competing-risks models under hybrid Type-I censoring, where observation stops at the random time ${\tau }_n=\min (X_{\left(r\right)},T_0)$. Using the counting-process martingale framework, we first prove the local asymptotic normality (LAN) of the model and identify the limiting Fisher information as a block-diagonal matrix composed of operational (${\tau }^{\ast }$-truncated) cause-specific informations. Unlike previous work, we derive the testing-function (Hellinger-affinity) condition required for posterior tail control from the standard regularity assumptions rather than imposing it as an extra hypothesis. The posterior distribution of $\sqrt{n}(\theta -{\widehat{\theta }}_n)$ is shown to converge in total variation to a Gaussian law with covariance $\mathcal{I}{\left({\theta }_0\right)}^{-1}$ for every prior positive and continuous at ${\theta }_0$. The convergence rate is $O_P({(\log n)}^{3/2}{n}^{-1/2})$; a fourth-order smoothness condition removes the logarithmic factor. The abstract conditions are verified for the exponential, Weibull, and Gompertz families, and a simulation study corroborates the asymptotic approximation and the nominal coverage of Bayesian credible sets. Full article

The standard models of stock-price dynamics and option valuation rest on stochastic processes postulated at the outset; here, we lay down an entropic-inference framework that derives these processes rather than assuming them, by making explicit the information each one encodes. A symmetry comes first: markets reward returns rather than price levels, which selects the logarithm of price as the dynamical variable. The price then evolves through two channels, a continuous one carrying the constraints of continuity and directionality, and a jump channel carrying the arrival rate and the first two moments of the jump size. Because these constraints act on disjoint parts of the microstate, the channels factorize as a theorem, and the dynamics is the Merton jump-diffusion, with Geometric Brownian Motion as its no-jump limit; the log-price density obeys a Kolmogorov–Feller equation, of which the Fokker–Planck equation is the no-jump limit. The same principle, now imposing no-arbitrage through the mean log-return, selects the Esscher transform from among the many martingale measures an incomplete market admits, here derived rather than borrowed; the premium then satisfies Merton’s partial integro-differential equation, and the risk-neutral mixture of lognormals generates the implied-volatility smile, the Black–Scholes results returning when jumps vanish. What changes from one model to the next is never the inference but the information supplied to it. Full article

We study the exact partial controllability of a semilinear stochastic fractional evolution equation of Caputo order $\alpha \in (1,2)$ in a separable Hilbert space, subject to a genuine state-dependent delay $y(t-\rho (t,y_t))$, impulsive effects, and Q-Wiener perturbations. To accommodate the state-dependent delay, we work in a Sobolev-type prehistory space that restores the local Lipschitz property lost in the space of continuous functions, and to accommodate the impulses, we use a piecewise-continuous mean-square path space; two initial data points are prescribed, as required for $\alpha \in (1,2)$. The existence and uniqueness of mild solutions are established by the Banach contraction principle on a Lipschitz ball equipped with a Bielecki-type weighted norm, which removes the smallness conditions of the contraction-based literature. The stochastic convolution is treated rigorously by the Da Prato factorization realized through the subordination of the fractional resolvent family, with the Burkholder inequality applied only to the genuine Itô integral and never to the non-martingale convolution itself. For controllability, we adopt the correct stochastic notion: the target is an ${\mathcal{F}}_T$-measurable random variable, and the adapted control is constructed through the martingale representation theorem, together with the projected controllability Gramian, yielding $Py\left(T\right)=\eta$ almost surely. A numerical study on a stochastic fractional wave equation verifies the main controllability theorem, driving the projected terminal state to the prescribed ${\mathcal{F}}_T$-measurable target at machine precision, and confirms the observability requirement. Full article

We study finite-horizon first-passage time and boundary non-crossing probabilities for Gaussian martingales, viewed as continuous local martingales obtained by running Brownian motion on a deterministic variance clock associated with deterministic volatility. Our aim is to quantify how the associated survival probability changes when the variance clock is perturbed. Using a deterministic time change representation, we reduce the problem to a Brownian boundary-crossing problem with a transformed horizon and a transformed boundary. This allows us to combine time change arguments with recent differentiability results for boundary-crossing probabilities. Under suitable regularity assumptions, we derive a first-order sensitivity formula with respect to the variance clock. The derivative splits naturally into two components: one produced by the deformation of the transformed boundary and one produced by the variation of the terminal transformed horizon. Several explicit examples are provided, including affine barriers and nonlinear deterministic clocks. These examples show in particular that, for nonconstant boundaries, redistributing variance over calendar time can change the finite-horizon survival probability even when the terminal variance is kept fixed. Full article

We present two models for incorporating the total effect of market friction noise into the dynamic pricing of assets and European options. The first model is developed under a continuous-time Black–Scholes–Merton framework. The second model is a discrete, binomial tree model developed as an extension of the static Grossman–Stiglitz model. Both models are market-complete and provide a unique equivalent martingale measure that establishes a unique map between parameters governing the risk-neutral and real-world price dynamics. We provide empirical examples to extract the coefficients of the model, in particular those coefficients characterizing the influence of the frictions on prices. In addition to isolating the impact of noise on the volatility, the discrete model enables us to extract the noise impact on the drift coefficient. We provide evidence for the primary market friction that we believe our empirical examples capture. Full article

This paper investigates an infinite-horizon optimal consumption and investment problem for an agent who consumes two types of goods: necessities and luxuries. The agent derives utility from both goods but faces a ratcheting constraint on luxury consumption, which prohibits any decline in its level over time. This constraint captures the irreversible nature of high living standards or luxury habits often observed in real economies. We formulate the problem in a complete financial market with a risk-free asset and a risky stock and solve it analytically using the dual–martingale method. The dual problem is shown to reduce to a family of optimal stopping problems, from which we derive explicit closed-form solutions for the value function and optimal policies. Our results reveal that the ratcheting constraint generates asymmetric consumption dynamics: necessities adjust freely, whereas luxuries exhibit downward rigidity. As a consequence, the marginal propensity to consume necessities declines with wealth, while luxury consumption and portfolio risk exposure increase more sharply compared to the benchmark case without ratcheting. The model provides a continuous-time microfoundation for persistent high consumption levels and greater risk-taking among wealthy individuals. Full article

We study a finite-horizon optimal consumption and investment problem in a complete continuous-time market where consumption is restricted within fixed upper and lower bounds. Assuming constant relative risk aversion (CRRA) preferences, we employ the dual-martingale approach to reformulate the problem and derive closed-form integral representations for the dual value function and its derivatives. These results yield explicit feedback formulas for the optimal consumption, portfolio allocation, and wealth processes. We establish the duality theorem linking the primal and dual value functions and verify the regularity and convexity properties of the dual solution. Our results show that the upper and lower consumption bounds transform the linear Merton rule into a piecewise policy: consumption equals L when wealth is low, follows the unconstrained Merton ratio in the interior region, and is capped at H when wealth is high. Full article

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

Using a Donsker-type construction, we prove the existence of a new class of processes, which we call the self-stabilizing processes. These processes have a particular property: the “local intensities of jumps” vary with the values. Moreover, we also show that the self-stabilizing processes have many other good properties, such as stochastic Hölder continuity and strong localizability. Such a self-stabilizing process is simultaneously a Markov process, a martingale (when the local index of stability is greater than 1), a self-scaling process and a self-regulating process. Full article

We develop a continuous-time model of optimal consumption, portfolio allocation, and early retirement that, to our knowledge, is the first to incorporate an implementation delay —a fixed lag $\delta$ between the retirement decision and the actual cessation of labor and income. Using a dual-martingale approach, we obtain closed-form solutions and quantify how $\delta$ affects optimal behavior. For example, when $\delta$ increases from $0.5$ to 2 years (baseline parameters: $\beta =0.04$, $r=0.02$, $\mu =0.08$, $\sigma =0.2$, $\gamma =3$, $k_B=0.3$, and $\epsilon =1$), optimal pre-retirement consumption rises by approximately 7%, the risky asset share falls by about 5 percentage points, the expected retirement time increases by over 1 year, and the retirement wealth threshold $x_R$ grows by roughly 10%. These results provide policy-relevant insights for retirement systems where procedural lags can distort incentives and reduce welfare. 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

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

This paper aims to develop optional semimartingale methods in risk theory to allow for a larger class of risk models. Optional semimartingales are left-continuous with right-limit stochastic processes defined on a probability space where the usual conditions—completeness and right-continuity of the filtration—are not assumed. Three risk models are formulated, accounting for inflation, interest rates, and claim occurrences. The first model extends the martingale approach to calculate ruin probabilities, the second employs the Gerber–Shiu function to evaluate the expected discounted penalty from financial oscillations or jumps, and the third introduces a Gaussian risk model using counting processes to capture premium and claim cash flow jumps in insurance companies. 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 study an optimal consumption, leisure, and investment problem over a finite horizon in a continuous-time financial market with partial borrowing constraints. The agent derives utility from consumption and leisure, with preferences represented by a Cobb–Douglas utility function. The agent allocates time between work and leisure, earning wage income based on working hours. A key feature of our model is a partial borrowing constraint that limits the agent’s debt capacity to a fraction of the present value of their maximum future labor income. We employ the dual-martingale approach to derive the optimal consumption, leisure, and investment strategies. The problem reduces to solving a variational inequality with a free boundary, which we analyze using analytical and numerical methods. We provide an integral equation representation of the free boundary and solve it numerically via a recursive integration method. Our results highlight the impact of the borrowing constraint on the agent’s optimal decisions and the interplay between labor supply, consumption, and portfolio choice. Full article