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We propose an extension of the Cox-Ross-Rubinstein (CRR) model based on q-binomial (or Kemp) random walks, with application to default with logistic failure rates. This model allows us to consider time-dependent switching probabilities varying according to a trend parameter on a non-self-similar binomial tree. In particular, it includes tilt and stretch parameters that control increment sizes. Option pricing formulas are written using q-binomial coefficients, and we study the convergence of this model to a Black-Scholes type formula in continuous time. A convergence rate of order O(N-1/2) is obtained. [ABSTRACT FROM AUTHOR]

We prove the existence of viscosity solutions for fractional semilinear elliptic PDEs on open balls with bounded exterior condition in dimension $ d\geq 1 $. Our approach relies on a tree-based probabilistic representation based on a $ (2s) $-stable branching processes for all $ s\in (0,1) $, and our existence results hold for sufficiently small exterior conditions and nonlinearity coefficients. In comparison with existing approaches, we consider a wide class of polynomial nonlinearities without imposing upper bounds on their maximal degree or number of terms. Numerical illustrations are provided in large dimensions. [ABSTRACT FROM AUTHOR]

We apply general moment identities for Poisson stochastic integrals with random integrands to the computation of the moments of Markovian growth–collapse processes. This extends existing formulas for mean and variance available in the literature to closed-form moment expressions of all orders. In comparison with other methods based on differential equations, our approach yields explicit summations in terms of the time parameter. We also treat the case of the associated embedded chain, and provide recursive codes in Maple and Mathematica for the computation of moments and cumulants of any order with arbitrary cut-off moment sequences and jump size functions. [ABSTRACT FROM AUTHOR]

This paper studies stochastic ordering under nonlinear expectations εG generated by solutions of G-Backward Stochastic Differential Equations (G-BSDEs) defined on G-expectation spaces. We derive sufficient conditions for the convex, increasing convex, and monotonic G-stochastic orderings of G-diffusion processes at terminal time. Our approach relies on comparison properties for G-Forward-Backward Stochastic Differential Equations (G-FBSDEs) and on relevant extensions of convexity, monotonicity and continuous dependence properties for the solutions of associated Hamilton-Jacobi-Bellman (HJB) equations. Applications of G-stochastic ordering to contingent claim superhedging price comparison under ambiguous coeffcients are provided. [ABSTRACT FROM AUTHOR]

We derive normal approximation bounds in the Wasserstein distance for sums of generalized U-statistics, based on a general distance bound for functionals of independent random variables of arbitrary distributions. Those bounds are applied to normal approximation for the combined weights of subgraphs in the Erdős–Rényi random graph, extending the graph counting results of Barbour et al. (A central limit theorem for decomposable random variables with applications to random graphs, J. Combin. Theory Ser. B 47(2) (1989), pp. 125–145) to the setting of weighted graphs. Our approach relies on a general stochastic analytic framework for functionals of independent random sequences. [ABSTRACT FROM AUTHOR]

We extend the construction of equilibria for linear-quadratic and mean-variance portfolio problems available in the literature to a large class of mean-field time-inconsistent stochastic control problems in continuous time. Our approach relies on a time discretization of the control problem via n-person games, which are characterized via the maximum principle using backward stochastic differential equations. The existence of equilibria is proved by applying weak convergence arguments to the solutions of n-person games. A numerical implementation is provided by approximating n-person games using finite Markov chains. [ABSTRACT FROM AUTHOR]

We derive a characterization of equilibrium controls in continuous-time, time-inconsistent control (TIC) problems using the Malliavin calculus. For this, the classical duality analysis of adjoint BSDEs is replaced with the Malliavin integration by parts. This results into a necessary and sufficient maximum principle which is applied to a linear-quadratic TIC problem, recovering previous results obtained by duality analysis in the mean-variance case, and extending them to the linear-quadratic setting. We also show that our results apply beyond the linear-quadratic case by treating the generalized Merton problem. [ABSTRACT FROM AUTHOR]