Nonlinear Filtering Equation of a Jump Process with Counting Observations

We study the filtering problem of an R ^d -valued pure jump process when the observations is a counting process. We assume that the dynamic of the state and the observations may be strongly dependent and that the two processes may jump together. Weak and pathwise uniqueness of solution of the Kushner–Stratonovich equation are discussed.     

On Hölder solutions of the integro-differential Zakai equation

The existence and uniqueness of solutions of the Cauchy problem to a stochastic parabolic integro-differential equation is investigated. The equation considered arises in a nonlinear filtering problem with a jump signal process and continuous observation.     

An Optimal Control Problem Associated with SDEs Driven by Lévy-Type Processes

Abstract

In this article, we consider an optimal control problem associated with jump type stochastic differential equations driven by Lévy-type processes. The problem arises from portfolio optimization for the pair of the wealth process and the cumulative consumption process in (incomplete) financial market models. We establish the existence and the uniqueness of (constrained) viscosity solutions to the associated the integro-differential Hamilton–Jacobi–Bellman equation.     

A Maximum Principle for Stochastic Control with Partial Information

Abstract

We study the problem of optimal control of a jump diffusion, that is, a process which is the solution of a stochastic differential equation driven by Lévy processes. It is required that the control process is adapted to a given subfiltration of the filtration generated by the underlying Lévy processes. We prove two maximum principles (one sufficient and one necessary) for this type of partial information control. The results are applied to a partial information mean-variance portfolio selection problem in finance.     

Hamilton-Jacobi-Bellman inequality for the average control of piecewise deterministic Markov processes

ABSTRACT

The main goal of this paper is to study the infinite-horizon long run average continuous-time optimal control problem of piecewise deterministic Markov processes (PDMPs) with the control acting continuously on the jump intensity λ and on the transition measure Q of the process. We provide conditions for the existence of a solution to an integro-differential optimality inequality, the so called Hamilton-Jacobi-Bellman (HJB) equation, and for the existence of a deterministic stationary optimal policy. These results are obtained by using the so-called vanishing discount approach, under some continuity and compactness assumptions on the parameters of the problem, as well as some non-explosive conditions for the process.     

Analytic solution to the problem of the temperature jump in a gas with rotational degrees of freedom

An exact solution is obtained for the first time for the problem of the temperature jump in a gas with allowance for internal (rotational) degrees of freedom. The treatment is based on a model collision integral proposed by the authors. The problem reduces to the solution of a boundary-value problem for a linear vector transport equation with matrix kernel. Separation of the variable leads to a characteristic equation for which eigenvectors are found in a space of generalized functions and the eigenvalue spectrum is investigated. An expansion of the solution to the problem with respect to eigenvectors of the continuous and discrete spectra is established. On the basis of the conditions of solvability of the vector Riemann-Hilbert boundary-value problem which arises in the process of the proof, an exact (in closed form) expression is obtained for the temperature jump.Moscow Pedagogical University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 3, pp. 530–540, June, 1993.     

The modified jump problem for the Laplace equation and singularities at the tips

The boundary value problem for the Laplace equation outside several cuts in a plane is studied. The jump of the solution of the Laplace equation and the boundary condition containing the jump of its normal derivative are specified of the cuts. The unique solution of this problem is obtained. The problem is reduced to the uniquely solvable Fredholm equation of the second kind and index zero. The singularities at the ends of the cuts are investigated.     

Existence of Optimal Controls for Partially Observed Jump Processes

A partially observable control problem for an R ^d -valued jump process with counting observations is studied. The state and the observations may be strongly dependent and, in particular, the two processes may jump together. An equivalent separated problem is introduced and the existence of an optimal control for the separated problem is obtained in the class of relaxed and generalized controls. Equivalence between the initial problem and the relaxed generalized separated control problem is discussed.     

The jump problem for the Helmholtz equation and singularities at the edges

The boundary value problem for the Helmholtz equation outside several cuts in a plane is studied. The jump of the solution of the Helmholtz equation and the jump of its normal derivative are specified of the cuts. The unique solution of this problem is constructed in the explicit form by means of single layer and angular potentials. The singularities at the ends of the cuts are investigated.     

基于跳扩散过程的欧式交换期权定价

考虑了股票价格服从带时滞泊松跳的跳扩散模型的欧式交换期权定价问题,运用无套利理论推导出期权价值微分方程,利用变换计价单位的方法,得到交换期权的显示定价公式.     

Sufficient Stochastic Maximum Principle for the Optimal Control of Semi-Markov Modulated Jump-Diffusion with Application to Financial Optimization

The finite state semi-Markov process is a generalization over the Markov chain in which the sojourn time distribution is any general distribution. In this article, we provide a sufficient stochastic maximum principle for the optimal control of a semi-Markov modulated jump-diffusion process in which the drift, diffusion, and the jump kernel of the jump-diffusion process is modulated by a semi-Markov process. We also connect the sufficient stochastic maximum principle with the dynamic programming equation. We apply our results to finite horizon risk-sensitive control portfolio optimization problem and to a quadratic loss minimization problem.     

The jump problem for the laplace equation

The boundary value problem for the Laplace equation outside several cuts in a plane is studied. The jump of the solution of the Laplace equation and the jump of its normal derivative are specified of the cuts. The problem is studied under different conditions at infinity, which lead to different uniqueness and existence theorems. The solution of this problem is constructed in the explicit form by means of single-layer and angular potentials. The singularities at the ends of the cuts are investigated.     

Finding discontinuities in the coefficients of the linear nonstationary transport equations

An X-ray tomography problem that is an inverse problem for the transport differential equation is set up and investigated. The absorption and single scattering of particles are taken into account. The transport equation is nonstationary (its coefficients and the unknown function depend on time), involves multiple energy levels, and its coefficients can undergo jump discontinuities with respect to the spatial variable (in other words, the medium in which the process proceeds is inhomogeneous). The sought object is the set on which the coefficients of the equation suffer a discontinuity, which corresponds to the search for the boundaries between the different substances composing the sensed medium.     

Bayesian sequential testing for Lévy processes with diffusion and jump components

We study the Bayesian problem of sequential testing of two simple hypotheses about the Lévy-Khintchine triplet of a Lévy process, having diffusion component, represented by a Brownian motion with drift, and jump component of finite variation. The method of proof consists of reducing the original optimal stopping problem to a free-boundary problem. We show it is characterized by a second order integro-differential equation, that the unknown value function solves on the continuation region, and by the smooth fit principle, which holds at the unknown boundary points. Several examples are presented.     

Modelling ultra-fast nanoparticle melting with the Maxwell–Cattaneo equation

The role of thermal relaxation in nanoparticle melting is studied using a mathematical model based on the Maxwell–Cattaneo equation for heat conduction. The model is formulated in terms of a two-phase Stefan problem. We consider the cases of the temperature profile being continuous or having a jump across the solid–liquid interface. The jump conditions are derived from the sharp-interface limit of a phase-field model that accounts for variations in the thermal properties between the solid and liquid. The Stefan problem is solved using asymptotic and numerical methods. The analysis reveals that the Fourier-based solution can be recovered from the classical limit of zero relaxation time when either boundary condition is used. However, only the jump condition avoids the onset of unphysical “supersonic” melting, where the speed of the melt front exceeds the finite speed of heat propagation. These results conclusively demonstrate that the jump condition, not the continuity condition, is the most suitable for use in models of phase change based on the Maxwell–Cattaneo equation. Numerical investigations show that thermal relaxation can increase the time required to melt a nanoparticle by more than a factor of ten. Thus, thermal relaxation is an important process to include in models of nanoparticle melting and is expected to be relevant in other rapid phase-change processes.     

Solvability of an inverse problem for discontinuous Sturm–Liouville operators

In this paper, we consider the Sturm–Liouville equation with the jump conditions inside the interval (0,π). The inverse problem is studied, which consists in recovering operator coefficients from two spectra, corresponding to different boundary conditions. We prove the uniqueness theorem and provide necessary and sufficient conditions for solvability of the inverse problem. We also obtain the oscillation theorem for the eigenfunctions of the considered discontinuous boundary value problem.     

Open-loop control of stochastic fluid systems and applications

We consider a stochastic process Q(t) satisfying a linear differential equation, driven by a jump process which is modulated by a binary sequence. We prove multimodular properties related to the process Q(t) and apply those results to optimal strategies in storage systems models and in ruin problems.     

Tracking a sharp crested wave front in hyperbolic heat transfer

In this article we investigate the numerical oscillations encountered when approximating the solution to the hyperbolic heat conduction equation. We consider a benchmark problem and show that it is not well-posed, unless a jump condition is specified. The alternative is to “smooth” the jump which leads to a sharp crested wave front, but with no discontinuity. To track the wave front we split the problem into auxiliary problems and solve these using different methods. The resulting solution is oscillation-free.     

The modified jump problem for the Helmholtz equation

The boundary value problem for the Helmholtz equation outside several cuts in a plane is studied. The 2 boundary conditions are given on the cuts. One of them specifies the jump of the unkown function. Another one contain the jump of the normal derivative of an unknown function and a limit value of this function on the cuts. The unique solution of this problem is reduced to the uniquely solvable Fredholm equation of the second kind and index zero by means of single layer and angular potentials. The singularities at the ends of the cuts are investigated.

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Sunto Nel presente lavoro si studia il problema al contorno per l'equazione di Helmholtz all'esterno di più tagli nel piano. Le due condizioni al contorno sono assegnate sui tagli. Una di queste prescrive il salto della funzione incognita, l'altra contiene il salto della derivata normale di una funzione incognita ed un valore limite di questa funzione sui tagli. La soluzione univoca di questo problema è ricondotta all'equazione di Fredholm di seconda specie ed indice zero, univocamente risolubiles, per mezzo dei potenziali di singolo strato ed angolare. Si studiano, inoltre, le singolarità agli estremi dei tagli.

     

Mean-Variance Hedging on Uncertain Time Horizon in a Market with a Jump

In this work, we study the problem of mean-variance hedging with a random horizon T∧τ, where T is a deterministic constant and τ is a jump time of the underlying asset price process. We first formulate this problem as a stochastic control problem and relate it to a system of BSDEs with a jump. We then provide a verification theorem which gives the optimal strategy for the mean-variance hedging using the solution of the previous system of BSDEs. Finally, we prove that this system of BSDEs admits a solution via a decomposition approach coming from filtration enlargement theory.