Strong Markov Process Research Articles

DOUBLE BARRIER OPTIONS IN REGIME-SWITCHING HYPER-EXPONENTIAL JUMP-DIFFUSION MODELS

We present a very fast and accurate algorithm for calculating prices of finite lived double barrier options with arbitrary terminal payoff functions under regime-switching hyper-exponential jump-diffusion (HEJD) models, which generalize the double-exponential jump-diffusion model pioneered by Kou and Lipton. Numerical tests demonstrate an excellent agreement of our results with those obtained using other methods, as well as a significant increase in computation speed (sometimes by a factor of 5). The first step of our approach is Carr's randomization, whose convergence we prove for barrier and double barrier options under strong Markov processes of a wide class. The resulting sequence of perpetual option pricing problems is solved using an efficient iteration algorithm and the Wiener-Hopf factorization.

On the orthogonal component of BSDEs in a Markovian setting

In this note we consider a quadratic growth backward stochastic differential equation (BSDE) driven by a continuous martingale M . We prove (in Theorem 3.2) that if M is a strong Markov process and if the BSDE has the form (2.2) with regular data then the unique solution ( Y , Z , N ) of the BSDE is reduced to ( Y , Z ) , i.e. the orthogonal martingale N is equal to zero, showing that in a Markovian setting the “usual” solution ( Y , Z ) (of a BSDE with regular data) has not to be completed by a strongly orthogonal component even if M does not enjoy the martingale representation property.

On the excursions of reflected local-time processes and stochastic fluid queues

In this paper we extend our previous work. We consider the local-time process L of a strong Markov process X, add negative drift to L, and reflect it à la Skorokhod to obtain a process Q. The reflection of X, together with Q, is, in some sense, a macroscopic model for a service system with two priorities. We derive an expression for the joint law of the duration of an excursion, the maximum value of the process on it, and the time between successive excursions. We work with a properly constructed stationary version of the process. Examples are also given in the paper.

On the excursions of reflected local-time processes and stochastic fluid queues

In this paper we extend our previous work. We consider the local-time process L of a strong Markov process X, add negative drift to L, and reflect it à la Skorokhod to obtain a process Q. The reflection of X, together with Q, is, in some sense, a macroscopic model for a service system with two priorities. We derive an expression for the joint law of the duration of an excursion, the maximum value of the process on it, and the time between successive excursions. We work with a properly constructed stationary version of the process. Examples are also given in the paper.

From resolvents to càdlàg processes through compact excessive functions and applications to singular SDE on Hilbert spaces

We present an approach for constructing càdlàg strong Markov processes given a resolvent of kernels. The conditions imposed on the resolvent are checkable in applications and allow the control of the set of admissible starting points of the process. The main application is to singular SDEs on Hilbert spaces.

ABSOLUTELY CONTINUOUS COMPENSATORS

We give sufficient conditions on the underlying filtration such that all totally inaccessible stopping times have compensators which are absolutely continuous. If a semimartingale, strong Markov process X has a representation as a solution of a stochastic differential equation driven by a Wiener process, Lebesgue measure, and a Poisson random measure, then all compensators of totally inaccessible stopping times are absolutely continuous with respect to the minimal filtration generated by X. However Çinlar and Jacod have shown that all semimartingale strong Markov processes, up to a change of time and slightly of space, have such a representation.

Large Deviations Principle for a Large Class of One-Dimensional Markov Processes

We study the large deviations principle for one-dimensional, continuous, homogeneous, strong Markov processes that do not necessarily behave locally as a Wiener process. Any strong Markov process X t in ℝ that is continuous with probability one, under some minimal regularity conditions, is governed by a generalized elliptic operator D v D u , where v and u are two strictly increasing functions, v is right-continuous and u is continuous. In this paper, we study large deviations principle for Markov processes whose infinitesimal generator is εD v D u where 0<ε≪1. This result generalizes the classical large deviations results for a large class of one-dimensional “classical” stochastic processes. Moreover, we consider reaction-diffusion equations governed by a generalized operator D v D u . We apply our results to the problem of wavefront propagation for these type of reaction-diffusion equations.

Deviation Inequalities for Centered Additive Functionals of Recurrent Harris Processes Having General State Space

Let X be a Harris recurrent strong Markov process in continuous time with general Polish state space E, having invariant measure μ. In this paper we use the regeneration method to derive non asymptotic deviation bounds for $$P_{x}\biggl(\biggl|\int_0^tf(X_s)\,ds\biggr|\geq t^{\frac{1}{2}+\eta}\varepsilon \biggr)$$ in the positive recurrent case, for nice functions f with μ(f)=0 (f must be a charge). We generalize these bounds to the fully null-recurrent case in the moderate deviations regime. We obtain a Gaussian concentration bound for all functions f which are a charge. The rate of convergence is expressed in terms of the deterministic equivalent of the process. The main ingredient of the proof is Nummelin splitting in continuous time, which allows one to introduce regeneration times for the process on an enlarged state space.

A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps

In this paper, we consider the following type of non-local (pseudo-differential) operators \mathcal{L} on \mathbb{R}^d : \begin{align*} \mathcal{L} u(x) =&amp; \frac{1}{2} \sum_{i, j=1}^d \frac{\partial}{\partial x_i} \Big(a_{ij}(x) \frac{\partial u(x)}{\partial x_j}\Big) \\ &amp;\ + \lim_{\varepsilon \downarrow 0} \int_{\{y\in \mathbb{R}^d: |y-x|&gt;\varepsilon\}} (u(y)-u(x)) J(x, y) dy, \end{align*} where A(x)=(a_{ij}(x))_{1\leq i,j\leq d} is a measurable d\times d matrix-valued function on \mathbb{R}^d that is uniformly elliptic and bounded and J is a symmetric measurable non-trivial non-negative kernel on \mathbb{R}^d \times \mathbb{R}^d satisfying certain conditions. Corresponding to \mathcal{L} is a symmetric strong Markov process X on \mathbb{R}^d that has both the diffusion component and pure jump component. We establish a priori Hölder estimate for bounded parabolic functions of \mathcal{L} and parabolic Harnack principle for positive parabolic functions of \mathcal{L} . Moreover, two-sided sharp heat kernel estimates are derived for such operator \mathcal{L} and jump-diffusion X . In particular, our results apply to the mixture of symmetric diffusion of uniformly elliptic divergence form operator and mixed stable-like processes on \mathbb{R}^d . To establish these results, we employ methods from both probability theory and analysis.

Random walks on strict partitions

We construct a diffusion process in the infinite-dimensional simplex consisting of all nonincreasing infinite sequences of nonnegative numbers with sum less than or equal to one. The process is constructed as a limit of a certain sequence of Markov chains. The state space of the nth chain is the set of all strict partitions of n (that is, partitions with distinct parts). As n →∞, these random walks converge to a continuous-time strong Markov process in the infinite-dimensional simplex. The process has continuous sample paths. The main result about the limit process is an expression for its pre-generator as a formal second-order differential operator in a polynomial algebra. Bibliography: 29 titles.

A Dirichlet process characterization of a class of reflected diffusions

For a class of stochastic differential equations with reflection for which a certain ${\mathbb{L}}^p$ continuity condition holds with $p>1$, it is shown that any weak solution that is a strong Markov process can be decomposed into the sum of a local martingale and a continuous, adapted process of zero $p$-variation. When $p=2$, this implies that the reflected diffusion is a Dirichlet process. Two examples are provided to motivate such a characterization. The first example is a class of multidimensional reflected diffusions in polyhedral conical domains that arise as approximations of certain stochastic networks, and the second example is a family of two-dimensional reflected diffusions in curved domains. In both cases, the reflected diffusions are shown to be Dirichlet processes, but not semimartingales.

Optimal Stopping Games and Nash Equilibrium

We show that the value function of the optimal stopping game for a right-continuous strong Markov process can be identified via equality between the smallest superharmonic and the largest subharmonic function lying between the gain functions (semiharmonic characterization) if and only if the Nash equilibrium holds (i.e., there exists a saddle point of optimal stopping times). When specialized to optimal stopping problems it is seen that the former identification reduces to the classic characterization of the value function in terms of superharmonic or subharmonic functions. The equivalence itself shows that finding the value function by “pulling a rope” between “two obstacles” is the same as establishing a Nash equilibrium. Further properties of the value function and the optimal stopping times are exhibited in the proof.

Non-local Dirichlet forms and symmetric jump processes

We consider the non-local symmetric Dirichlet form $(\mathcal {E}, \mathcal {F})$ given by \[ \mathcal {E} (f,f)=\int \limits _{\mathbb {R}^d} \int \limits _{\mathbb {R}^d} (f(y)-f(x))^2 J(x,y) dx dy \] with $\mathcal {F}$ the closure with respect to $\mathcal {E}_1$ of the set of $C^1$ functions on $\mathbb {R}^d$ with compact support, where $\mathcal {E}_1 (f, f):=\mathcal {E} (f, f)+\int _{\mathbb {R}^d} f(x)^2 dx$, and where the jump kernel $J$ satisfies \[ \kappa _1|y-x|^{-d-\alpha } \leq J(x,y) \leq \kappa _2|y-x|^{-d-\beta } \] for $0<\alpha < \beta <2, |x-y|<1$. This assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to $(\mathcal {E}, \mathcal {F})$. We prove a parabolic Harnack inequality for non-negative functions that solve the heat equation with respect to $\mathcal {E}$. Finally we construct an example where the corresponding harmonic functions need not be continuous.

CONSTRUCTION OF ELLIPTIC DIFFUSIONS WITH REFLECTING BOUNDARY CONDITION AND AN APPLICATION TO CONTINUOUS N-PARTICLE SYSTEMS WITH SINGULAR INTERACTIONS

AbstractWe give a Dirichlet form approach for the construction and analysis of elliptic diffusions in $\bar{\varOmega}\subset\mathbb{R}^n$ with reflecting boundary condition. The problem is formulated in an $L^2$-setting with respect to a reference measure $\mu$ on $\bar{\varOmega}$ having an integrable, $\mathrm{d} x$-almost everywhere (a.e.) positive density $\varrho$ with respect to the Lebesgue measure. The symmetric Dirichlet forms $(\mathcal{E}^{\varrho,a},D(\mathcal{E}^{\varrho,a}))$ we consider are the closure of the symmetric bilinear forms\begin{gather*} \mathcal{E}^{\varrho,a}(f,g)=\sum_{i,j=1}^n\int_{\varOmega}\partial_ifa_{ij} \partial_jg\,\mathrm{d}\mu,\quad f,g\in\mathcal{D}, \\ \mathcal{D}=\{f\in C(\bar{\varOmega})\mid f\in W^{1,1}_{\mathrm{loc}}(\varOmega),\ \mathcal{E}^{\varrho,a}(f,f)\lt\infty\}, \end{gather*}in $L^2(\bar{\varOmega},\mu)$, where $a$ is a symmetric, elliptic, $n\times n$-matrix-valued measurable function on $\bar{\varOmega}$. Assuming that $\varOmega$ is an open, relatively compact set with boundary $\partial\varOmega$ of Lebesgue measure zero and that $\varrho$ satisfies the Hamza condition, we can show that $(\mathcal{E}^{\varrho,a},D(\mathcal{E}^{\varrho,a}))$ is a local, quasi-regular Dirichlet form. Hence, it has an associated self-adjoint generator $(L^{\varrho,a},D(L^{\varrho,a}))$ and diffusion process $\bm{M}^{\varrho,a}$ (i.e. an associated strong Markov process with continuous sample paths). Furthermore, since $1\in D(\mathcal{E}^{\varrho,a})$ (due to the Neumann boundary condition) and $\mathcal{E}^{\varrho,a}(1,1)=0$, we obtain a conservative process $\bm{M}^{\varrho,a}$ (i.e. $\bm{M}^{\varrho,a}$ has infinite lifetime). Additionally, assuming that $\sqrt{\varrho}\in W^{1,2}(\varOmega)\cap C(\bar{\varOmega})$ or that $\varrho$ is bounded, $\varOmega$ is convex and $\{\varrho=0\}$ has codimension at least 2, we can show that the set $\{\varrho=0\}$ has $\mathcal{E}^{\varrho,a}$-capacity zero. Therefore, in this case we can even construct an associated conservative diffusion process in $\{\varrho&gt;0\}$. This is essential for our application to continuous $N$-particle systems with singular interactions. Note that for the construction of the self-adjoint generator $(L^{\varrho,a},D(L^{\varrho,a}))$ and the Markov process $\bm{M}^{\varrho,a}$ we do not need to assume any differentiability condition on $\varrho$ and $a$. We obtain the following explicit representation of the generator for $\sqrt{\varrho}\in W^{1,2}(\varOmega)$ and $a\in W^{1,\infty}(\varOmega)$:$$ L^{\varrho,a}=\sum_{i,j=1}^n\partial_i(a_{ij}\partial_j)+\partial_i(\log\varrho)a_{ij}\partial_j. $$Note that the drift term can be singular, because we allow $\varrho$ to be zero on a set of Lebesgue measure zero. Our assumptions in this paper even allow a drift that is not integrable with respect to the Lebesgue measure.

Non-Exclusive Real Options with First-Mover Advantages

This paper analyses the exercise decision of non-exclusive real options in a two-player setting. A general model of non-exclusive real options, allowing the underlying asset to follow any strong Markov process is developed, thus extending the existing literature, which is mainly based on one-dimensional geometric Brownian motion. For games with a first-mover advantage it is proved that an equilibrium with the rent-equalization property exists. It is also shown that many more exercise scenarios are possible than in deterministic timing games. As an example, a duopoly where two firms can adopt a new technology, whose profitability follows a two-dimensional, correlated geometric Brownian motion is studied.

Subgeometric rates of convergence of f-ergodic strong Markov processes

We provide a condition in terms of a supermartingale property for a functional of the Markov process, which implies (a) f -ergodicity of strong Markov processes at a subgeometric rate, and (b) a moderate deviation principle for an integral (bounded) functional. An equivalent condition in terms of a drift inequality on the extended generator is also given. Results related to ( f , r ) -regularity of the process, of some skeleton chains and of the resolvent chain are also derived. Applications to specific processes are considered, including elliptic stochastic differential equations, Langevin diffusions, hypoelliptic stochastic damping Hamiltonian systems and storage models.

A priori estimates for integro-differential operators with measurable kernels

The aim of this work is to develop a localization technique and to establish a regularity result for non-local integro-differential operators \({\fancyscript{L}}\) of order \({\alpha\in (0,2)}\) . Thereby we extend the De Giorgi–Nash–Moser theory to non-local integro-differential operators. The operators \({\fancyscript{L}}\) under consideration generate strong Markov processes via the theory of Dirichlet forms. As is well known, regularity properties of the resolvents are important for many aspects of the corresponding stochastic process. Therefore, this work is related to probability theory and analysis, especially partial differential equations, at the same time.

Intensity process and compensator: A new filtration expansion approach and the Jeulin–Yor theorem

Let $(X_t)_{t\ge0}$ be a continuous-time, time-homogeneous strong Markov process with possible jumps and let $\tau$ be its first hitting time of a Borel subset of the state space. Suppose $X$ is sampled at random times and suppose also that $X$ has not hit the Borel set by time $t$. What is the intensity process of $\tau$ based on this information? This question from credit risk encompasses basic mathematical problems concerning the existence of an intensity process and filtration expansions, as well as some conceptual issues for credit risk. By revisiting and extending the famous Jeulin--Yor [Lecture Notes in Math. 649 (1978) 78--97] result regarding compensators under a general filtration expansion framework, a novel computation methodology for the intensity process of a stopping time is proposed. En route, an analogous characterization result for martingales of Jacod and Skorohod [Lecture Notes in Math. 1583 (1994) 21--35] under local jumping filtration is derived.

A Theory of Non-Exclusive Real Options With First-Mover Advantages

This paper analyses the exercise decision of non-exclusive real options in a two-player setting. A general model of non-exclusive real options, allowing the underlying asset to follow any strong Markov process is developed, thus extending the existing literature, which is mainly based on one-dimensional geometric Brownian motion. For games with a first-mover advantage it is proved that an equilibrium with the rent-equalisation property exists. As an example, a duopoly where two firms can adopt a new technology, whose profitability follows a two-dimensional, correlated geometric Brownian motion is studied.

Perturbation of symmetric Markov processes

We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower-order perturbation of the L 2-infinitesimal generator $$\mathcal {L}$$ of a general symmetric Markov process. An illuminating concrete example for $$\mathcal {L}$$ is $$\Delta_D-(-\Delta)^s_D$$ , where D is a bounded Euclidean domain in $$\mathbb {R}^d, s\in [0, 1], \Delta_D$$ is the Laplace operator in D with zero Dirichlet boundary condition and $$-(-\Delta)^s_D$$ is the fractional Laplacian in D with zero exterior condition. The strong Markov process corresponding to $$\mathcal {L}$$ is a Lévy process that is the sum of Brownian motion in $$\mathbb {R}^d$$ and an independent symmetric (2s)-stable process in $$\mathbb {R}^d$$ killed upon exiting the domain D. This probabilistic representation is a combination of Feynman-Kac and Girsanov formulas. Crucial to the development is the use of an extension of Nakao’s stochastic integral for zero-energy additive functionals and the associated Itô formula, both of which were recently developed in Chen et al. [Stochastic calculus for Dirichlet processes (preprint)(2006)].