arXiv subjects
Publications and source records attributed to Andrzej Rozkosz.
We consider solutions of the Cauchy problem for semilinear equations with (possibly) different Lévy operators. We provide various results on their convergence under the assumption that symbols of the involved operators converge to the symbol of some Lévy operator. Some results are proved for a more general class of pseudodifferential operators.
We show identities of Hardy-Stein type for harmonic functions relative to integro-differential operators corresponding to general symmetric regular Dirichlet forms satisfying the absolute continuity condition. The novelty is that we consider operators of mixed type containing both local and nonlocal component. Moreover, the identities are proved for compositions of harmonic functions and general convex functions. We also provide some conditional identities, i.e. identities for ratios of harmonic functions. As an application we give a characterization of norms in harmonic Hardy spaces and prove Littlewood--Paley type estimates for square functions. To illustrate general results, we discuss in some details the case of divergence form operator and purely nonlocal operator defined by some jump kernel. Our proofs are rather short and use mainly probabilistic methods.
We consider Neumann problem for linear elliptic equations involving integro-differential operators of Levy-type. We show that suitably defined viscosity solutions have probabilistic representations given in terms of the reflected stochastic Skorokhod equation associated with an Ito process and an independent pure-jump Levy process. As an application of the representation we show that viscosity solutions arise a limits of some penalized equations and give some stability results for the viscosity solutions. Our proofs are based on new limit theorems for solutions of penalized stochastic equations with jumps and new estimates on the bounded variation parts of the solutions.
We consider the Robin problem for a uniformly elliptic divergence operator with measure data on the right-hand side of the equation and an absorption term on the boundary involving blowing up terms. We prove the existence of a positive renormalized solution and provide it stochastic representation, which can be viwed as a generalized nonlinear Feynman-Kac formula. From this representation we derive some additional regularity results for the solution and some uniqueness results.
We study the Dirichlet problem for semilinear equations on general open sets with measure data on the right-hand side and irregular boundary data. For this purpose we develop the classical method of orthogonal projection. We treat in a unified form equations with operators belonging to the broad class of integro-differential operators associated with symmetric regular Dirichlet forms.
We consider the Dirichlet problem for equation involving a general operator associated with a symmetric transient regular Dirichlet form and bounded Borel measure on the right-hand side of the equation. We introduce a new function space (depending on the form) which allows us to distinguish between solutions with diffuse measure and with general Borel measure. This new space can by characterized analytically in terms of the Poisson kernel associated with the underlying operator or probabilistically by using the notion of Doob class (D) of processes naturally associated with the operator. We also prove a reconstruction formula describing, in terms of the carré du champ operator and jump measure associated with the underlying form, the behaviour of the solution on the set where it is very large.
We show, that under natural assumptions, solutions of Dirichlet problems for uniformly elliptic divergence form operator can be approximated pointwise by solutions of some versions of Robin problems. The proof is based on stochastic representation of solutions and properties of reflected diffusions corresponding to divergence form operators.
We provide general conditions ensuring that the value functions of some nonlinear stopping problems with finite horizon converge to the value functions of the corresponding problems with infinite horizon. Our result can be formulated as result on stability, with respect to time horizon, of nonlinear $f$-expectations. We also study the rate of convergence. Many examples are given to illustrate our results. They include the analysis of time asymptotics of the fair prices of American options in a multidimensional exponential Lévy model.
We consider the problem of pricing perpetual American options written on dividend-paying assets whose price dynamics follow a multidimensional Black and Scholes model. For convex Lipschitz continuous reward functions, we give a probabilistic characterization of the fair price in terms of a reflected BSDE, and an analytical one in terms of an obstacle problem. We also provide the early exercise premium formula.
Let $Q=(0,T)\timesΩ$, where $Ω$ is a bounded open subset of $\mathbb{R}^d$. We consider the parabolic $p$-capacity on $Q$ naturally associated with the usual $p$-Laplacian. Droniou, Porretta and Prignet have shown that if a bounded Radon measure $μ$ on $Q$ is diffuse, i.e. charges no set of zero $p$-capacity, $p>1$, then it is of the form $μ=f+\mbox{div}(G)+g_t$ for some $f\in L^1(Q)$, $G\in (L^{p'}(Q))^d$ and $g\in L^p(0,T;W^{1,p}_0(Ω)\cap L^2(Ω))$. We show the converse of this result: if $p>1$, then each bounded Radon measure $μ$ on $Q$ admitting such a decomposition is diffuse.
In the paper, we first propose a definition of renormalized solution of semilinear elliptic equation involving operator corresponding to a general (possibly nonlocal) symmetric regular Dirichlet form satisfying the so-called absolute continuity condition and general (possibly nonsmooth) measure data. Then we analyze the relationship between our definition and other concepts of solutions considered in the literature (probabilistic solutions, solution defined via the resolvent kernel of the underlying Dirichlet form, Stampacchia's definition by duality). We show that under mild integrability assumption on the data all these concepts coincide.
We consider a family $\{L_t,\, t\in [0,T]\}$ of closed operators generated by a family of regular (non-symmetric) Dirichlet forms $\{(B^{(t)},V),t\in[0,T]\}$ on $L^2(E;m)$. We show that a bounded (signed) measure $μ$ on $(0,T)\times E$ is smooth, i.e. charges no set of zero parabolic capacity associated with $\frac{\partial}{\partial t}+L_t$, if and only if $μ$ is of the form $μ=f\cdot m_1+g_1+\partial_tg_2$ with $f\in L^1((0,T)\times E;dt\otimes m)$, $g_1\in L^2(0,T;V')$, $g_2\in L^2(0,T;V)$. We apply this decomposition to the study of the structure of additive functionals in the Revuz correspondence with smooth measures. As a by-product, we also give some existence and uniqueness results for solutions of semilinear equations involving the operator $\frac{\partial}{\partial t}+L_t$ and a functional from the dual $\mathcal{W}'$ of the space $\mathcal{W}=\{u\in L^2(0,T;V):\partial_t u\in L^2(0,T;V')\}$ on the right-hand side of the equation.
We consider semilinear equation of the form $-Lu=f(x,u)+μ$, where $L$ is the operator corresponding to a transient symmetric regular Dirichlet form ${\mathcal E}$, $μ$ is a diffuse measure with respect to the capacity associated with ${\mathcal E}$, and the lower-order perturbing term $f(x,u)$ satisfies the sign condition in $u$ and some weak integrability condition (no growth condition on $f(x,u)$ as a function of $u$ is imposed). We prove the existence of a solution under mild additional assumptions on ${\mathcal E}$. We also show that the solution is unique if $f$ is nonincreasing in $u$.
We study large time behaviour of solutions of the Cauchy problem for equations of the form $\partial_tu-L u+λu=f(x,u)+g(x,u)\cdotμ$, where $L$ is the operator associated with a regular lower bounded semi-Dirichlet form ${\mathcal{E}}$ and $μ$ is a nonnegative bounded smooth measure with respect to the capacity determined by ${\mathcal{E}}$. We show that under the monotonicity and some integrability assumptions on $f,g$ as well as some assumptions on the form ${\mathcal{E}}$, $u(t,x)\rightarrow v(x)$ as $t\rightarrow\infty$ for quasi-every $x$, where $v$ is a solution of some elliptic equation associated with our parabolic equation. We also provide the rate convergence. Some examples illustrating the utility of our general results are given.
We consider the problem of valuation of American options written on dividend-paying assets whose price dynamics follows a multidimensional exponential Levy model. We carefully examine the relation between the option prices, related partial integro-differential variational inequalities and reflected backward stochastic differential equations. In particular, we prove regularity results for the value function and obtain the early exercise premium formula for broad class of payoff functions.
We consider a quasi-regular Dirichlet form. We show that a bounded signed measure charges no set of zero capacity associated with the form if and only if the measure can be decomposed into the sum of an integrable function and a bounded linear functional on the domain of the form. The decomposition allows one to describe explicitly the set of bounded measures charging no sets of zero capacity for interesting classes of Dirichlet forms. By way of illustration, some examples are given.
We consider a system of seminlinear parabolic variational inequalities with time-dependent convex obstacles. We prove the existence and uniqueness of its solution. We also provide a stochastic representation of the solution and show that it can be approximated by the penalization method. Our proofs are based upon probabilistic methods from the theory of Markov processes and the theory of backward stochastic differential equations.
We study the problem of existence, uniqueness and regularity of probabilistic solutions of the Cauchy problem for nonlinear stochastic partial differential equations involving operators corresponding to regular (nonsymmetric) Dirichlet forms. In proofs we combine the methods of backward doubly stochastic differential equations with those of probabilistic potential theory and Dirichlet forms.