Let $L$ be a Lévy-type generator whose Lévy measure is controlled from below by that of a non-degenerate $α$-stable ($0<α<2$) process. In this paper, we study the martingale problem for the operator $\mathcal{L}_{t}=L+K_{t}$, with $K_{t}$ being a time-dependent non-local operator defined by \[ K_{t}f(x):=\int_{\mathbb{R}^{d}\backslash\{0\}}[f(x+y)-f(x)-\mathbf{1}_{α>1}\mathbf{1}_{\{|y|\le1\}}y\cdot\nabla f(x)]M(t,x,dy), \] where $M(t,x,\cdot)$ is a Lévy measure on $\mathbb{R}^{d}\backslash\{0\}$ for each $(t,x)\in \mathbb{R}_+ \times \mathbb{R}^{d}$. We show that if \[ \sup_{t\geq0,x\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}\backslash\{0\}}1\wedge|y|^βM(t,x,dy)<\infty \] for some $0<β<α$, then the martingale problem for $\mathcal{L}_{t}$ is well-posed.
We introduce and discuss Lévy-type cylindrical martingale problems on separable reflexive Banach spaces. Our main observations are the following: Cylindrical martingale problems have a one-to-one relation to weak solutions of stochastic partial differential equations. Moreover, well-posed problems possess the strong Markov property and a Cameron-Martin-Girsanov-type formula holds. As applications, we derive existence and uniqueness results.
We prove limit theorems for cylindrical martingale problems associated to Lévy generators. Furthermore, we give sufficient and necessary conditions for the Feller property of well-posed problems with continuous coefficients. We discuss two applications. First, we derive continuity and linear growth conditions for the existence of weak solutions to infinite-dimensional stochastic differential equations driven by Lévy noise. Second, we derive continuity, local boundedness and linear growth conditions for limit theorems and the Feller property of weak solutions to stochastic partial differential …
The dynamics of a Markov process are often specified by its infinitesimal generator or, equivalently, its symbol. This paper contains examples of analytic symbols which do not determine the law of the corresponding Markov process uniquely. These examples also show that the law of a polynomial process is not necessarily determined by its generator. On the other hand, we show that a combination of smoothness of the symbol and ellipticity warrants uniqueness in law. The proof of this result is based on proving stability of univariate marginals relative to some properly chosen distance.
This paper is devoted to the study of a certain type of martingale problems associated to general operators corresponding to processes which have finite lifetime. We analyse several properties and in particular the weak convergence of sequences of solutions for an appropriate Skorokhod topology setting. We point out the Feller-type features of the associated solutions to this type of martingale problem. Then localisation theorems for well-posed martingale problems or for corresponding generators are proved.
Under suitable assumptions of regularity and non-degeneracy on the covariance of the driving additive noise, any Markov solution to the stochastic Navier-Stokes equations has an associated generator of the diffusion and is the unique solution to the corresponding martingale problem. Some elementary examples are discussed to interpret these results.
We construct a family of non-Gaussian martingales the marginals of which are all Gaussian. We give the predictable quadratic variation of these processes and show they do not have continuous paths. These processes are Markovian and inhomogeneous in time, and we give their infinitesimal generators. Within this family we find a class of piecewise deterministic pure jump processes and describe the laws of jumps and times between the jumps.
We develop a martingale approach for a class of singular stochastic PDEs of Burgers type (including fractional and multi-component Burgers equations) by constructing a domain for their infinitesimal generators. It was known that the domain must have trivial intersection with the usual cylinder test functions, and to overcome this difficulty we import some ideas from paracontrolled distributions to an infinite dimensional setting in order to construct a domain of controlled functions. Using the new domain, we are able to prove existence and uniqueness for the Kolmogorov backward equation and th…
We study a backward stochastic differential equation whose terminal condition is an integrable function of a local martingale and generator has bounded growth in $z$. When the local martingale is a strict local martingale, the BSDE admits at least two different solutions. Other than a solution whose first component is of class D, there exists another solution whose first component is not of class D and strictly dominates the class D solution. Both solutions are $\mathbb{L}^p$ integrable for any $0<p<1$. These two different BSDE solutions generate different viscosity solutions to the associated…
This paper establishes an existence and uniqueness result for the adapted solution of a general time interval multidimensional backward stochastic differential equation (BSDE), where the generator $g$ satisfies a weak stochastic-monotonicity condition and a general growth condition in the state variable $y$, and a stochastic-Lipschitz condition in the state variable $z$. This unifies and strengthens some known works. In order to prove this result, we develop some ideas and techniques employed in \citet{XiaoFan2017Stochastics} and \citet{LiuLiFan2019CAM}. In particular, we put forward and prove…
Constrained Markov processes, such as reflecting diffusions, behave as an unconstrained process in the interior of a domain but upon reaching the boundary are controlled in some way so that they do not leave the closure of the domain. In this paper, the behavior in the interior is specified by a generator of a Markov process, and the constraints are specified by a controlled generator. Together, the generators define a "constrained martingale problem". The desired constrained processes are constructed by first solving a simpler "controlled martingale problem" and then obtaining the desired pro…
With an emphasis on generators with quadratic growth in the control variable we consider measure solutions of BSDE, a solution concept corresponding to the notion of risk neutral measure in mathematical finance. In terms of measure solutions, solving a BSDE reduces to martingale representation with respect to an underlying filtration. Measure solutions related to measures equivalent to the historical one provide classical solutions. We derive the existence of measure solutions in scenarios in which the generating functions are just continuous, of at most linear growth in the control variable (…
A study of a non-linear parabolic SPDEs of the form $\partial_{t}u=\mathcal{L}\,u + σ(u)f(B_t^x,t)\dot{w}$ with $\dot{w}$ as the space-time white noise and $f(B_t^x,t)$ a space-time harmonic function was done. The function $σ:\mathbb{R}\rightarrow\mathbb{R}$ is Lipschitz continuous and $\mathcal{L}$ the $L^2$-generator of a Lévy process. Some precise condition for existence and uniqueness of the solution were given and we show that the solution grows weakly(in law/distribution) in time (for large $t$) at most a precise exponential rate for the $\mathcal{L}$; and grows in time at most a precise…
We show that, under certain smoothness conditions, a Brownian martingale at a fixed time can be represented as an exponential of its value at a later time. The time-dependent generator of this exponential operator is equal to one half times the Malliavin derivative. This result can also be seen as a generalization of the semi-group theory of parabolic partial differential equations to the parabolic path-dependent partial differential equations introduced by Dupire (2009) and Cont and Founié (2011). The exponential operator can be calculated explicitly in a series expansion, which resembles the…
We study a stochastic optimal control problem for forward-backward control systems with quadratic generators. In order to establish the first and second-order variational and adjoint equations, we obtain a new estimate for one-dimensional linear BSDEs with unbounded stochastic Lipschitz coefficients involving bounded mean oscillation martingales (BMO-martingales for short) and prove the solvability for a class of multi-dimensional BSDEs with this type. Finally, a new global stochastic maximum principle is deduced.
Let $(\mathbb{P}^{s,x})_{(s,x)\in[0,T]\times E}$ be a family of probability measures, where $E$ is a Polish space,defined on the canonical probability space ${\mathbb D}([0,T],E)$ of $E$-valued cadlag functions. We suppose that a martingale problem with respect to a time-inhomogeneous generator $a$ is well-posed. We consider also an associated semilinear {\it Pseudo-PDE} with generator $a$ for which we introduce a notion of so called {\it decoupled mild} solution and study the equivalence with the notion of martingale solution introduced in a companion paper. We also investigate well-posedness…
We establish the existence of martingale solutions to a class of stochastic conservation equations. The underlying models correspond to random perturbations of kinetic models for collective motion such as the Cucker-Smale and Motsch-Tadmor models. By regularizing the coefficients, we first construct approximate solutions obtained as the mean-field limit of the corresponding particle systems. We then establish the compactness in law of this family of solutions by relying on a stochastic averaging lemma. This extends the results obtained by Karper, Mellet and Trivisa in the deterministic case.
Let $\{d_k\}_{k \geq 0}$ be a complex martingale difference in $L^p[0,1],$ where $1<p<\infty,$ and $\{\e_k\}_{k \geq 0}$ a sequence in $\{\pm 1\}.$ We obtain the following generalization of Burkholder's famous result. If $τ\in [-\frac 12, \frac 12]$ and $n \in \Z_+$ then $$|\sum_{k=0}^n{(\{c} \e_k τ) d_k}|_{L^p([0,1], \C^2)} \leq ((p^*-1)^2 + τ^2)^{\frac 12}|\sum_{k=0}^n{d_k}|_{L^p([0,1], \C)},$$ where $((p^*-1)^2 + τ^2)^{\frac 12}$ is sharp and $p^*-1 = \max\{p-1, \frac 1{p-1}\}.$ For $2\leq p<\infty$ the result is also true with sharp constant for $τ\in \R.$
We show that, under certain smoothness conditions, a Brownian martingale, when evaluated at a fixed time, can be represented via an exponential formula at a later time. The time-dependent generator of this exponential operator only depends on the second order Malliavin derivative operator evaluated along a "frozen path". The exponential operator can be expanded explicitly to a series representation, which resembles the Dyson series of quantum mechanics. Our continuous-time martingale representation result can be proven independently by two different methods. In the first method, one constructs…
We consider a general piecewise deterministic Markov process (PDMP) $X=\{X_t\}_{t\geqslant 0}$ with measure-valued generator $\mathcal{A}$, for which the conditional distribution function of the inter-occurrence time is not necessarily absolutely continuous. A general form of the exponential martingales is presented as $$M^f_t=\frac{f(X_t)}{f(X_0)}\left[\mathrm{Sexp}\left(\int_{(0,t]}\frac{\mathrm{d}L(\mathcal{A}f)_s}{f(X_{s-})}\right)\right]^{-1}.$$ Using this exponential martingale as a likelihood ratio process, we define a new probability measure. It is shown that the original process remai…
In this note we prove that the local martingale part of a convex function f of a d-dimensional semimartingale X = M + A can be written in terms of an It^o stochastic integral \int H(X)dM, where H(x) is some particular measurable choice of subgradient of f at x, and M is the martingale part of X. This result was first proved by Bouleau in [2]. Here we present a new treatment of the problem. We first prove the result for X' = X + eB, e > 0, where B is a standard Brownian motion, and then pass to the limit as e tends to 0, using results in [1] and [4].
We study perturbations of Feller generators under `lower order terms' with measurable coefficients. We investigate which properties of the original semigroup -- such as positivity, conservativeness and the Feller property -- are passed to the perturbed semigroup. We give several examples and discuss applications in the theory of martingale problems and stochastic differential equations with measurable coefficients.
We study existence of probability measure valued jump-diffusions described by martingale problems. We develop a simple device that allows us to embed Wasserstein spaces and other similar spaces of probability measures into locally compact spaces where classical existence theory for martingale problems can be applied. The method allows for general dynamics including drift, diffusion, and possibly infinite-activity jumps. We also develop tools for verifying the required conditions on the generator, including the positive maximum principle and certain continuity and growth conditions. To illustra…
We consider the martingale problem associated to the Navier-Stokes in dimension 2 or 3. Existence is well known and it has been recently shown that markovian transition semi group associated to these equations can be constructed. We study the Kolmogorov operator associated to these equations. It can be defined formally as a differential operator on an infinite dimensional Hilbert space. It can be also defined in an abstract way as the infinitesimal generator of the transition semi group. We explicit cores for these abstract operators and identify them with the concrete differential operators o…
We deal with backward stochastic differential equations with time delayed generators. In this new type of equations, a generator at time t can depend on the values of a solution in the past, weighted with a time delay function for instance of the moving average type. We prove existence and uniqueness of a solution for a sufficiently small time horizon or for a sufficiently small Lipschitz constant of a generator. We give examples of BSDE with time delayed generators that have multiple solutions or that have no solutions. We show for some special class of generators that existence and uniquenes…
We prove that weakly continuous solutions to martingale problems admit a canonical regular conditional probability distribution. This allows for the construction of time consistent convex dynamic procedures in a non dominated setting. Making use of the martingale problem approach for continuous diffusions and diffusions with Levy generator, we give an explicit construction of such procedures having furthermore a Feller property. These procedures lead to viscosity solution of fully non linear second order partial differential equations in case of continuous diffusions. In case of diffusions wit…
In this article we study and classify optimal martingales in the dual formulation of optimal stopping problems. In this respect we distinguish between weakly optimal and surely optimal martingales. It is shown that the family of weakly optimal and surely optimal martingales may be quite large. On the other hand it is shown that the Doob-martingale, that is, the martingale part of the Snell envelope, is in a certain sense the most robust surely optimal martingale under random perturbations. This new insight leads to a novel randomized dual martingale minimization algorithm that doesn't require …
In this paper, we give several new results on solvability of a quadratic BSDE whose generator depends also on the mean of both variables. First, we consider such a BSDE using John-Nirenberg's inequality for BMO martingales to estimate its contribution to the evolution of the first unknown variable. Then we consider the BSDE having an additive expected value of a quadratic generator in addition to the usual quadratic one. In this case, we use a deterministic shift transformation to the first unknown variable, when the usual quadratic generator depends neither on the first variable nor its mean,…
We consider a piecewise-deterministic Markov process (PDMP) with general conditional distribution of inter-occurrence time, which is called a general PDMP here. Our purpose is to establish the theory of measure-valued generator for general PDMPs. The additive functional of a semi-dynamic system (SDS) is introduced firstly, which presents us an analytic tool for the whole paper. The additive functionals of a general PDMP are represented in terms of additive functionals of the SDS. The necessary and sufficient conditions of being a local martingale or a special semimartingale for them are given.…
We prove that if we are given a generator of a cadlag Markov process and an open domain $G$ in the state space, on which the generator has the local property expressed in a suitable way on a class $\mathcal{C}$ of test functions that is sufficiently rich, then the Markov process has continuous paths when it passes through $G$. The result holds for any Markov process which is associated with the generator merely on $\mathcal{C}$. This points out that the path continuity of the process is an a priori property encrypted by the generator acting on enough test functions, and this property can be ea…
We consider backward stochastic differential equations (BSDE) with nonlinear generators typically of quadratic growth in the control variable. A measure solution of such a BSDE will be understood as a probability measure under which the generator is seen as vanishing, so that the classical solution can be reconstructed by a combination of the operations of conditioning and using martingale representations. In case the terminal condition is bounded and the generator fulfills the usual continuity and boundedness conditions, we show that measure solutions with equivalent measures just reinterpret…
The loop measure is associated with a Markov generator. We compute the variation of the loop measure induced by an in nitesimal variation of the generator a ecting the killing rates or the jumping rates.
The problem of finding a martingale on a manifold with a fixed random terminal value can be solved by considering BSDEs with a generator with quadratic growth. We study here a generalization of these equations and we give uniqueness and existence results in two different frameworks, using differential geometry tools. Applications to PDEs are given, including a certain class of Dirichlet problems on manifolds.
We develop a technique based on Malliavin-Bismut calculus ideas, for asymptotic expansion of dual control problems arising in connection with exponential indifference valuation of claims, and with minimisation of relative entropy, in incomplete markets. The problems involve optimisation of a functional of Brownian paths on Wiener space, with the paths perturbed by a drift involving the control. In addition there is a penalty term in which the control features quadratically. The drift perturbation is interpreted as a measure change using the Girsanov theorem, leading to a form of the integratio…
We study the sensitivity of the expected utility maximization problem in a continuous semi-martingale market with respect to small changes in the market price of risk. Assuming that the preferences of a rational economic agent are modeled with a general utility function, we obtain a second-order expansion of the value function, a first-order approximation of the terminal wealth, and construct trading strategies that match the indirect utility function up to the second order. If a risk-tolerance wealth process exists, using it as a numéraire and under an appropriate change of measure, we reduce…
In an incomplete model, where under an appropriate numéraire, the stock price process is driven by a sigma-bounded semimartingale, we investigate the behavior of the expected utility maximization problem under small perturbations of the numéraire. We establish a quadratic approximation of the value function and a first-order expansion of the terminal wealth. Relying on a description of the base return process in terms of its semimartingale characteristics, we also construct wealth processes and nearly optimal strategies that allow for matching the primal value function up to the second order. …
First essential m-dissipativity of an infinite-dimensional Ornstein-Uhlenbeck operator $N$, perturbed by the gradient of a potential, on a domain $\mathcal{F}C_b^{\infty}$ of finitely based, smooth and bounded functions, is shown. Our considerations allow unbounded diffusion operators as coefficients. We derive corresponding second order regularity estimates for solutions $f$ of the Kolmogorov equation $αf-Nf=g$, $α\in (0,\infty)$, generalizing some results of Da Prato and Lunardi. Second we prove essential m-dissipativity for generators $(L_Φ,\mathcal{F}C_b^{\infty})$ of infinite-dimensional …
We prove that the Green function of a generator of isotropic unimodal Lévy processes with the weak lower scaling order bigger than one and the Green function of its gradient perturbations are comparable for bounded smooth open sets if the drift function is from an appropriate Kato class.
We prove that the Green function of a generator of symmetric unimodal Lévy processes with the weak lower scaling order bigger than one and the Green function of its gradient perturbations are comparable for bounded $C^{1,1}$ subsets of the real line if the drift function is from an appropriate Kato class.
