Malliavin Derivative Research Articles

Backward doubly stochastic differential equations with polynomial growth coefficients

In this paper we study the solvability of backward doubly stochastic differential equations (BDSDEs for short) with polynomial growth coefficients and their connections with SPDEs. The corresponding SPDE is in a very general form, which may depend on the derivative of the solution. We use Wiener-Sobolev compactness arguments to derive a strongly convergent subsequence of approximating SPDEs. For this, we prove some new estimates to the solution and its Malliavin derivative of the corresponding approximating BDSDEs. These estimates lead to the verifications of the conditions in the Wiener-Sobolev compactness theorem and the solvability of the BDSDEs and the SPDEs with polynomial growth coefficients.

Honest and dishonest stochastic control for Lévy processes with application to property market

This work weighs the optimal profit of an honest agent who has access to additional information such as property maintenance, adequate security measures, than the one given by the development of the market and who takes advantage of this to optimise his position in the market. This is driven by the forward anticipating calculus while Malliavin derivatives model the dishonest agent, who is limited in information as a result of his inadequacy in property maintenance. A function f(t)is non-anticipative with respect to the Brownian motion z(t) if the value of the function at time t on the path of z(t) is determined by the past history of z(t) up to time t. It is non-anticipative in respect of an agent who may have: (i) less information from the general information generated by the market noise; (ii) a delayed information as such, we look at the flops of a dishonest agent and the rapid sales of property by the honest agent.

Adaptive Wick--Malliavin Approximation to Nonlinear SPDEs with Discrete Random Variables

We propose an adaptive Wick--Malliavin (WM) expansion in terms of the Malliavin derivative of order $Q$ to simplify the propagator of general polynomial chaos (gPC) of order $P$ (a system of deterministic equations for the coefficients of gPC) and to control the error growth with respect to time. Specifically, we demonstrate the effectiveness of the WM method by solving a stochastic reaction equation and a Burgers equation with several discrete random variables. Exponential convergence is shown numerically with respect to $Q$ when $Q \geq P-1$. We also analyze the computational complexity of the WM method and identify a significant speedup with respect to gPC, especially in high dimensions.

Explicit solution of relative entropy weighted control

We consider the minimization over probability measures of the expected value of a random variable, regularized by relative entropy with respect to a given probability distribution. In the general setting we provide a complete characterization of the situations in which a finite optimal value exists and the situations in which a minimizing probability distribution exists. Specializing to the case where the underlying probability distribution is Wiener measure, we characterize finite relative entropy changes of measure in terms of square integrability of the corresponding change of drift. For the optimal change of measure for the relative entropy weighted optimization, an expression involving the Malliavin derivative of the cost random variable is derived. The theory is illustrated by its application to several examples, including the case where the cost variable is the maximum of a standard Brownian motion over a finite time horizon. For this example we obtain an exact optimal drift, as well as an approximation of the optimal drift through a Monte-Carlo algorithm.

Function Spaces and Operators with Applications

The theory of function spaces and operators along with their various applications is an active branch of mathematics and there were a large number of papers published in the relative areas and many major discoveries have been obtained. We are happy that this special issue has received the attention of many researchers in these areas. The special issue consisted of twelve research papers covering commutators of multiliner Fourier multiplier operators on weighted Lebesgue spaces, fractional Hardy operators and their commutators on Hardy spaces, Marcinkiewicz integrals in weighted weak Hardy spaces, geometry of normed linear spaces (including Banach spaces and Hilbert spaces), ArzelaAscoli theorem for demilinear mappings in topological linear spaces, Malliavin derivatives in variable exponent spaces, the theory of Young measures in variable exponent Sobolev spaces and applications, the applications of reproducing kernel spaces on binary singular problems and nonlinear fourth-order differential equations with integral boundary conditions, and weighted Lorentz spaces and operators. We hope that readers interested in functional analysis will find useful information in this special issue. At the same time we also hope that in the near future we can see new papers to be published based on this special issue.

Convertible Bonds with Higher Loan Rate: Model, Valuation, and Optimal Strategy

We study the pricing problem for convertible bonds via backward stochastic differential equations (BSDEs). By virtue of reflected BSDEs and Malliavin derivatives, we establish the formulae for the fair price of convertible bonds and the hedging portfolio strategy explicitly. We also obtain the optimal conversion time when there is no dividends-paying for underlying common stocks. Furthermore, we consider the case that the loan rate is higher than riskless interest rate in a financial market, and conclude that it does not affect the price of convertible bonds actually. To illustrate our results, some numerical simulations are given and discussed at last.

Malliavin Derivatives in Spaces with Variable Exponents

Spaces with variable exponentsLpx(H,μ)andLpx(H,μ;H)are introduced. After discussing some approximation results ofLpx(H,μ), Sobolev spaces onHwith variable exponents are introduced. At last, we define Malliavin derivatives inLpx(H,μ)and discuss some properties of Malliavin derivatives inLpx(H,μ).

BSDEs with terminal conditions that have bounded Malliavin derivative

We show existence and uniqueness of solutions to BSDEs of the formYt=ξ+∫tTf(s,Ys,Zs)ds−∫tTZsdWs in the case where the terminal condition ξ has bounded Malliavin derivative. The driver f(s,y,z) is assumed to be Lipschitz continuous in y but only locally Lipschitz continuous in z. In particular, it can grow arbitrarily fast in z. If in addition to having bounded Malliavin derivative, ξ is bounded, the driver needs only be locally Lipschitz continuous in y. In the special case where the BSDE is Markovian, we obtain existence and uniqueness results for semilinear parabolic PDEs with non-Lipschitz nonlinearities. We discuss the case where there is no lateral boundary as well as lateral boundary conditions of Dirichlet and Neumann type.

Comparison inequalities on Wiener space

We define a covariance-type operator on Wiener space: for F and G two random variables in the Gross–Sobolev space D1,2 of random variables with a square-integrable Malliavin derivative, we let ΓF,G≔〈DF,−DL−1G〉, where D is the Malliavin derivative operator and L−1 is the pseudo-inverse of the generator of the Ornstein–Uhlenbeck semigroup. We use Γ to extend the notion of covariance and canonical metric for vectors and random fields on Wiener space, and prove corresponding non-Gaussian comparison inequalities on Wiener space, which extend the Sudakov–Fernique result on comparison of expected suprema of Gaussian fields, and the Slepian inequality for functionals of Gaussian vectors. These results are proved using a so-called smart-path method on Wiener space, and are illustrated via various examples. We also illustrate the use of the same method by proving a Sherrington–Kirkpatrick universality result for spin systems in correlated and non-stationary non-Gaussian random media.

Smooth density for the solution of scalar SDEs with locally Lipschitz coefficients under Hörmander condition

Abstract In this paper the existence of a smooth density is proved for the solution of an SDE, with locally Lipschitz coefficients and semi-monotone drift, under Hormander condition. We prove the nondegeneracy condition for the solution of the SDE, from it an integration by parts formula would result in the Wiener space. To this end we construct a sequence of SDEs with globally Lipschitz coefficients whose solutions converge to the original one and use some Lyapunov functions to show the uniform boundedness of the p -moments of the solutions and their Malliavin derivatives with respect to n .

Malliavin regularity of solutions to mixed stochastic differential equations

For a mixed stochastic differential equation driven by independent fractional Brownian motions and Wiener processes, the existence and integrability of the Malliavin derivative of the solution are established. It is also proved that the solution possesses exponential moments.

Chaos expansion methods of stochastic processes for Malliavin-type equations

Abstract In this short survey we present the application of the Askey-scheme of orthogonal polynomials to define several discrete and continuous distribution types in frame-work of T. Hidaʼs white noise analysis. The results are applied to define fractional versions of the distributions and to solve stochastic differential equations involving the Malliavin derivative, Skorokhod integral and Ornstein-Uhlenbeck operator.

Tools for Malliavin Calculus in UMD Banach Spaces

In this paper we study the Malliavin derivatives and Skorohod integrals for processes taking values in an infinite dimensional space. Such results are motivated by their applications to SPDEs and in particular financial mathematics. Vector-valued Malliavin theory in Banach space E is naturally restricted to spaces E which have the so-called umd property, which arises in harmonic analysis and stochastic integration theory. We provide several new results and tools for the Malliavin derivatives and Skorohod integrals in an infinite dimensional setting. In particular, we prove weak characterizations, a chain rule for Lipschitz functions, a sufficient condition for pathwise continuity and an Itô formula for non-adapted processes.

Simulation of BSDEs by Wiener chaos expansion

We present an algorithm to solve BSDEs based on Wiener chaos expansion and Picard's iterations. We get a forward scheme where the conditional expectations are easily computed thanks to chaos decomposition formulas. We use the Malliavin derivative to compute $Z$. Concerning the error, we derive explicit bounds with respect to the number of chaos and the discretization time step. We also present numerical experiments. We obtain very encouraging results in terms of speed and accuracy.

BV-regularity for the Malliavin derivative of the maximum of the Wiener process

We show that, on the classical Wiener space, the random variable $M = \sup_{0\le t \le T} W_t$ admits a measure as second Malliavin derivative, whose total variation measure is finite and singular w.r.t. the Wiener measure.

Stochastic Burgers Equation with Random Initial Velocities: A Malliavin Calculus Approach

In this article we prove an existence theorem for solutions of the stochastic Burgers equation (SBE) on the unit interval with Dirichlet boundary conditions and anticipating initial velocities. The SBE is driven by affine (additive + linear) noise. In order to establish the existence theorem, we adopt a somewhat counterintuitive perspective in which stochastic dynamical systems ideas lead to the existence of solutions rather than vice versa. More specifically, our approach uses the Malliavin calculus and is based on the existence and regularity of a perfect cocycle on the energy space for the SBE. The proof of the existence theorem requires Malliavin regularity of the infinite-dimensional initial velocity field together with new spatial estimates on the cocycle, its Frechet and Malliavin derivatives. The existence theorem provides a dynamic characterization of solutions of the nonanticipating SBE on its unstable invariant manifolds. Furthermore, as a corollary of the existence theorem, we show that random...

Chaotic extensions and the lent particle method for Brownian motion

In previous works, we have developed a new Malliavin calculus on the Poisson space based on the lent particle formula. The aim of this work is to prove that, on the Wiener space for the standard Ornstein-Uhlenbeck structure, we also have such a formula which permits to calculate easily and intuitively the Malliavin derivative of a functional. Our approach uses chaos extensions associated to stationary processes of rotations of normal martingales.

The stochastic wave equation in high dimensions: Malliavin differentiability and absolute continuity

We consider the class of non-linear stochastic partial differential equations studied in [Conus-Dalang, 2008]. Equivalent formulations using integration with respect to a cylindrical Brownian motion and also the Skorohod integral are established. It is proved that the random field solution to these equations at any fixed point $(t,x)\in[0,T]\times \mathbb{R}^d$ is differentiable in the Malliavin sense. For this, an extension of the integration theory in [Conus-Dalang, 2008] to Hilbert space valued integrands is developed, and commutation formulae of the Malliavin derivative and stochastic and pathwise integrals are proved. In the particular case of equations with additive noise, we establish the existence of density for the law of the solution at $(t,x)\in]0,T]\times\mathbb{R}^d$. The results apply to the stochastic wave equation in spatial dimension $d\ge 4$.

Functional Itô calculus and stochastic integral representation of martingales

We develop a nonanticipative calculus for functionals of a continuous semimartingale, using an extension of the Ito formula to path-dependent functionals which possess certain directional derivatives. The construction is based on a pathwise derivative, introduced by Dupire, for functionals on the space of right-continuous functions with left limits. We show that this functional derivative admits a suitable extension to the space of square-integrable martingales. This extension defines a weak derivative which is shown to be the inverse of the Ito integral and which may be viewed as a nonanticipative "lifting" of the Malliavin derivative. These results lead to a constructive martingale representation formula for Ito processes. By contrast with the Clark-Haussmann-Ocone formula, this representation only involves nonanticipative quantities which may be computed pathwise.

Maximum Principle for General Controlled Systems Driven by Fractional Brownian Motions

We obtain a maximum principle for stochastic control problem of general controlled stochastic differential systems driven by fractional Brownian motions (of Hurst parameter H>1/2). This maximum principle specifies a system of equations that the optimal control must satisfy (necessary condition for the optimal control). This system of equations consists of a backward stochastic differential equation driven by both fractional Brownian motions and the corresponding underlying standard Brownian motions. In addition to this backward equation, the maximum principle also involves the Malliavin derivatives. Our approach is to use conditioning and Malliavin calculus. To arrive at our maximum principle we need to develop some new results of stochastic analysis of the controlled systems driven by fractional Brownian motions via fractional calculus. Our approach of conditioning and Malliavin calculus is also applied to classical system driven by standard Brownian motions while the controller has only partial information. As a straightforward consequence, the classical maximum principle is also deduced in this more natural and simpler way.