Malliavin Calculus Research Articles

Pointwise Second-order Necessary Conditions for Stochastic Optimal Controls, Part I: The Case of Convex Control Constraint

This paper is the first part of our series of work to establish pointwise second-order necessary conditions for stochastic optimal controls. In this part, both drift and diffusion terms may contain the control variable but the control region is assumed to be convex. Under some assumptions in terms of the Malliavin calculus, we establish the desired necessary conditions for stochastic singular optimal controls in the classical sense.

A simple relationship between Greeks for Asian options

Computation of Greeks is an important task in financial risk management. In the last decade, there have been many studies on the computational methods for a variety of financial options. The Malliavin calculus approach has become one of the main approaches to derive Greeks for these options, for example. In this article, we show a new relationship between Greeks (vega, rho, and theta) for Asian options. This result is obtained using elementary mathematics.

Parameter estimation based on discrete observations of fractional Ornstein–Uhlenbeck process of the second kind

Fractional Ornstein–Uhlenbeck process of the second kind \((\text {fOU}_{2})\) is a solution of the Langevin equation \(\mathrm {d}X_t = -\theta X_t\,\mathrm {d}t+\mathrm {d}Y_t^{(1)}, \theta >0\) with a Gaussian driving noise \( Y_t^{(1)} := \int ^t_0 e^{-s} \,\mathrm {d}B_{a_s}\), where \( a_t= H e^{\frac{t}{H}}\) and \(B\) is a fractional Brownian motion with Hurst parameter \(H \in (0,1)\). In this article we consider the case \(H>\frac{1}{2}\), and by using the ergodicity of \(\text {fOU}_{2}\) process we construct consistent estimators for the drift parameter \(\theta \) based on discrete observations in two possible cases: \((i)\) the Hurst parameter \(H\) is known and \((ii)\) the Hurst parameter \(H\) is unknown. Moreover, using Malliavin calculus techniques we prove central limit theorems for our estimators which are valid for the whole range \(H \in (\frac{1}{2},1)\).

Asymptotic behavior of weighted cubic variation of sub-fractional brownian motion

ABSTRACTIn this article, we investigate the convergence of renormalized weighted cubic variation of a sub-fractional Brownian motion SH with Hurst index H. When , we prove by means of Malliavin calculus that the convergence holds in L2 toward an explicit limit which only depends on SH. We also numerically simulate the sample paths of such a type of sub-fractional Brownian motion regulated by different Hurst index H.

The stochastic filtering problem: a brief historical account

Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this paper is to give a brief historical account of the subject concentrating on the continuous-time framework.

The stochastic filtering problem: a brief historical account

Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this paper is to give a brief historical account of the subject concentrating on the continuous-time framework.

Functionals of a Lévy Process on Canonical and Generic Probability Spaces

We develop an approach to Malliavin calculus for Lévy processes from the perspective of expressing a random variable \(Y\) by a functional \(F\) mapping from the Skorohod space of càdlàg functions to \(\mathbb {R}\), such that \(Y=F(X)\) where \(X\) denotes the Lévy process. We also present a chain-rule-type application for random variables of the form \(f(\omega ,Y(\omega ))\). An important tool for these results is a technique which allows us to transfer identities proved on the canonical probability space (in the sense of Solé et al.) associated to a Lévy process with triplet \((\gamma ,\sigma ,\nu )\) to an arbitrary probability space \((\varOmega ,\mathcal {F},\mathbb {P})\) which carries a Lévy process with the same triplet.

Derivative formulae for SDEs driven by multiplicative [formula omitted]-stable-like processes

By using Bismut’s approach to the Malliavin calculus with jumps, we establish a derivative formula of Bismut–Elworthy–Li’s type for SDEs driven by multiplicative Lévy noises, whose Lévy measure satisfies some order conditions. In particular, α-stable-like noises are allowed. Moreover, we also obtain the sharp gradient estimate in short time for the corresponding transition semigroup provided α∈(1,2).

Optimal investment, consumption and proportional reinsurance under model uncertainty

This paper considers the optimal investment, consumption and proportional reinsurance strategies for an insurer under model uncertainty. The surplus process of the insurer before investment and consumption is assumed to be a general jump–diffusion process. The financial market consists of one risk-free asset and one risky asset whose price process is also a general jump–diffusion process. We transform the problem equivalently into a two-person zero-sum forward–backward stochastic differential game driven by two-dimensional Lévy noises. The maximum principles for a general form of this game are established to solve our problem. Some special interesting cases are studied by using Malliavin calculus so as to give explicit expressions of the optimal strategies.

An Exponential Integrator Scheme for Time Discretization of Nonlinear Stochastic Wave Equation

This work is devoted to convergence analysis of an exponential integrator scheme for semi-discretization in time of nonlinear stochastic wave equation. A unified framework is first set forth, which covers important cases of additive and multiplicative noises. Within this framework, the proposed scheme is shown to converge uniformly in the strong $L^p$-sense with precise convergence rates given. The abstract results are then applied to several concrete examples. Further, weak convergence rates of the scheme are examined for the case of additive noise. To analyze the weak error for the nonlinear case, techniques based on the Malliavin calculus were usually exploited in the literature. Under certain appropriate assumptions on the nonlinearity, this paper provides a weak error analysis, which does not rely on the Malliavin calculus. The rates of weak convergence can, as expected, be improved in comparison with the strong rates. Both strong and weak convergence results obtained here show that the proposed method achieves higher convergence rates than the implicit Euler and Crank-Nicolson time discretizations. Numerical results are finally reported to confirm our theoretical findings.

Solving a Nonlinear Fractional Stochastic Partial Differential Equation with Fractional Noise

In this article, we will prove the existence, uniqueness and Holder regularity of the solution to the fractional stochastic partial differential equation of the form $$\begin{aligned} \frac{\partial }{\partial t}u(t,x)=\mathfrak {D}(x,D)u(t,x)+\frac{\partial f}{\partial x}(t,x,u(t,x))+\frac{\partial ^2 W^H}{\partial t\partial x}(t,x), \end{aligned}$$ where \(\mathfrak {D}(x,D)\) denotes the Markovian generator of stable-like Feller process, \(f:[0,T]\times \mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}\) is a measurable function, and \(\frac{\partial ^2 W^H}{\partial t\partial x}(t,x)\) is a double-parameter fractional noise. In addition, we establish lower and upper Gaussian bounds for the probability density of the mild solution via Malliavin calculus and the new tool developed by Nourdin and Viens (Electron J Probab 14:2287–2309, 2009).

LAN property for a simple Lévy process

In this paper, we consider a simple Lévy process given by a Brownian motion and a compensated Poisson process, whose drift and diffusion parameters as well as its intensity are unknown. Supposing that the process is observed discretely at high frequency, we derive the local asymptotic normality (LAN) property. In order to obtain this result, Malliavin calculus and Girsanov's theorem are applied in order to write the log-likelihood ratio in terms of sums of conditional expectations, for which a central limit theorem for triangular arrays can be applied.

Normal Approximation of Poisson Functionals in Kolmogorov Distance

Peccati, Solè, Taqqu, and Utzet recently combined Stein’s method and Malliavin calculus to obtain a bound for the Wasserstein distance of a Poisson functional and a Gaussian random variable. Convergence in the Wasserstein distance always implies convergence in the Kolmogorov distance at a possibly weaker rate. But there are many examples of central limit theorems having the same rate for both distances. The aim of this paper was to show this behavior for a large class of Poisson functionals, namely so-called U-statistics of Poisson point processes. The technique used by Peccati et al. is modified to establish a similar bound for the Kolmogorov distance of a Poisson functional and a Gaussian random variable. This bound is evaluated for a U-statistic, and it is shown that the resulting expression is up to a constant the same as it is for the Wasserstein distance.

Optimal investment, consumption and proportional reinsurance for an insurer with option type payoff

This paper is devoted to the study of optimization of investment, consumption and proportional reinsurance for an insurer with option type payoff at the terminal time under the criterion of exponential utility maximization. The surplus process of the insurer and the financial risky asset process are assumed to be diffusion processes driven by Brownian motions which are non-Markovian in general. Very general constraints are imposed on the investment and the proportional reinsurance processes. Based on the martingale optimization principle, we use BSDE and BMO martingale techniques to derive the optimal strategy and the optimal value function. Some interesting particular cases are studied in which the explicit expressions for the optimal strategy are given by using the Malliavin calculus.

Optimal Portfolio and Consumption in Anticipative Electricity Futures Markets

We derive utility maximizing portfolios and consumption rates in electricity futures markets under anticipative information modeled by enlarged filtrations. The emerging optimization exercises are solved by point-wise maximization and a sufficient stochastic maximum principle. We provide rigorous economical interpretation of our mathematical results and distinguish optimal trading behavior for small and large investors with insider knowledge. On the mathematical side we are concerned with jump processes, (B)SDEs, enlarged filtrations, forward integrals, stochastic maximum principles and Malliavin calculus.

Densities for SDEs driven by degenerate $\alpha$-stable processes

In this work, by using the Malliavin calculus, under H\"ormander's condition, we prove the existence of distributional densities for the solutions of stochastic differential equations driven by degenerate subordinated Brownian motions. Moreover, in a special degenerate case, we also obtain the smoothness of the density. In particular, we obtain the existence of smooth heat kernels for the following fractional kinetic Fokker-Planck (nonlocal) operator: \[\mathscr{L}^{(\alpha)}_b:=\Delta^{\alpha/2}_{\mathrm{v}}+\mathrm {v}\cdot \nabla_x+b(x,\mathrm{v})\cdot \nabla_{\mathrm{v}},\qquad x,\mathrm{v}\in\mathbb{R}^d,\] where $\alpha\in(0,2)$ and $b:\mathbb{R}^d\times\mathbb{R}^d\to \mathbb{R}^d$ is smooth and has bounded derivatives of all orders.

Calculations of Greeks for Jump Diffusion Processes

We investigate the computation of Greeks for the jump diffusion process. In particular, we consider the Heston model and the Levy process. For the Heston model, we first consider the case where no jumps are involved in the model. We then introduce jumps in both the price process and the volatility and compute the Greeks. In all this, we use Malliavin calculus. Finally, we consider the Greeks for the Levy process. In particular, we compute Δ of an approximation of a Levy process and then bypass through the limit arguments to obtain Δ for the original Levy process.

Small-time expansions for local jump-diffusion models with infinite jump activity

We consider a Markov process $X$, which is the solution of a stochastic differential equation driven by a L\'{e}vy process $Z$ and an independent Wiener process $W$. Under some regularity conditions, including non-degeneracy of the diffusive and jump components of the process as well as smoothness of the L\'{e}vy density of $Z$ outside any neighborhood of the origin, we obtain a small-time second-order polynomial expansion for the tail distribution and the transition density of the process $X$. Our method of proof combines a recent regularizing technique for deriving the analog small-time expansions for a L\'{e}vy process with some new tail and density estimates for jump-diffusion processes with small jumps based on the theory of Malliavin calculus, flow of diffeomorphisms for SDEs, and time-reversibility. As an application, the leading term for out-of-the-money option prices in short maturity under a local jump-diffusion model is also derived.

On the eigenvalue process of a matrix fractional Brownian motion

We investigate the process of eigenvalues of a symmetric matrix-valued process which upper diagonal entries are independent one-dimensional Hölder continuous Gaussian processes of order γ∈(1/2,1). Using the stochastic calculus with respect to the Young integral we show that these eigenvalues do not collide at any time with probability one. When the matrix process has entries that are fractional Brownian motions with Hurst parameter H∈(1/2,1), we find a stochastic differential equation in a Malliavin calculus sense for the eigenvalues of the corresponding matrix fractional Brownian motion. A new generalized version of the Itô formula for the multidimensional fractional Brownian motion is first established.

Two-Sided Bounds for Degenerate Processes with Densities Supported in Subsets of ℝ N

We obtain bounds for the density of \(\left (x_{1,n}+W_{t}, x_{n+1}+{{\int }_{0}^{t}} |x_{1,n}+\right .\\\left .W_{s}|^{k}ds\right )\) where (Wt)t≥0 is a standard Brownian motion of ℝn, k∈ℕ∗ is even and x=(x1,n,xn+1)∈ℝn+1. This process satisfies a weak Hormander condition but the support of its density is not the whole space. Also, the Density has various asymptotic regimes depending on the starting/final points considered (which are as well related to the number of brackets needed to span the space in Hormander’s theorem). The proofs of lower and upper bounds are based on Harnack inequalities and Malliavin calculus respectively. The case of the joint law of Brownian motion and the integral of odd powers of its coordinates is also considered.