In this article, we prove integration by parts (IBP) formulas concerning maxima of solutions to some stochastic differential equations (SDEs). We will deal with three types of maxima. First, we consider discrete time maximum, and then continuous time maximum in the case of one-dimensional SDEs. Finally, we deal with the maximum of the components of a solution to multi-dimensional SDEs. Applications to study their probability density functions by means of the IBP formulas are also discussed.

Integration by parts formulas concerning maxima of some SDEs with applications to study on density functions

In this paper, we consider convex sets $K_r = \{g \ge r\}$ in an infinite dimensional Hilbert space, where $g$ is suitably related to a reference Gaussian measure $\mu$ in $H$. We first show how to define a surface measure on the level sets $\{g = r\}$ that is related to $\mu$. This allows to introduce an integration-by-parts formula in $H$. This formula can be applied in several important constructions, as for instance the case where $\mu$ is the law of a (Gaussian) stochastic process and $H$ is the space of its trajectories

Construction of a surface integral under local Malliavin assumption and integration by parts formulae

In this paper, we consider the sets $$K_r = \{g \ge r\}$$
 defined in an infinite-dimensional Hilbert space, where g is suitably related to a reference Gaussian measure $$\mu $$
 in H. We first show how to define a surface measure that is related to $$\mu $$
 . This allows to introduce an integration-by-parts formula in $$K_r$$
 , which can be applied in several important constructions, as is the case where $$\mu $$
 is the law of a (Gaussian) stochastic process and H is the space of its trajectories.

Construction of a surface integral under local Malliavin assumptions, and related integration by parts formulas

Avikainen provided a sharp upper bound of the difference $\mathbb{E}[|g(X)-g(\widehat{X})|^{q}]$ by the moments of $|X-\widehat{X}|$ for any one-dimensional random variables $X$ with bounded density and $\widehat{X}$, and function of bounded variation $g$. In this article, we generalize this estimate to any one-dimensional random variable $X$ with Holder continuous distribution function. As applications, we provide the rate of convergence for numerical schemes for solutions of one-dimensional stochastic differential equations (SDEs) driven by Brownian motion and symmetric $\alpha$-stable with $\alpha \in (1,2)$, fractional Brownian motion with drift and Hurst parameter $H \in (0,1/2)$, and stochastic heat equations (SHEs) with Dirichlet boundary conditions driven by space--time white noise, with irregular coefficients. We also consider a numerical scheme for maximum and integral type functionals of SDEs driven by Brownian motion with irregular coefficients and payoffs which are related to multilevel Monte Carlo method.

A generalized Avikainen’s estimate and its applications

In this paper, we establish a probabilistic representation for two integration by parts formulas, one being of Bismut-Elworthy-Li’s type, for the marginal law of a one-dimensional diffusion process killed at a given level. These formulas are established by combining a Markovian perturbation argument with a tailor-made Malliavin calculus for the underlying Markov chain structure involved in the probabilistic representation of the original marginal law. Among other applications, an unbiased Monte Carlo path simulation method for both integration by parts formula stems from the previous probabilistic representations.

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Integration by parts formula for killed processes: A point of view from approximation theory

Absolute continuity of the laws of a multi-dimensional stochastic differential equation with coefficients dependent on the maximum

In this paper, we give a decomposition formula to calculate the vega index (sensitivity with respect to changes in volatility) for options with prices that depend on the extrema (maximum or minimum) and terminal value of the underlying stock price; this is assumed to follow a one-dimensional perturbed diffusion process. As a numerical application, we compute the vega index for lookback, European and up-in call options under the Black–Scholes model perturbed with a constant elasticity of variance model-type perturbation. We compare these values with the standard nonperturbed Black–Scholes model, which, interestingly, turn out to be very different.

Volatility risk structure for options depending on extrema

In this article, we shall prove an integration by parts (IBP) type formula for stopping times. In order to obtain the formula, we will first construct a process which works as if it is an “alarm clock” telling us whether the stopping times are already achieved or not. Then, we shall use the Girsanov theorem. Applications of the formula to the numerical computation of the risk called the delta for options depending on the stopping times will be also considered and show the gain of efficiency compared with a classical method.

An Integration by Parts Type Formula for Stopping Times and its Application

This article proves some properties of the probability density function concerning maxima of a solution to one-dimensional stochastic differential equations. We first obtain lower and upper bounds on the density function of the discrete time maximum of the solution. We then prove that the density function of the discrete time maximum converges to that of the continuous time maximum of the solution. Finally, we prove the positivity of the density function of the continuous time maximum and a relationship between the density functions of the continuous time maximum and the solution itself.

Tomonori Nakatsu

Papers

Absolute continuity of the laws of a multi-dimensional stochastic differential equation with coefficients dependent on the maximum

Integration by parts formulas concerning maxima of some SDEs with applications to study on density functions

Volatility risk structure for options depending on extrema

An Integration by Parts Type Formula for Stopping Times and its Application

Some Properties of Density Functions on Maxima of Solutions to One-Dimensional Stochastic Differential Equations