We study the numerical evaluation of several functions appearing in the small time expansion of the distribution of the time-integral of the geometric Brownian motion as well as its joint distribution with the terminal value of the underlying Brownian motion. A precise evaluation of these distributions is relevant for the simulation of stochastic volatility models with log-normally distributed volatility, and Asian option pricing in the Black-Scholes model. We derive series expansions for these distributions, which can be used for numerical evaluations. Using tools from complex analysis, we determine the convergence radius and large order asymptotics of the coefficients in these expansions. We construct an efficient numerical approximation of the joint distribution of the time-integral of the gBM and its terminal value, and illustrate its application to Asian option pricing in the Black-Scholes model.
The density distribution function of the integral of the exponential Brownian motion is determined explicitly in the form of a rapidly convergent series.
In this paper, we combine Hida distribution theory and Sobolev-Watanabe-Kree spaces in order to study finely the link between forward integrals obtained by regularization and Wick-Itô integrals with respect to fractional Brownian motion and the Rosenblatt process. The new methodology developed in this paper allows to retrieve results for fractional Brownian motion and to obtain new results regarding the Rosenblatt process. In particular, an Itô formula for functionals of the Rosenblatt process is obtained which holds in the space of square-integrable random variables.
We prove change of variables formulas [Itô formulas] for functions of both arithmetic and geometric averages of geometric fractional Brownian motion. They are valid for all convex functions, not only for smooth ones. These change of variables formulas provide us integral representations of functions of average in the sense of generalized Lebesgue-Stieltjes integral.
We show that if a random variable is a final value of an adapted Holder continuous process, then it can be represented as a stochastic integral with respect to fractional Brownian motion, and the integrand is an adapted process, continuous up to the final point.
In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area) A(T), covered by the process in the time interval [0,T]. The Laplace transform of A(T) follows as a consequence. The main proof involves a double Laplace transform of A(T) and is based on excursion theory and local time for Brownian motion.
We derive general sufficient conditions for the existence of Riemann-Stieltjes integrals $\int_a^b Yd X$. Our results extend the classical conditions of L.C.Young and improve some recent results that deal with integrals involving a fractional Brownian motion with the Hurst index $H>1/2$.
We find an explicit expression for the cross-covariance between stochastic integral processes with respect to a $d$-dimensional fractional Brownian motion (fBm) $B_t$ with Hurst parameter $H>1/2$, where the integrands are vector fields applied to $B_t$. It provides, for example, a direct alternative proof of Y. Hu and D. Nualart's result that the stochastic integral component in the fractional Bessel process decomposition is not itself a fractional Brownian motion.
In this paper, we study the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion (GSDEs) with integral-Lipschitz conditions on their coefficients.
Wiener integral for the coordinate process is defined under the $ σ$-finite measure unifying Brownian penalisations, which has been introduced by Najnudel, Roynette and Yor. Its decomposition before and after last exit time from 0 is studied. This study prepares for the author's recent study of Cameron--Martin formula for the $ σ$-finite measure.
The objective of this paper is to derive a representation of symmetric G-martingales as stochastic integrals with respect to the G-Brownian motion. For this end, we first study some extensions of stochastic calculus with respect to G-martingales under the sublinear expectation spaces.
Let $B$ be a fractional Brownian motion with Hurst parameter $H=1/6$. It is known that the symmetric Stratonovich-style Riemann sums for $\int g(B(s))\,dB(s)$ do not, in general, converge in probability. We show, however, that they do converge in law in the Skorohod space of càdlàg functions. Moreover, we show that the resulting stochastic integral satisfies a change of variable formula with a correction term that is an ordinary Itô integral with respect to a Brownian motion that is independent of $B$.
The purpose of this note is to prove a central limit theorem for the $L^2$-modulus of continuity of the Brownian local time obtained in \cite{CLMR}, using techniques of stochastic analysis. The main ingredients of the proof are an asymptotic version of Knight's theorem and the Clark-Ocone formula for the $L^2$-modulus of the Brownian local time.
We show that a pathwise stochastic integral with respect to fractional Brownian motion with an adapted integrand $g$ can have any prescribed distribution, moreover, we give both necessary and sufficient conditions when random variables can be represented in this form. We also prove that any random variable is a value of such integral in some improper sense. We discuss some applications of these results, in particular, to fractional Black--Scholes model of financial market.
We consider Riemann sum approximations of stochastic integrals with respect to the fractional Browian motion of index $H\geq \frac12$. We show the convergence of these schemes at first and second order. The processes obtained in the limit in the second case are stochastic integrals with respect to the Rosenblatt process if $H >\frac34$ and the standard Brownian motion otherwise. These results are obtained under the assumption that the integrand is a `controlled' process. We provide many examples of such processes, in particular fractional semimartingales and multiple Wiener-Itô integrals
We construct a class of iterated stochastic integrals with respect to Brownian motion on an abstract Wiener space which allows for the definition of Brownian motions on a general class of infinite-dimensional nilpotent Lie groups based on abstract Wiener spaces. We then prove that a Cameron--Martin type quasi-invariance result holds for the associated heat kernel measures in the non-degenerate case, and give estimates on the associated Radon--Nikodym derivative. We also prove that a log Sobolev estimate holds in this setting.
A regime-switching geometric Brownian motion is used to model a geometric Brownian motion with its coefficients changing randomly according to a Markov chain. In this work, we give a complete characterization of the recurrent property of this process. The long time behavior of this process such as its $p$-th moment is also studied. Moreover, the quantitative properties of the regime-switching geometric Brownian motion with two-state switching are investigated to show the difference between geometric Brownian motion with switching and without switching. At last, some estimates of its first pass…
The aim of this paper is to establish the weak convergence, in the topology of the Skorohod space, of the $ν$-symmetric Riemann sums for functionals of the fractional Brownian motion when the Hurst parameter takes the critical value $H=(4\ell+2)^{-1}$, where $\ell=\ell(ν)\geq 1$ is the largest natural number satisfying $\int_0^1 α^{2j}ν(dα)=(2j+1)^{-1}$ for all $j=0,\ldots,\ell-1$. As a consequence, we derive a change-of-variable formula in distribution, where the correction term is a stochastic integral with respect to a Brownian motion that is independent of the fractional Brownian motion.
For Brownian motion in a (two-dimensional) wedge with negative drift and oblique reflection on the axes, we derive an explicit formula for the Laplace transform of its stationary distribution (when it exists), in terms of Cauchy integrals and generalized Chebyshev polyno-mials. To that purpose we solve a Carleman-type boundary value problem on a hyperbola, satisfied by the Laplace transforms of the boundary stationary distribution.
We study the asymptotic behaviour of the cross-variation of two-dimensional processes having the form of a Young integral with respect to a fractional Brownian motion of index $H \textgreater{} 1/ 2$. When $H$ is smaller than or equal to $3 / 4$, we show asymptotic mixed normality. When $H$ is strictly bigger than $3/4$, we obtain a limit that is expressed in terms of the difference of two independent Rosenblatt processes.
In this note we prove an existence and uniqueness result of solution for stochastic Volterra integral equations driven by a fractional Brownian motion with Hurst parameter H > 1/2, showing also that the solution has finite moments. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann-Stieltjes integral.
We show that if a random variable is the final value of an adapted log-Hölder continuous process, then it can be represented as a stochastic integral with respect to a fractional Brownian motion with adapted integrand. In order to establish this representation result, we extend the definition of the fractional integral.
In this article, an uniform discretization of stochastic integrals $\int_{0}^{1} f'_-(B_t)\ud B_t$, with respect to fractional Brownian motion with Hurst parameter $H \in (1/2,1)$, for a large class of convex functions $f$ is considered. In Statistics & Decisions, 27, 129-143, for any convex function $f$, the almost sure convergence of uniform discretization to such stochastic integral is proved. Here we prove $L^r$- convergence of uniform discretization to stochastic integral. In addition, we obtain a rate of convergence. It turns out that the rate of convergence can be brought as closely as …
In this paper we investigate classical solution of a semi-linear system of backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. By proving an Itô-Wentzell formula for jump diffusions as well as an abstract result of stochastic evolution equations, we obtain the stochastic integral partial differential equation for the inverse of the stochastic flow generated by a stochastic differential equation driven by a Brownian motion and a Poisson point process. By composing the random field generated by the solution of a backward stochastic…
This paper concerns the almost sure time dependent local extinction behavior for super-coalescing Brownian motion $X$ with $(1+β)$-stable branching and Lebesgue initial measure on $\bR$. We first give a representation of $X$ using excursions of a continuous state branching process and Arratia's coalescing Brownian flow. For any nonnegative, nondecreasing and right continuous function $g$, put τ:=\sup \{t\geq 0: X_t([-g(t),g(t)])>0 \}. We prove that $\bP\{τ=\infty\}=0$ or 1 according as the integral $\int_1^\infty g(t)t^{-1-1/β} dt$ is finite or infinite.
Let v be a bounded function with bounded support in R^d, d>=3. Let x,y in R^d. Let Z(t) denote the path integral of v along the path of a Brownian bridge in R^d which runs for time t, starting at x and ending at y. As t->infty, it is perhaps evident that the distribution of Z(t) converges weakly to that of the sum of the integrals of v along the paths of two independent Brownian motions, starting at x and y and running forever.Here we prove a stronger result, namely convergence of the corresponding moment generating functions and of moments. This result is needed for applications in physics.
We show how to build an immersion coupling of a two-dimensional Brownian motion $(W_1, W_2)$ along with $\binom{n}{2} + n= \tfrac12n(n+1)$ integrals of the form $\int W_1^iW_2^j \circ dW_2$, where $j=1,\ldots,n$ and $i=0, \ldots, n-j$ for some fixed $n$. The resulting construction is applied to the study of couplings of certain hypoelliptic diffusions (driven by two-dimensional Brownian motion using polynomial vector fields). This work follows up previous studies concerning coupling of Brownian stochastic areas and time integrals (Ben Arous, Cranston and Kendall (1995), Kendall and Price (2004…
We derive a nonlinear integral equation to calculate Root's solution of the Skorokhod embedding problem for atom-free target measures. We then use this to efficiently generate bounded time-space increments of Brownian motion and give a parabolic version of Muller's classic "Random walk over spheres" algorithm.
The first passage time problem for Brownian motions hitting a barrier has been extensively studied in the literature. In particular, many incarnations of integral equations which link the density of the hitting time to the equation for the barrier itself have appeared. Most interestingly, Peskir(2002b) demonstrates that a master integral equation can be used to generate a countable number of new equations via differentiation or integration by parts. In this article, we generalize Peskir's results and provide a more powerful unifying framework for generating integral equations through a new cla…
Estimating dynamic correlation between a pair of time series is of importance in many applications. We present new estimators for the dynamic correlation between a pair of correlated Brownian motions and separately for dynamic correlation between a pair of correlated Geometric Brownian motions. We show that, as the sample size increases, all estimators presented in this paper converge in probability to the underlying true dynamic correlation.
In this paper, we study the $\frac{1}{H}$-variation of stochastic divergence integrals $X_t = \int_0^t u_s δB_s$ with respect to a fractional Brownian motion $B$ with Hurst parameter $H < \frac{1}{2}$. Under suitable assumptions on the process u, we prove that the $\frac{1}{H}$-variation of $X$ exists in $L^1(Ω)$ and is equal to $e_H \int_0^T|u_s|^H ds$, where $e_H = \mathbb{E}|B_1|^H$. In the second part of the paper, we establish an integral representation for the fractional Bessel Process $\|B_t\|$, where $B_t$ is a $d$-dimensional fractional Brownian motion with Hurst parameter $H < \frac{…
By using Malliavin calculus and multiple Wiener-Itô integrals, we study the existence and the regularity of stochastic currents defined as Skorohod (divergence) integrals with respect to the Brownian motion and to the fractional Brownian motion. We consider also the multidimensional multiparameter case and we compare the regularity of the current as a distribution in negative Sobolev spaces with its regularity in Watanabe space.
For classical finite time horizon stopping problems driven by a Brownian motion \[V(t,x) = \sup_{t\leqτ\leq0}E_{(t,x)}[g(τ,W_τ)],\] we derive a new class of Fredholm type integral equations for the stopping set. For large problem classes of interest, we show by analytical arguments that the equation uniquely characterizes the stopping boundary of the problem. Regardless of the uniqueness, we use the representation to rigorously find the limit behavior of the stopping boundary close to the terminal time. Interestingly, it turns out that the leading-order coefficient is universal for wide classe…
We consider convolution-type stochastic Volterra equations with additive Hilbert-valued fractional Brownian motion, $0<H<1$. We find the weak solution to this stochastic Volterra equation, and study its stochastic integral part, the stochastic convolution, which we show to be mean-zero Gaussian. We develop an Itô isometry for stochastic integrals with respect to a Hilbert-valued fractional Brownian motion, and use it to compute the covariance of the stochastic convolution. This formula, which uses fractional integrals and derivatives, generalizes the well-known formula from the case $H=1/2$.
The integrated Brownian motion is sometimes known as the Langevin process. Lachal studied several excursion laws induced by the latter. Here we follow a different point of view developed by Pitman for general stationary processes. We first construct a stationary Langevin process and then determine explicitly its stationary excursion measure. This is then used to provide new descriptions of Ito's excursion measure of the Langevin process reflected at a completely inelastic boundary, which has been introduced recently by Bertoin.
Let $(B(t))_{t\in [0,1]}$ be the linear Brownian motion and $(X_n(t))_{t\in [0,1]}$ be the $(n-1)$-fold integral of Brownian motion, $n$ being a positive integer: $$ X_n(t)=\int_0^t \frac{(t-s)^{n-1}}{(n-1)!} \,\dd B(s) for any $t\in[0,1]$. $$ In this paper we construct several bridges between times 0 and 1 of the process $(X_n(t))_{t\in [0,1]}$ involving conditions on the successive derivatives of $X_n$ at times 0 and 1. For this family of bridges, we make a correspondance with certain boundary value problems related to the one-dimensional polyharmonic operator. We also study the classical pr…
The main result is a counterpart of the theorem of Monroe [\emph{Ann. Probability} \textbf{6} (1978) 42--56] for a geometric Brownian motion: A process is equivalent to a time change of a geometric Brownian motion if and only if it is a nonnegative supermartingale. We also provide a link between our main result and Monroe [\emph{Ann. Math. Statist.} \textbf{43} (1972) 1293--1311]. This is based on the concept of a \emph{minimal} stopping time, which is characterised in Monroe [\emph{Ann. Math. Statist.} \textbf{43} (1972) 1293--1311] and Cox and Hobson [\emph{Probab. Theory Related Fields} \te…
We study integral representations of random variables with respect to general Hölder continuous processes and with respect to two particular cases; fractional Brownian motion and mixed fractional Brownian motion. We prove that arbitrary random variable can be represented as an improper integral, and that the stochastic integral can have any distribution. If in addition the random variable is a final value of an adapted Hölder continuous process, then it can be represented as a proper integral. It is also shown that in the particular case of mixed fractional Brownian motion, any adapted random …
We develop the functional Itô/path-dependent calculus with respect to fractional Brownian motion with Hurst parameter $H> \frac{1}{2}$. Firstly, two types of integrals are studied. The first type is Stratonovich integral, and the second type is Wick-Itô integral. Then we establish the functional Itô formulas for fractional Brownian motion, which extend the functional Itô formulas in Dupire (2009) and Cont-Fournié (2013) to the case of non-semimartingale. Finally, as an application, we deal with a class of fractional backward stochastic differential equations (BSDEs). A relation between fractio…
In this paper we consider stochastic differential equations with non-negativity constraints, driven by a fractional Brownian motion with Hurst parameter $H>\1/2$. We first study an ordinary integral equation where the integral is defined in the Young sense and then we apply this result pathwise to solve the stochastic problem.
This paper studies the regularity properties of the density of the exit measure for super-Brownian motion with (1+β)-stable branching mechanism. It establishes the continuity of the density in dimension d=2 and the unboundedness of the density in all other dimensions where the density exists. An alternative description of the exit measure and its density is also given via a stochastic integral representation. Results are applied to the probabilistic representation of nonnegative solutions of the partial differential equation Δu=u^{1+β}.
We consider the integral of fractional Brownian motion (IFBM) and its functionals $ξ_T$ on the intervals $(0,T)$ and $(-T,T)$ of the following types: the maximum $M_T$, the position of the maximum, the occupation time above zero etc. We show how the asymptotics of $P(ξ_T<1)=p_T, T\to \infty$, is related to the Hausdorff dimension of Lagrangian regular points for the inviscid Burgers equation with FBM initial velocity. We produce computational evidence in favor of a power asymptotics for $p_T$. The data do not reject the hypothesis that the exponent $θ$ of the power law is related to the simila…
We give a new estimate on Stieltjes integrals of Hölder continuous functions and use it to prove an existence-uniqueness theorem for solutions of ordinary differential equations with Hölder continuous forcing. We construct stochastic integrals with respect to fractional Brownian motion, and establish sufficient conditions for its existence. We prove that stochastic differential equations with fractional Brownian motion have a unique solution with probability 1 in certain classes of Hölder-continuous functions. We give tail estimates of the maximum of stochastic integrals from tail estimates of…
In this note we prove the existence of a density for the law of the solution for 1-dimensional stochastic delay differential equations with normal reflection. The equations are driven by a fractional Brownian motion with Hurst parameter $H > 1/2$. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann-Stieltjes integral.
We construct in this article a rough path over fractional Brownian motion with arbitrary Hurst index by (i) using the Fourier normal ordering algorithm introduced in \cite{Unt-Holder} to reduce the problem to that of regularizing tree iterated integrals and (ii) applying the Bogolioubov-Parasiuk-Hepp-Zimmermann (BPHZ) renormalization algorithm to Feynman diagrams representing tree iterated integrals.
Fractional Brownian motion belongs to a class of long memory Gaussian processes that can be represented as linear functionals of an infinite dimensional Markov process. This representation leads naturally to: - An efficient algorithm to approximate the process. - An infinite dimensional ergodic theorem which applies to functionals of the type $integral_0^t phi(V_h(s)) ds $ where $V_h(s)=integral_0^t h(t-u) dB_u$ and $B$ is a standard Brownian motion.
In this note we prove an existence and uniqueness result of solution for stochastic differential delay equations with hereditary drift driven by a fractional Brownian motion with Hurst parameter $H > 1/2$. Then, we show that, when the delay goes to zero, the solutions to these equations converge, almost surely and in $L^p$, to the solution for the equation without delay. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann-Stieltjes integral.
We study the anticipative backward stochastic differential equations (BSDEs, for short) driven by fractional Brownian motion with Hurst parameter H greater than 1/2. The stochastic integral used throughout the paper is the divergence operator type integral. We obtain the existence and uniqueness of solutions to these equations. A comparison theorem for this type of anticipative BSDEs is also established.
In this paper, we studied the functional ergodic limits of the site-dependent branching Brownian motions in R. The results show that the limiting processes are non-degenerate if and only if the variance functions of branching laws are integrable. When the functions are integrable, although the limiting processes will vary according to the integrals, they are always positive, infinitely divisible and self-similar, and their marginal distributions are determined by a kind of 1/2-fractional integral equations. As a byproduct, the unique non-negative solutions of the integral equations are explici…
We consider equidistant approximations of stochastic integrals driven by Hölder continuous Gaussian processes of order $H>\frac12$ with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the $L^1$-distance and provide examples with different drivers. It turns out that the exact rate of convergence is proportional to $n^{1-2H}$ that is twice better compared to the best known results in the case of discontinuous integrands, and corresponds to the known rate in the case of smooth integrands. The novelty of our approach is that, instead of using mu…
Starting from the construction of a geometric rough path associated with a fractional Brownian motion with Hurst parameter $H\in]{1/4}, {1/2}[$ given by Coutin and Qian (2002), we prove a large deviation principle in the space of geometric rough paths, extending classical results on Gaussian processes. As a by-product, geometric rough paths associated to elements of the reproducing kernel Hilbert space of the fractional Brownian motion are obtained and an explicit integral representation is given.
For a parameter 0<q<1, we use the Jackson q-integral to define integration with respect to the so called q-Brownian motion. Our main results are the q-analogs of the L_2-isometry and of the Ito formula for polynomial integrands. We also indicate how the L_2-isometry extends the integral to more general functions.
Let $τ$ be the first hitting time of the point 1 by the geometric Brownian motion $X(t)= x \exp(B(t)-2μt)$ with drift $μ\geq 0$ starting from $x>1$. Here $B(t)$ is the Brownian motion starting from 0 with $E^0 B^2(t) = 2t$. We provide an integral formula for the density function of the stopped exponential functional $A(τ)=\int_0^τX^2(t) dt$ and determine its asymptotic behaviour at infinity. Although we basically rely on methods developed in \cite{BGS}, the present paper also covers the case of arbitrary drifts $μ\geq 0$ and provides a significant unification and extension of results of the ab…
Let $B^H$ be a fractional Brownian motion with Hurst index $0<H<1$ and the weighted local time ${\mathscr L}^H(\cdot,t)$. In this paper, we consider the integral functional $$ {\mathcal C}^H_t(a):=\lim_{\varepsilon\downarrow 0}\int_0^t1_{\{|B^H_s-a|>\varepsilon\}}\frac1{B^H_s-a}ds^{2H}\equiv \frac1π{\mathscr H}{\mathscr L}^H(\cdot,t)(a) $$ in $L^2(Ω)$ with $ a\in {\mathbb R}, t\geq 0$ and ${\mathscr H}$ denoting the Hilbert transform. We show that $$ {\mathcal C}^H_t(a)=2\left((B^H_t-a)\log|B^H_t-a|-B^H_t+a\log|a| -\int_0^t\log|B^H_s-a|δB^H_s\right) $$ for all $a\in {\mathbb R}, t\geq 0$ which…
