We find a simple expression for the probability density of $\int \exp (B_s - s/2) ds$ in terms of its distribution function and the distribution function for the time integral of $\exp (B_s + s/2)$. The relation is obtained with a change of measure argument where expectations over events determined by the time integral are replaced by expectations over the entire probability space. We develop precise information concerning the lower tail probabilities for these random variables as well as for time integrals of geometric Brownian motion with arbitrary constant drift. In particular, $E[ \exp\big(θ/ \int \exp (B_s)ds\big) ]$ is finite iff $θ< 2$. We present a new formula for the price of an Asian call option.
We study the problem of optimal approximation of a fractional Brownian motion by martingales. We prove that there exist a unique martingale closest to fractional Brownian motion in a specific sense. It shown that this martingale has a specific form. Numerical results concerning the approximation problem are given.
We describe the class of functions $f: R^n\to R^m$ which transform a vector Brownian Motion into a martingale and use this description to give martingale characterization of the general measurable solution of the multidimensional Cauchy functional equation.
We derive the explicit form of the martingale representation for square-integrable processes that are martingales with respect to the natural filtration of the super-Brownian motion. This is done by using a weak extension of the Dupire derivative for functionals of superprocesses.
In this paper, inspired by the fractional Brownian sheet of Riemann-Liouville type, we introduce the operator fractional Brownian sheet of Riemman-Liouville type, and study some properties of it. We also present an approximation in law to it based on the martingale differences.
It is well known that martingale difference sequences are very useful in applications and theory. On the other hand, the operator fractional Brownian motion as an extension of the well-known fractional Brownian motion also plays important role in both applications and theory. In this paper, we study the relationship between them. We will construct an approximation sequence of operator fractional Brownian motion based on a martingale difference sequence.
We describe the classes of functions $f=(f(x), x\in R)$, for which processes $f(W_t)-Ef(W_t)$ and $f(W_t)/Ef(W_t)$ are martingales. We apply these results to give a martingale characterization of general solutions of the quadratic and the D'Alembert functional equations. We study also the time-dependent martingale transformations of a Brownian Motion.
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.
Hamza-Klebaner posed the problem of constructing martingales with Brownian marginals that differ from Brownian motion, so called fake Brownian motions. Besides its theoretical appeal, the problem represents the quintessential version of the ubiquitous fitting problem in mathematical finance where the task is to construct martingales that satisfy marginal constraints imposed by market data. Non-continuous solutions to this challenge were given by Madan-Yor, Hamza-Klebaner, Hobson, and Fan-Hamza-Klebaner whereas continuous (but non-Markovian) fake Brownian motions were constructed by Oleszkiewic…
In this paper we give necessary and sufficient conditions for a cylindrical continuous local martingale to be the stochastic integral with respect to a cylindrical Brownian motion. In particular we consider the class of cylindrical martingales with closed operator-generated covariations. We also prove that for every cylindrical continuous local martingale $M$ there exists a time change $τ$ such that $M\circ τ$ is Brownian representable.
We consider a branching Brownian motion in $\mathbb{R}^d$. We prove that there exists a random subset $Θ$ of $\mathbb{S}^{d-1}$ such that the limit of the derivative martingale exists simultaneously for all directions $θ\in Θ$ almost surely. This allows us to define a random measure on $\mathbb{S}^{d-1}$ whose density is given by the derivative martingale. The proof is based on first moment arguments: we approximate the martingale of interest by a series of processes, which do not take into account the particles that travelled too far away. We show that these new processes are uniformly integr…
In this paper we introduce the notion of fractional martingale as the fractional derivative of order $α$ of a continuous local martingale, where $α\in(-{1/2},{1/2})$, and we show that it has a nonzero finite variation of order $\frac{2}{1+2α}$, under some integrability assumptions on the quadratic variation of the local martingale. As an application we establish an extension of Lévy's characterization theorem for the fractional Brownian motion.
We prove the max-martingale conjecture given in recent article with Marc Yor. We show that for a continuous local martingale $(N\_t:t\ge 0)$ and a function $H:R x R\_+\to R$, $H(N\_t,\sup\_{s\leq t}N\_s)$ is a local martingale if and only if there exists a locally integrable function $f$ such that $H(x,y)=\int\_0^y f(s)ds-f(y)(x-y)+H(0,0)$. This implies readily, via Levy's equivalence theorem, an analogous result with the maximum process replaced by the local time at 0.
We find the best approximation of the fractional Brownian motion with the Hurst index $H\in (0,1/2)$ by Gaussian martingales of the form $\int _0^ts^γdW_s$, where $W$ is a Wiener process, $γ>0$.
In this paper the whole family of fractional Brownian motions is constructed as a single Gaussian field indexed by time and the Hurst index simultaneously. The field has a simple covariance structure and it is related to two generalizations of fractional Brownian motion known as multifractional Brownian motions. A mistake common to the existing literature regarding multifractional Brownian motions is pointed out and corrected. The Gaussian field, due to inherited ``duality'', reveals a new way of constructing martingales associated with the odd and even part of a fractional Brownian motion and…
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…
Let $G$ be a Lie Group with a left invariant connection such that its connection function is skew-symmetric. Our main goal is to show a version of Pluzhnikov's Theorem for this kind of connection. To this end, we use the stochastic logarithm. More exactly, the stochastic logarithm gives characterizations for Brownian motions and Martingales in $G$, and these characterzations are used to prove Pluzhnikov's Theorem.
The paper studies a non-linear transformation between Brownian martingales, which is given by the inverse of the pricing operator in the mathematical finance terminology. Subsequently, the solvability of systems of equations corresponding to such transformations is investigated. The latter give rise to novel monotone pathwise couplings of an arbitrary number of certain diffusion processes with varying diffusion coefficients. In the case that there is an uncountable number of these diffusion processes and that the index set is an interval such couplings can be viewed as models for the growth of…
We study the optional projection of a standard Brownian motion on the natural filtration of certain kinds of observation processes. The observation process, $Y$, is defined as a solution of a stochastic differential equation such that it reveals some (possibly noisy) information about the signs of the Brownian motion when $Y$ hits 0. As such, the associated optional projections are related to Azéma's martingales which are obtained by projecting the Brownian motion onto the filtration generated by observing its signs.
In this article we establish a large deviation principle for the family {ν_ε:ε\in (0,1)} of distributions of the scaled stochastic processes {P_{-\log\sqrtε}Z_t}_{t\leq 1}, where (Z_t)_{t\in \lbrack 0,1]} is a square-integrable martingale over Brownian filtration and (P_t)_{t\geq 0} is the Ornstein-Uhlenbeck semigroup. The rate function is identified as well in terms of the Wiener-Itô chaos decomposition of the terminal value Z_{1}. The result is established by developing a continuity theorem for large deviations, together with two essential tools, the hypercontractivity of the Ornstein-Uhlenb…
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…
Evans (1992) described the semi-group of a superprocess with quadratic branching mechanism under a martingale change of measure in terms of the semi-group of an immortal particle and the semigroup of the superprocess prior to the change of measure. This result, commonly referred to as the spine decomposition, alludes to a pathwise decomposition in which independent copies of the original process `immigrate' along the path of the immortal particle. For branching particle diffusions the analogue of this decomposition has already been demonstrated in the pathwise sense, see for example Hardy and …
We study the limits of the additive and derivative martingales of one-dimensional branching Brownian motion in a periodic environment. Then we prove the existence of pulsating travelling wave solutions of the corresponding F-KPP equation in the supercritical and critical cases by representing the solutions probabilistically in terms of the limits of the additive and derivative martingales. We also prove that there is no pulsating travelling wave solution in the subcritical case. Our main tools are the spine decomposition and martingale change of measures.
This paper is concerned with the connection between G-Brownian Motion and analytic functions. We introduce the complex version of sublinear expectation, and then do the stochastic analysis in this framework. Furthermore, the conformal G-Brownian Motion is introduced together with a representation, and the corresponding conformal invariance is shown.
Clifford analysis has been the field of active research for several decades resulting in various methods to solve problems in pure and applied mathematics. However, the area of stochastic analysis has not been addressed in its full generality in the Clifford setting, since only a few contributions have been presented so far. Considering that the tools of stochastic analysis play an important role in the study of objects, such as positive definite functions, reproducing kernels and partial differential equations, it is important to develop tools for the study of these objects in the context of …
In this paper, we establish Girsanov's formula for $G$-Brownian motion. Peng (2007, 2008) constructed $G$-Brownian motion on the space of continuous paths under a sublinear expectation called $G$-expectation; as obtained by Denis et al. (2011), $G$-expectation is represented as the supremum of linear expectations with respect to martingale measures of a certain class. Our argument is based on this representation with an enlargement of the associated class of martingale measures, and on Girsanov's formula for martingales in the classical stochastic analysis. The methodology differs from that of…
Let the process Y(t) be a Skorohod integral process with respect to Brownian motion. We use a recent result by Tudor (2004), to prove that Y(t) can be represented as the limit of linear combinations of processes that are products of forward and backward Brownian martingales. Such a result is a further step towards the connection between the theory of continuous-time (semi)martingales, and that of anticipating stochastic integration. We establish an explicit link between our results and the classic characterization, due to Duc and Nualart (1990), of the chaotic decomposition of Skorohod integra…
We consider the additive martingale $W_t(λ)$ and the derivative martingale $\partial W_t(λ)$ for one-dimensional supercritical super-Brownian motions with general branching mechanism. In the critical case $λ=λ_0$, we prove that $\sqrt{t}W_t(λ_0)$ converges in probability to a positive limit, which is a constant multiple of the almost sure limit $\partial W_\infty(λ_0)$ of the derivative martingale $\partial W_t(λ_0)$. We also prove that, on the survival event, $\limsup_{t\to\infty}\sqrt{t}W_t(λ_0)=\infty$ almost surely.
We observe that the probability distribution of the Brownian motion with drift $-c \frac x {1-t}$ where $c\not =1$ is singular with respect to that of the classical Brownian bridge measure on $[0,1]$, while their Cameron-Martin spaces are equal set-wise if and only if $c> \frac 12$, providing also examples of exponential martingales on $[0,1)$ not extendable to a continuous martingale on $[0,1]$. Other examples of generalised Brownian bridges are also studied.
It is well known that upward conditioned Brownian motion is a three-dimensional Bessel process, and that a downward conditioned Bessel process is a Brownian motion. We give a simple proof for this result, which generalizes to any continuous local martingale and clarifies the role of finite versus infinite time in this setting. As a consequence, we can describe the law of regular diffusions that are conditioned upward or downward.
We introduce a transform on the class of stochastic exponentials for d-dimensional Brownian motions. Each stochastic exponential generates another stochastic exponential under the transform. The new exponential process is often merely a supermartingale even in cases where the original process is a martingale. We determine a necessary and sufficient condition for the transform to be a martingale process. The condition links expected values of the transformed stochastic exponential to the distribution function of certain time-integrals.
In this paper we present a martingale related to the exit measures of super-Brownian motion. By changing measure with this martingale in the canonical way we have a new process associated with the conditioned exit measure. This measure is shown to be identical to a measure generated by a non-homogeneous branching particle system with immigration of mass. An application is given to the problem of conditioning the exit measure to hit a number of specified points on the boundary of a domain. The results are similar in flavor to the "immortal particle" picture of conditioned super-Brownian motion …
The aim of this paper is to represent any continuous local martingale as an almost sure limit of a nested sequence of simple, symmetric random walks, time changed by a discrete quadratic variation process. One basis of this is a similar construction of Brownian motion. The other major tool is a representation of continuous local martingales given by Dambis, Dubins and Schwarz (DDS) in terms of Brownian motion time-changed by the quadratic variation. Rates of convergence (which are conjectured to be nearly optimal in the given setting) are also supplied. A necessary and sufficient condition for…
In this paper, we study the martingale property for a Scott correlated stochastic volatility model, when the correlation coefficient between the Brownian motion driving the volatility and the one driving the asset price process is arbitrary. For this study we verify the martingale property by using the necessary and sufficient conditions given by Bernard \emph{et al.} \cite{Bernard}. Our main results are to prove that the price process is a true and uniformly integrable martingale if and only if $ρ\in [-1,0]$ for two transformations of Brownian motion describing the dynamics of the underling a…
In this article, we consider the properties of hitting times for $G$-martingale and the stopped processes. We prove that the stopped processes for $G$-martingales are still $G$-martingales and that the hitting times for a class of $G$-martingales including $G$-Brownian motion are quasi-continuous. As an application, we improve the $G$-martingale representation theorems in [Song10].
In this paper non-asymptotic exponential estimates are derived for tail of maximum martingale distribution by naturally norming in the spirit of the classical Law of Iterated Logarithm. Key words: Martingales, exponential estimations, moment, Banach spaces of random variables, tail of distribution, conditional expectation.
We consider functional equations (Cauchy's, Abel's and some other functional equations) and show that to find general solution of these equations is equivalent to establish that a space-transformation of a Brownian Motion by suitable function (or functions) is a martingale.
In this paper we study dynamic backward problems, with the computation of conditional expectations as a main objective, in a framework where the (forward) state process satisfies a Volterra type SDE, with fractional Brownian motion as a typical example. Such processes are neither Markov processes nor semimartingales, and most notably, they feature a certain time inconsistency which makes any direct application of Markovian ideas, such as flow properties, impossible without passing to a path-dependent framework. Our main result is a functional Itô formula, extending the seminal work of Dupire \…
In this paper we introduce the concept of conic martingales}. This class refers to stochastic processes having the martingale property, but that evolve within given (possibly time-dependent) boundaries. We first review some results about the martingale property of solution to driftless stochastic differential equations. We then provide a simple way to construct and handle such processes. Specific attention is paid to martingales in $[0,1]$. One of these martingales proves to be analytically tractable. It is shown that up to shifting and rescaling constants, it is the only martingale (with the …
Let $(Z_t)_{t\geq 0}$ denote the derivative martingale of branching Brownian motion, i.e.\@ the derivative with respect to the inverse temperature of the normalized partition function at critical temperature. A well-known result by Lalley and Sellke [\textit{Ann. Probab.}, 15(3):1052--1061, 1987] says that this martingale converges almost surely to a limit $Z_\infty$, positive on the event of survival. In this paper, our concern is the fluctuations of the derivative martingale around its limit. A corollary of our results is the following convergence, confirming and strengthening a conjecture b…
We construct a fake exponential Brownian motion, a continuous martingale different from classical exponential Brownian motion but with the same marginal distributions, thus extending results of Albin and Oleszkiewicz for fake Brownian motions. The ideas extend to other diffusions.
Contributions of the present paper consist of two parts. In the first one, we contribute to the theory of stochastic calculus for signed measures. For instance, we provide some results permitting to characterize martingales and Brownian motion both defined under a signed measure. We also prove that the uniformly integrable martingales (defined with respect to a signed measure) can be expressed as relative martingales and we provide some new results to the study of the class $Σ(H)$ which appeared for the first time in \cite{f} and studied in \cite{f,e,o}. The second part is devoted to the const…
Let $ψ:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be a smooth and continuous real function and $ψ\in\mathrm{L}^{2}(\mathbb{R}^{+})$. Let ${B}(t)$ be a standard Brownian motion defined with respect to a probability space $(Ω,\mathscr{F},{\mathsf{P}})$ and where $d{B}(t)=ξ(t)dt$ and $t\in\mathbb{R}^{+}$. The process $ξ(t)$ is a Gaussian white noise with expectation $\mathbf{\mathsf{E}}~ξ(t)=0$ and with covariance ${\mathsf{E}}~ξ(t)ξ(s)=δ(t-s)$. The Dolean-Dades stochastic exponential ${Z}(t)$ is the solution to the linear stochastic differential equation describing a geometric Brownian motion such…
We show that solutions of free stochastic differential equations with regular drifts and diffusion coefficients, when considered backwards in time, still satisfy free SDEs for an explicit free Brownian motion and drift. We also study the stochastic integral part with respect to this reversed free Brownian motion of canonical closed martingales. We deduce that conjugate variables computed along a free Brownian motion, an example of such a reversed martingale appearing in the definition of non-microstates free entropy, are in the $L^2$ domain of corresponding free difference quotients for almost…
In this paper a martingale problem for super-Brownian motion with interactive branching is derived. The uniqueness of the solution to the martingale problem is obtained by using the pathwise uniqueness of the solution to a corresponding system of SPDEs with proper boundary conditions. The existence of the solution to the martingale problem and the Holder continuity of the density process are also studied.
For self-normalized martingales with conditionally symmetric differences, de la Peña [A general class of exponential inequalities for martingales and ratios. Ann. Probab. 27, No.1, 537-564] established the Gaussian type exponential inequalities. Bercu and Touati [Exponential inequalities for self-normalized martingales with applications. Ann. Appl. Probab. 18: 1848-1869] extended de la Peña's inequalities to martingales with differences heavy on left. In this paper, we establish Bernstein type exponential inequalities for self-normalized martingales with differences bounded from below. Moreove…
We provide a probabilistic approach to studying minimal surfaces in three-dimensional Euclidean space. Following a discussion of the basic relationship between Brownian motion on a surface and minimality of the surface, we introduce a way of coupling Brownian motions on two minimal surfaces. This coupling is then used to study two classes of results in the theory of minimal surfaces, maximum principle-type results, such as weak and strong halfspace theorems and the maximum principle at infinity, and Liouville theorems.
In this work, Bernstein's concentration inequalities for squared integrable matrix-valued discrete-time martingales are obtained. Based on Lieb's theory and Bernstein's condition, a suitable supermartingale can be constructed. Our proof is largely based on this new exponential supermartingale, Freedman's method, and Doob's stopping theorem. Our result can be regarded as an extension of Tropp's work (ECP, 2012).
We give sufficient criteria for the Doléans-Dade exponential of a stochastic integral with respect to a counting process local martingale to be a true martingale. The criteria are adapted particularly to the case of counting processes and are sufficiently weak to be useful and verifiable, as we illustrate by several examples. In particular, the criteria allow for the construction of for example nonexplosive Hawkes processes as well as counting processes with stochastic intensities depending on diffusion processes.
