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Introduction to stochastic calculus with applications
This text presents a concise and rigorous treatment of stochastic calculus. It also gives its main applications in finance, biology and engineering.
Book
A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria
We analyze a class of nonlinear partial differential equations (PDEs) defined on
eBook
Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation
The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are
given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its highest curve, the
Airy
In this paper, we employ the Brownian Gibbs property to make a close
comparison between the Airy line ensemble’s curves after affine shift and Brownian bridge, proving the finiteness of a superpolynomially
growing moment bound on Radon-Nikodym derivatives.
We also determine the value of a natural exponent describing in Brownian last
passage percolation the decay in probability for the existence of several near geodesics that are disjoint except for their common
endpoints, where the notion of ‘near’ refers to a small deficit in scaled geodesic energy, with the parameter specifying this nearness
tending to zero.
To prove both results, we introduce a technique that may be useful elsewhere for finding upper bounds on
probabilities of events concerning random systems of curves enjoying the Brownian Gibbs property.
Several results in this article
play a fundamental role in a further study of Brownian last passage percolation in three companion papers (Hammond 2017a,b,c), in which
geodesic coalescence and geodesic energy profiles are investigated in scaled coordinates.
eBook
The Regularity of the Linear Drift in Negatively Curved Spaces
We show that the linear drift of the Brownian motion on the universal cover of a closed connected smooth Riemannian manifold is
eBook
Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs
We introduce a decoupling method on the Wiener space to define a wide class of anisotropic Besov spaces. The decoupling method is
based on a general distributional approach and not restricted to the Wiener space.
The class of Besov spaces we introduce
contains the traditional isotropic Besov spaces obtained by the real interpolation method, but also new spaces that are designed to
investigate backwards stochastic differential equations (BSDEs). As examples we discuss the Besov regularity (in the sense of our
spaces) of forward diffusions and local times. It is shown that among our newly introduced Besov spaces there are spaces that
characterize quantitative properties of directional derivatives in the Malliavin sense without computing or accessing these Malliavin
derivatives explicitly.
Regarding BSDEs, we deduce regularity properties of the solution processes from the Besov regularity of
the initial data, in particular upper bounds for their
Among other tools, we use methods from harmonic analysis. As a
by-product, we improve the asymptotic behaviour of the multiplicative constant in a generalized Fefferman inequality and verify the
optimality of the bound we established.
eBook
Stochastic flows in the Brownian web and net
It is known that certain one-dimensional nearest-neighbor random walks in i.i.d. random space-time environments have diffusive
scaling limits. Here, in the continuum limit, the random environment is represented by a ‘stochastic flow of kernels’, which is a
collection of random kernels that can be loosely interpreted as the transition probabilities of a Markov process in a random
environment. The theory of stochastic flows of kernels was first developed by Le Jan and Raimond, who showed that each such flow is
characterized by its
Our main result gives a
graphical construction of general Howitt-Warren flows, where the underlying random environment takes on the form of a suitably marked
Brownian web. This extends earlier work of Howitt and Warren who showed that a special case, the so-called ‘erosion flow’, can be
constructed from two coupled ‘sticky Brownian webs’. Our construction for general Howitt-Warren flows is based on a Poisson marking
procedure developed by Newman, Ravishankar and Schertzer for the Brownian web. Alternatively, we show that a special subclass of the
Howitt-Warren flows can be constructed as random flows of mass in a Brownian net, introduced by Sun and Swart.
Using these
constructions, we prove some new results for the Howitt-Warren flows. In particular, we show that the kernels spread with a finite speed
and have a locally finite support at deterministic times if and only if the flow is embeddable in a Brownian net. We show that the
kernels are always purely atomic at deterministic times, but, with the exception of the erosion flows, exhibit random times when the
kernels are purely non-atomic. We moreover prove ergodic statements for a class of measure-valued processes induced by the Howitt-Warren
flows.
Our work also yields some new results in the theory of the Brownian web and net. In particular, we prove several new
results about coupled sticky Brownian webs and about a natural coupling of a Brownian web with a Brownian net. We also introduce a
‘finite graph representation’ which gives a precise description of how paths in the Brownian net move between deterministic times.
eBook