281 research outputs found
Stochastic integrals and conditional full support
We present conditions that imply the conditional full support (CFS) property,
introduced by Guasoni, R\'asonyi, and Schachermayer [Ann. Appl. Probab., 18
(2008), pp. 491--520], for processes Z := H + K \cdot W, where W is a Brownian
motion, H is a continuous process, and processes H and K are either progressive
or independent of W. Moreover, in the latter case under an additional
assumption that K is of finite variation, we present conditions under which Z
has CFS also when W is replaced with a general continuous process with CFS. As
applications of these results, we show that several stochastic volatility
models and the solutions of certain stochastic differential equations have CFS.Comment: 19 pages, v3: almost entirely rewritten, new result
Random Time Forward Starting Options
We introduce a natural generalization of the forward-starting options, first
discussed by M. Rubinstein. The main feature of the contract presented here is
that the strike-determination time is not fixed ex-ante, but allowed to be
random, usually related to the occurrence of some event, either of financial
nature or not. We will call these options {\bf Random Time Forward Starting
(RTFS)}. We show that, under an appropriate "martingale preserving" hypothesis,
we can exhibit arbitrage free prices, which can be explicitly computed in many
classical market models, at least under independence between the random time
and the assets' prices. Practical implementations of the pricing methodologies
are also provided. Finally a credit value adjustment formula for these OTC
options is computed for the unilateral counterparty credit risk.Comment: 19 pages, 1 figur
Optimal Multi-Modes Switching Problem in Infinite Horizon
This paper studies the problem of the deterministic version of the
Verification Theorem for the optimal m-states switching in infinite horizon
under Markovian framework with arbitrary switching cost functions. The problem
is formulated as an extended impulse control problem and solved by means of
probabilistic tools such as the Snell envelop of processes and reflected
backward stochastic differential equations. A viscosity solutions approach is
employed to carry out a finne analysis on the associated system of m
variational inequalities with inter-connected obstacles. We show that the
vector of value functions of the optimal problem is the unique viscosity
solution to the system. This problem is in relation with the valuation of firms
in a financial market
Low Mach number effect in simulation of high Mach number flow
In this note, we relate the two well-known difficulties of Godunov schemes:
the carbuncle phenomena in simulating high Mach number flow, and the inaccurate
pressure profile in simulating low Mach number flow. We introduced two simple
low-Mach-number modifications for the classical Roe flux to decrease the
difference between the acoustic and advection contributions of the numerical
dissipation. While the first modification increases the local numerical
dissipation, the second decreases it. The numerical tests on the double-Mach
reflection problem show that both modifications eliminate the kinked Mach stem
suffered by the original flux. These results suggest that, other than
insufficient numerical dissipation near the shock front, the carbuncle
phenomena is strongly relevant to the non-comparable acoustic and advection
contributions of the numerical dissipation produced by Godunov schemes due to
the low Mach number effect.Comment: 9 pages, 1 figur
G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion
The present paper is devoted to the study of sample paths of G-Brownian
motion and stochastic differential equations (SDEs) driven by G-Brownian motion
from the view of rough path theory. As the starting point, we show that
quasi-surely, sample paths of G-Brownian motion can be enhanced to the second
level in a canonical way so that they become geometric rough paths of roughness
2 < p < 3. This result enables us to introduce the notion of rough differential
equations (RDEs) driven by G-Brownian motion in the pathwise sense under the
general framework of rough paths. Next we establish the fundamental relation
between SDEs and RDEs driven by G-Brownian motion. As an application, we
introduce the notion of SDEs on a differentiable manifold driven by GBrownian
motion and construct solutions from the RDE point of view by using pathwise
localization technique. This is the starting point of introducing G-Brownian
motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin.
The last part of this paper is devoted to such construction for a wide and
interesting class of G-functions whose invariant group is the orthogonal group.
We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian
motion of independent interest
Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
This paper is devoted to a deeper understanding of the heat flow and to the
refinement of calculus tools on metric measure spaces (X,d,m). Our main results
are:
- A general study of the relations between the Hopf-Lax semigroup and
Hamilton-Jacobi equation in metric spaces (X,d).
- The equivalence of the heat flow in L^2(X,m) generated by a suitable
Dirichlet energy and the Wasserstein gradient flow of the relative entropy
functional in the space of probability measures P(X).
- The proof of density in energy of Lipschitz functions in the Sobolev space
W^{1,2}(X,d,m).
- A fine and very general analysis of the differentiability properties of a
large class of Kantorovich potentials, in connection with the optimal transport
problem.
Our results apply in particular to spaces satisfying Ricci curvature bounds
in the sense of Lott & Villani [30] and Sturm [39,40], and require neither the
doubling property nor the validity of the local Poincar\'e inequality.Comment: Minor typos corrected and many small improvements added. Lemma 2.4,
Lemma 2.10, Prop. 5.7, Rem. 5.8, Thm. 6.3 added. Rem. 4.7, Prop. 4.8, Prop.
4.15 and Thm 4.16 augmented/reenforced. Proof of Thm. 4.16 and Lemma 9.6
simplified. Thm. 8.6 corrected. A simpler axiomatization of weak gradients,
still equivalent to all other ones, has been propose
Drift dependence of optimal trade execution strategies under transient price impact
We give a complete solution to the problem of minimizing the expected
liquidity costs in presence of a general drift when the underlying market
impact model has linear transient price impact with exponential resilience. It
turns out that this problem is well-posed only if the drift is absolutely
continuous. Optimal strategies often do not exist, and when they do, they
depend strongly on the derivative of the drift. Our approach uses elements from
singular stochastic control, even though the problem is essentially
non-Markovian due to the transience of price impact and the lack in Markovian
structure of the underlying price process. As a corollary, we give a complete
solution to the minimization of a certain cost-risk criterion in our setting
On arbitrages arising from honest times
In the context of a general continuous financial market model, we study
whether the additional information associated with an honest time gives rise to
arbitrage profits. By relying on the theory of progressive enlargement of
filtrations, we explicitly show that no kind of arbitrage profit can ever be
realised strictly before an honest time, while classical arbitrage
opportunities can be realised exactly at an honest time as well as after an
honest time. Moreover, stronger arbitrages of the first kind can only be
obtained by trading as soon as an honest time occurs. We carefully study the
behavior of local martingale deflators and consider no-arbitrage-type
conditions weaker than NFLVR.Comment: 25 pages, revised versio
Two refreshing views of Fluctuation Theorems through Kinematics Elements and Exponential Martingale
In the context of Markov evolution, we present two original approaches to
obtain Generalized Fluctuation-Dissipation Theorems (GFDT), by using the
language of stochastic derivatives and by using a family of exponential
martingales functionals. We show that GFDT are perturbative versions of
relations verified by these exponential martingales. Along the way, we prove
GFDT and Fluctuation Relations (FR) for general Markov processes, beyond the
usual proof for diffusion and pure jump processes. Finally, we relate the FR to
a family of backward and forward exponential martingales.Comment: 41 pages, 7 figures; version2: 45 pages, 7 figures, minor revisions,
new results in Section
- …