33 research outputs found
Multiple G-It\^{o} integral in the G-expectation space
In this paper, motivated by mathematic finance we introduce the multiple
G-It\^{o} integral in the G-expectation space, then investigate how to
calculate. We get the the relationship between Hermite polynomials and multiple
G-It\^{o} integrals which is a natural extension of the classical result
obtained by It\^{o} in 1951.Comment: 9 page
Minimal Supersolutions of BSDEs with Lower Semicontinuous Generators
We study the existence and uniqueness of minimal supersolutions of backward
stochastic differential equations with generators that are jointly lower
semicontinuous, bounded below by an affine function of the control variable and
satisfy a specific normalization property
A comparison principle for PDEs arising in approximate hedging problems: application to Bermudan options
In a Markovian framework, we consider the problem of finding the minimal
initial value of a controlled process allowing to reach a stochastic target
with a given level of expected loss. This question arises typically in
approximate hedging problems. The solution to this problem has been
characterised by Bouchard, Elie and Touzi in [1] and is known to solve an
Hamilton-Jacobi-Bellman PDE with discontinuous operator. In this paper, we
prove a comparison theorem for the corresponding PDE by showing first that it
can be rewritten using a continuous operator, in some cases. As an application,
we then study the quantile hedging price of Bermudan options in the non-linear
case, pursuing the study initiated in [2].
[1] Bruno Bouchard, Romuald Elie, and Nizar Touzi. Stochastic target problems
with controlled loss. SIAM Journal on Control and Optimization,
48(5):3123-3150,2009. [2] Bruno Bouchard, Romuald Elie, Antony R\'eveillac, et
al. Bsdes with weak terminal condition. The Annals of Probability,
43(2):572-604,2015
Options hedging under liquidity costs
Following the framework of Cetin, Jarrow and Protter (CJP) we study the problem of super-replication in presence of liquidity costs under additional restrictions on the gamma of the hedging strategies in a generalized Black-Scholes economy. We find that the minimal super-replication price is different than the one suggested by the Black-Scholes formula and is the unique viscosity solution of the associated dynamic programming equation. This is in contrast with the results of CJP who find that the arbitrage free price of a contingent claim coincides with the Black-Scholes price. However, in CJP a larger class of admissible portfolio processes is used and the replication is achieved in the L^2 approximating
sense
A backward dual representation for the quantile hedging of Bermudan options
Within a Markovian complete financial market, we consider the problem of
hedging a Bermudan option with a given probability. Using stochastic target and
duality arguments, we derive a backward numerical scheme for the Fenchel
transform of the pricing function. This algorithm is similar to the usual
American backward induction, except that it requires two additional Fenchel
transformations at each exercise date. We provide numerical illustrations
Second order backward stochastic differential equations and fully non-linear parabolic PDEs
We introduce a class of second order backward stochastic differential
equations and show relations to fully non-linear parabolic PDEs. In particular,
we provide a stochastic representation result for solutions of such PDEs and
discuss Monte Carlo methods for their numerical treatment.Comment: 26 page
Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions
We study a class of stochastic target games where one player tries to find a
strategy such that the state process almost-surely reaches a given target, no
matter which action is chosen by the opponent. Our main result is a geometric
dynamic programming principle which allows us to characterize the value
function as the viscosity solution of a non-linear partial differential
equation. Because abstract mea-surable selection arguments cannot be used in
this context, the main obstacle is the construction of measurable
almost-optimal strategies. We propose a novel approach where smooth
supersolutions are used to define almost-optimal strategies of Markovian type,
similarly as in ver-ification arguments for classical solutions of
Hamilton--Jacobi--Bellman equations. The smooth supersolutions are constructed
by an exten-sion of Krylov's method of shaken coefficients. We apply our
results to a problem of option pricing under model uncertainty with different
interest rates for borrowing and lending.Comment: To appear in MO
Lyapunov stabilizability of controlled diffusions via a superoptimality principle for viscosity solutions
We prove optimality principles for semicontinuous bounded viscosity solutions
of Hamilton-Jacobi-Bellman equations. In particular we provide a representation
formula for viscosity supersolutions as value functions of suitable obstacle
control problems. This result is applied to extend the Lyapunov direct method
for stability to controlled Ito stochastic differential equations. We define
the appropriate concept of Lyapunov function to study the stochastic open loop
stabilizability in probability and the local and global asymptotic
stabilizability (or asymptotic controllability). Finally we illustrate the
theory with some examples.Comment: 22 page