568 research outputs found
Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management
We formulate and investigate a general stochastic control problem under a
progressive enlargement of filtration. The global information is enlarged from
a reference filtration and the knowledge of multiple random times together with
associated marks when they occur. By working under a density hypothesis on the
conditional joint distribution of the random times and marks, we prove a
decomposition of the original stochastic control problem under the global
filtration into classical stochastic control problems under the reference
filtration, which are determined in a finite backward induction. Our method
revisits and extends in particular stochastic control of diffusion processes
with finite number of jumps. This study is motivated by optimization problems
arising in default risk management, and we provide applications of our
decomposition result for the indifference pricing of defaultable claims, and
the optimal investment under bilateral counterparty risk. The solutions are
expressed in terms of BSDEs involving only Brownian filtration, and remarkably
without jump terms coming from the default times and marks in the global
filtration
Long time asymptotics for optimal investment
This survey reviews portfolio selection problem for long-term horizon. We
consider two objectives: (i) maximize the probability for outperforming a
target growth rate of wealth process (ii) minimize the probability of falling
below a target growth rate. We study the asymptotic behavior of these criteria
formulated as large deviations control pro\-blems, that we solve by duality
method leading to ergodic risk-sensitive portfolio optimization problems.
Special emphasis is placed on linear factor models where explicit solutions are
obtained
Feynman-Kac representation of fully nonlinear PDEs and applications
The classical Feynman-Kac formula states the connection between linear
parabolic partial differential equations (PDEs), like the heat equation, and
expectation of stochastic processes driven by Brownian motion. It gives then a
method for solving linear PDEs by Monte Carlo simulations of random processes.
The extension to (fully)nonlinear PDEs led in the recent years to important
developments in stochastic analysis and the emergence of the theory of backward
stochastic differential equations (BSDEs), which can be viewed as nonlinear
Feynman-Kac formulas. We review in this paper the main ideas and results in
this area, and present implications of these probabilistic representations for
the numerical resolution of nonlinear PDEs, together with some applications to
stochastic control problems and model uncertainty in finance
Optimal investment with counterparty risk: a default-density modeling approach
We consider a financial market with a stock exposed to a counterparty risk
inducing a drop in the price, and which can still be traded after this default
time. We use a default-density modeling approach, and address in this
incomplete market context the expected utility maximization from terminal
wealth. We show how this problem can be suitably decomposed in two optimization
problems in complete market framework: an after-default utility maximization
and a global before-default optimization problem involving the former one.
These two optimization problems are solved explicitly, respectively by duality
and dynamic programming approaches, and provide a fine understanding of the
optimal strategy. We give some numerical results illustrating the impact of
counterparty risk and the loss given default on optimal trading strategies, in
particular with respect to the Merton portfolio selection problem
Impulse control problem on finite horizon with execution delay
We consider impulse control problems in finite horizon for diffusions with
decision lag and execution delay. The new feature is that our general framework
deals with the important case when several consecutive orders may be decided
before the effective execution of the first one. This is motivated by financial
applications in the trading of illiquid assets such as hedge funds. We show
that the value functions for such control problems satisfy a suitable version
of dynamic programming principle in finite dimension, which takes into account
the past dependence of state process through the pending orders. The
corresponding Bellman partial differential equations (PDE) system is derived,
and exhibit some peculiarities on the coupled equations, domains and boundary
conditions. We prove a unique characterization of the value functions to this
nonstandard PDE system by means of viscosity solutions. We then provide an
algorithm to find the value functions and the optimal control. This easily
implementable algorithm involves backward and forward iterations on the domains
and the value functions, which appear in turn as original arguments in the
proofs for the boundary conditions and uniqueness results
Optimal High Frequency Trading with limit and market orders
We propose a framework for studying optimal market making policies in a limit order book (LOB). The bid-ask spread of the LOB is modelled by a Markov chain with finite values, multiple of the tick size, and subordinated by the Poisson process of the tick-time clock. We consider a small agent who continuously submits limit buy/sell orders and submits market orders at discrete dates. The objective of the market maker is to maximize her expected utility from revenue over a short term horizon by a tradeoff between limit and market orders, while controlling her inventory position. This is formulated as a mixed regime switching regular/ impulse control problem that we characterize in terms of quasi-variational system by dynamic programming methods. In the case of a mean-variance criterion with martingale reference price or when the asset price follows a Levy process and with exponential utility criterion, the dynamic programming system can be reduced to a system of simple equations involving only the inventory and spread variables. Calibration procedures are derived for estimating the transition matrix and intensity parameters for the spread and for Cox processes modelling the execution of limit orders. Several computational tests are performed both on simulated and real data, and illustrate the impact and profit when considering execution priority in limit orders and market ordersMarket making; limit order book; inventory risk; point process; stochastic control
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