32 research outputs found
Universal L^s -rate-optimality of L^r-optimal quantizers by dilatation and contraction
Let . For a given probability measure on , let
be a sequence of (asymptotically) - optimal
quantizers. For all and for every , one
defines the sequence by : . In this paper, we are interested in the asymptotics
of the -quantization error induced by the sequence . We show that for a wide family of distributions, the
sequence is -rate-optimal. For the
Gaussian and the exponential distributions, one shows how to choose the
parameter such that satisfies
the empirical measure theorem and probably be asymptotically -optimal.Comment: 26 page
Conditional hitting time estimation in a nonlinear filtering model by the Brownian bridge method
The model consists of a signal process which is a general Brownian
diffusion process and an observation process , also a diffusion process,
which is supposed to be correlated to the signal process. We suppose that the
process is observed from time 0 to at discrete times and aim to
estimate, conditionally on these observations, the probability that the
non-observed process crosses a fixed barrier after a given time . We
formulate this problem as a usual nonlinear filtering problem and use optimal
quantization and Monte Carlo simulations techniques to estimate the involved
quantities
An application to credit risk of a hybrid Monte Carlo-Optimal quantization method
In this paper we use a hybrid Monte Carlo-Optimal quantization method to approximate the conditional survival probabilities of a firm, given a structural model for its credit defaul, under partial information. We consider the case when the firm's value is a non-observable stochastic process and inverstors in the market have access to a process , whose value at each time t is related to . We are interested in the computation of the conditional survival probabilities of the firm given the "investor information". As a application, we analyse the shape of the credit spread curve for zero coupon bonds in two examples.credit risk, structural approach, survival probabilities, partial information, filtering, optimal quantization, Monte Carlo method.
Universal -rate-optimality of -optimal quantizers by dilatation and contraction
http://www.esaim-ps.org/International audienc
Pricing of barrier options by marginal functional quantization
This paper is devoted to the pricing of Barrier options by optimal quadratic
quantization method. From a known useful representation of the premium of
barrier options one deduces an algorithm similar to one used to estimate
nonlinear filter using quadratic optimal functional quantization. Some
numerical tests are fulfilled in the Black-Scholes model and in a local
volatility model and a comparison to the so called Brownian Bridge method is
also done
Product Markovian quantization of an R^d -valued Euler scheme of a diffusion process with applications to finance
We introduce a new approach to quantize the Euler scheme of an
-valued diffusion process. This method is based on a Markovian
and componentwise product quantization and allows us, from a numerical point of
view, to speak of {\em fast online quantization} in dimension greater than one
since the product quantization of the Euler scheme of the diffusion process and
its companion weights and transition probabilities may be computed quite
instantaneously. We show that the resulting quantization process is a Markov
chain, then, we compute the associated companion weights and transition
probabilities from (semi-) closed formulas. From the analytical point of view,
we show that the induced quantization errors at the -th discretization step
is a cumulative of the marginal quantization error up to time .
Numerical experiments are performed for the pricing of a Basket call option,
for the pricing of a European call option in a Heston model and for the
approximation of the solution of backward stochastic differential equations to
show the performances of the method
Recursive marginal quantization of the Euler scheme of a diffusion process
We propose a new approach to quantize the marginals of the discrete Euler
diffusion process. The method is built recursively and involves the conditional
distribution of the marginals of the discrete Euler process. Analytically, the
method raises several questions like the analysis of the induced quadratic
quantization error between the marginals of the Euler process and the proposed
quantizations. We show in particular that at every discretization step
of the Euler scheme, this error is bounded by the cumulative quantization
errors induced by the Euler operator, from times to time . For
numerics, we restrict our analysis to the one dimensional setting and show how
to compute the optimal grids using a Newton-Raphson algorithm. We then propose
a closed formula for the companion weights and the transition probabilities
associated to the proposed quantizations. This allows us to quantize in
particular diffusion processes in local volatility models by reducing
dramatically the computational complexity of the search of optimal quantizers
while increasing their computational precision with respect to the algorithms
commonly proposed in this framework. Numerical tests are carried out for the
Brownian motion and for the pricing of European options in a local volatility
model. A comparison with the Monte Carlo simulations shows that the proposed
method may sometimes be more efficient (w.r.t. both computational precision and
time complexity) than the Monte Carlo method.Comment: 29 page
Universal
We investigate in this paper the properties of some dilatations or contractions of a sequence (αn)n≥1 of Lr-optimal quantizers of an -valued random vector defined in the probability space with distribution . To be precise, we investigate the Ls-quantization rate of sequences when or s ∈ (r, +∞) and . We show that for a wide family of distributions, one may always find parameters (θ,µ) such that (αnθ,µ)n≥1 is Ls-rate-optimal. For the Gaussian and the exponential distributions we show the existence of a couple (θ*,µ*) such that (αθ*,µ*)n≥1 also satisfies the so-called Ls-empirical measure theorem. Our conjecture, confirmed by numerical experiments, is that such sequences are asymptotically Ls-optimal. In both cases the sequence (αθ*,µ*)n≥1 is incredibly close to Ls-optimality. However we show (see Rem. 5.4) that this last sequence is not Ls-optimal (e.g. when s = 2, r = 1) for the exponential distribution