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Register a new userWhen a Stochastic Exponential is a True Martingale. Extension of a Method of Bene^s
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Let $mathfrak{z}$ be a stochastic exponential, i.e., $mathfrak{z}_t=1+int_0^tmathfrak{z}_{s-}dM_s$, of a local martingale $M$ with jumps $triangle M_t>-1$. Then $mathfrak{z}$ is a nonnegative local martingale with $Emathfrak{z}_tle 1$. If $Emathfrak{z}_T= 1$, then $mathfrak{z}$ is a martingale on the time interval $[0,T]$. Martingale property plays an important role in many applications. It is therefore of interest to give natural and easy verifiable conditions for the martingale property. In this paper, the property $Emathfrak{z}_{_T}=1$ is verified with the so-called linear growth conditions involved in the definition of parameters of $M$, proposed by Girsanov cite{Girs}. These conditions generalize the Bene^s idea, cite{Benes}, and avoid the technology of piece-wise approximation. These conditions are applicable even if Novikov, cite{Novikov}, and Kazamaki, cite{Kaz}, conditions fail. They are effective for Markov processes that explode, Markov processes with jumps and also non Markov processes. Our approach is different to recently published papers cite{CFY} and cite{MiUr}.
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It is known the Girsanov exponent $mathfrak{z}_t$, being solution of Doleans-Dade equation $ mathfrak{z_t}=1+int_0^talpha(omega,s)dB_s $ generated by Brownian motion $B_t$ and a random process $alpha(omega,t)$ with $int_0^talpha^2(omega,s)ds<infty$ a.s., is the martingale provided that the Bene${rm check{s}}$ condition $$ |alpha(omega,t)|^2le text{rm const.}big[1+sup_{sin[0,t]}B^2_sbig], forall t>0, $$ holds true. In this paper, we show $B_t$ can be replaced by by a homogeneous purely discontinuous square integrable martingale $M_t$ with independent increments and paths from the Skorokhod space $ mathbb{D}_{[0,infty)} $ having positive jumps $triangle M_t$ with $Esum_{sin[0,t]}(triangle M_s)^3<infty$. A function $alpha(omega,t)$ is assumed to be nonnegative and predictable. Under this setting $mathfrak{z}_t$ is the martingale provided that $$ alpha^2(omega,t)le text{rm const.}big[1+sup_{sin[0,t]}M^2_{s-}big], forall t>0. $$ The method of proof differs from the original Bene${rm check{s}}$ one and is compatible for both setting with $B_t$ and $M_t$.
This work is devoted to a vast extension of Sanovs theorem, in Laplace principle form, based on alternatives to the classical convex dual pair of relative entropy and cumulant generating functional. The abstract results give rise to a number of probabilistic limit theorems and asymptotics. For instance, widely applicable non-exponential large deviation upper bounds are derived for empirical distributions and averages of i.i.d. samples under minimal integrability assumptions, notably accommodating heavy-tailed distributions. Other interesting manifestations of the abstract results include new results on the rate of convergence of empirical measures in Wasserstein distance, uniform large deviation bounds, and variational problems involving optimal transport costs, as well as an application to error estimates for approximate solutions of stochastic optimization problems. The proofs build on the Dupuis-Ellis weak convergence approach to large deviations as well as the duality theory for convex risk measures.
We discuss the detection of gravitational-wave backgrounds in the context of Bayesian inference and suggest a practical definition of what it means for a signal to be considered stochastic---namely, that the Bayesian evidence favors a stochastic signal model over a deterministic signal model. A signal can further be classified as Gaussian-stochastic if a Gaussian signal model is favored. In our analysis we use Bayesian model selection to choose between several signal and noise models for simulated data consisting of uncorrelated Gaussian detector noise plus a superposition of sinusoidal signals from an astrophysical population of gravitational-wave sources. For simplicity, we consider co-located and co-aligned detectors with white detector noise, but the method can be extended to more realistic detector configurations and power spectra. The general trend we observe is that a deterministic model is favored for small source numbers, a non-Gaussian stochastic model is preferred for intermediate source numbers, and a Gaussian stochastic model is preferred for large source numbers. However, there is very large variation between individual signal realizations, leading to fuzzy boundaries between the three regimes. We find that a hybrid, trans-dimensional model comprised of a deterministic signal model for individual bright sources and a Gaussian-stochastic signal model for the remaining confusion background outperforms all other models in most instances.
We study the stochastic growth process in discrete time $x_{i+1} = (1 + mu_i) x_i$ with growth rate $mu_i = rho e^{Z_i - frac12 var(Z_i)}$ proportional to the exponential of an Ornstein-Uhlenbeck (O-U) process $dZ_t = - gamma Z_t dt + sigma dW_t$ sampled on a grid of uniformly spaced times ${t_i}_{i=0}^n$ with time step $tau$. Using large deviation theory methods we compute the asymptotic growth rate (Lyapunov exponent) $lambda = lim_{nto infty} frac{1}{n} log mathbb{E}[x_n]$. We show that this limit exists, under appropriate scaling of the O-U parameters, and can be expressed as the solution of a variational problem. The asymptotic growth rate is related to the thermodynamical pressure of a one-dimensional lattice gas with attractive exponential potentials. For $Z_t$ a stationary O-U process the lattice gas coincides with a system considered previously by Kac and Helfand. We derive upper and lower bounds on $lambda$. In the large mean-reversion limit $gamma n tau gg 1$ the two bounds converge and the growth rate is given by a lattice version of the van der Waals equation of state. The predictions are tested against numerical simulations of the stochastic growth model.
We consider the sums $S_n=xi_1+cdots+xi_n$ of independent identically distributed random variables. We do not assume that the $xi$s have a finite mean. Under subexponential type conditions on distribution of the summands, we find the asymptotics of the probability ${bf P}{M>x}$ as $xtoinfty$, provided that $M=sup{S_n, nge1}$ is a proper random variable. Special attention is paid to the case of tails which are regularly varying at infinity. We provide some sufficient conditions for the integrated weighted tail distribution to be subexponential. We supplement these conditions by a number of examples which cover both the infinite- and the finite-mean cases. In particular, we show that subexponentiality of distribution $F$ does not imply subexponentiality of its integrated tail distribution $F^I$.
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