arXiv subjects

P. K. Friz

Publications and source records attributed to P. K. Friz.

5 recordsLinked to original sources

arXiv2016

Support theorem for a singular semilinear stochastic partial differential equation

We consider the generalized parabolic Anderson equation (gPAM) in 2 dimensions with periodic boundary. This is an example of a singular semilinear stochastic partial differential equations, solutions of which require renormalization and have only be understood recently via Hairer's regularity structures and, in some cases equivalently, paracontrollled distributions due to Gubinelli, Imkeller and Perkowski. In the present paper we describe the law of gPAM, by establishing a Stroock{Varadhan type support theorem in suitable Holder{Besov spaces.

math.PR↗

arXiv2013

Robust filtering: Correlated noise and multidimensional observation

In the late seventies, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721-734, Sijthoff & Noordhoff] pointed out that it would be natural for $π_t$, the solution of the stochastic filtering problem, to depend continuously on the observed data $Y=\{Y_s,s\in[0,t]\}$. Indeed, if the signal and the observation noise are independent one can show that, for any suitably chosen test function $f$, there exists a continuous map $θ^f_t$, defined on the space of continuous paths $C([0,t],\mathbb{R}^d)$ endowed with the uniform convergence topology such that $π_t(f)=θ^f_t(Y)$, almost surely; see, for example, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721-734, Sijthoff & Noordhoff], Clark and Crisan [Probab. Theory Related Fields 133 (2005) 43-56], Davis [Z. Wahrsch. Verw. Gebiete 54 (1980) 125-139], Davis [Teor. Veroyatn. Primen. 27 (1982) 160-167], Kushner [Stochastics 3 (1979) 75-83]. As shown by Davis and Spathopoulos [SIAM J. Control Optim. 25 (1987) 260-278], Davis [In Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Proc. NATO Adv. Study Inst. Les Arcs, Savoie, France 1980 505-528], [In The Oxford Handbook of Nonlinear Filtering (2011) 403-424 Oxford Univ. Press], this type of robust representation is also possible when the signal and the observation noise are correlated, provided the observation process is scalar. For a general correlated noise and multidimensional observations such a representation does not exist. By using the theory of rough paths we provide a solution to this deficiency: the observation process $Y$ is "lifted" to the process $\mathbf{Y}$ that consists of $Y$ and its corresponding Lévy area process, and we show that there exists a continuous map $θ_t^f$, defined on a suitably chosen space of Hölder continuous paths such that $π_t(f)=θ_t^f(\mathbf{Y})$, almost surely.

math.PR↗

arXiv2013

Marginal density expansions for diffusions and stochastic volatility, part I: Theoretical Foundations

Density expansions for hypoelliptic diffusions $(X^1,...,X^d)$ are revisited. In particular, we are interested in density expansions of the projection $(X_T^1,...,X_T^l)$, at time $T>0$, with $l \leq d$. Global conditions are found which replace the well-known "not-in-cutlocus" condition known from heat-kernel asymptotics. Our small noise expansion allows for a "second order" exponential factor. As application, new light is shed on the Takanobu--Watanabe expansion of Brownian motion and Levy's stochastic area. Further applications include tail and implied volatility asymptotics in some stochastic volatility models, discussed in a compagnion paper.

math.PR↗

arXiv2013

Marginal density expansions for diffusions and stochastic volatility, part II: Applications [to the Stein--Stein model]

In the compagnion paper [Marginal density expansions for diffusions and stochastic volatility, part I] we discussed density expansions for multidimensional diffusions $(X^1,...,X^d)$, at fixed time $T$ and projected to their first $l$ coordinates, in the small noise regime. Global conditions were found which replace the well-known "not-in-cutlocus" condition known from heat-kernel asymptotics. In the present paper we discuss financial applications; these include tail and implied volatility asymptotics in some correlated stochastic volatility models. In particular, we solve a problem left open by A. Gulisashvili and E.M. Stein (2009).

math.PR↗

arXiv2007

The Bismut-Elworthy-Li formula for jump-diffusions and applications to Monte Carlo pricing in finance

We extend the Bismut-Elworthy-Li formula to non-degenerate jump diffusions and "payoff" functions depending on the process at multiple future times. In the spirit of Fournie et al [13] and Davis and Johansson [9] this can improve Monte Carlo numerics for stochastic volatility models with jumps. To this end one needs so-called Malliavin weights and we give explicit formulae valid in presence of jumps: (a) In a non-degenerate situation, the extended BEL formula represents possible Malliavin weights as Ito integrals with explicit integrands; (b) in a hypoelliptic setting we review work of Arnaudon and Thalmaier [1] and also find explicit weights, now involving the Malliavin covariance matrix, but still straight-forward to implement. (This is in contrast to recent work by Forster, Lutkebohmert and Teichmann where weights are constructed as anticipating Skorohod integrals.) We give some financial examples covered by (b) but note that most practical cases of poor Monte Carlo performance, Digital Cliquet contracts for instance, can be dealt with by the extended BEL formula and hence without any reliance on Malliavin calculus at all. We then discuss some of the approximations, often ignored in the literature, needed to justify the use of the Malliavin weights in the context of standard jump diffusion models. Finally, as all this is meant to improve numerics, we give some numerical results with focus on Cliquets under the Heston model with jumps.

math.PR↗