# Seminar on Stochastics and PDEs

joint seminar of the Stochastics' group with the group of Mikko Parviainen

### Next talk:

Thu Nov 23, 14:15-16:00 @MaA 203
Eija Laukkarinen
Malliavin smoothness on the Lévy space with Hölder continuous or BV functionals

Abstract: Suppose X is a pure jump Lévy process and f is bounded and Hölder continuous or BV. We show that Malliavin smoothness of f (X1) depends both on the regularity of f and the Blumenthal-Getoor index of the Lévy measure.

### Autumn 2017:

• Tue Dec 05, 14:15-16:00 @MaD 302
Christian Bender
TBA
• We Nov 29, 15:30-17:00 @MaD 302
Philippe Briand
TBA
• Thu Nov 23, 14:15-16:00 @MaA 203
Eija Laukkarinen
Malliavin smoothness on the Lévy space with Hölder continuous or BV functionals

Abstract: Suppose X is a pure jump Lévy process and f is bounded and Hölder continuous or BV. We show that Malliavin smoothness of f (X1) depends both on the regularity of f and the Blumenthal-Getoor index of the Lévy measure.

• Thu Oct 26, 14:15-16:00 @MaA 203
Mikko Parviainen
Recent progress in normalized equations
• Thu Oct 19, 14:15-16:00 @MaA 203
Christel Geiss
Stochastic control and dynamic programming: an example
• Thu Oct 05, 14:15-16:00 @MaA 203
Amal Attouchi
The Cordes-Nirenberg condition and its applications

Abstract: An operator of the form aij(x) uij(x)=f is of Cordes type if the eigenvalues of the matrix aij are not too dispersed. In 2 dimensions every elliptic operator is of Cordes type but for dimension strictly larger than 2, this condition is a special one. In general it is not possible to obtain solvability in the class of strong solutions with the mere assumption of uniform ellipticity. Using some analytic techniques, we will see how the Cordes condition implies strong solvability and regularity for boundary value problems for linear uniformly elliptic problems with discontinuous coefficients, and some applications to a priori estimates for the L2 norm of the Hessian of solutions of some p-Laplacian problems.

• Thu Sept 28, 14:15-16:00 @MaA 203
Stefan Geiss
On a generalization of a result of Klass about randomly stopped sums
• Thu Sept 21, 14:15-16:00 @MaA 203
Amal Attouchi
Hölder regularity for the gradient and Hessian integrability for viscosity solutions of degenerate or singular p-Laplacian type equations in non divergence form

Abstract: Regularity theory for viscosity solutions to elliptic equations in non divergence form is a central topic in the field of analysis of PDEs since the works of Krylov-Safonov, Evans and Caffarelli. In this talk we consider the special case of the possibly degenerate or singular equation modelled on the p-Laplacian: |Du|γ ΔNpu=f including the special cases of the standard (γ= p-2) and normalised (γ=0) p-Laplacian. We review some recent regularity results and analyze the Hölder regularity of the gradient and the integrability of the Hessian.

• Thu Sept 14, 14:15-16:00 @MaA 203
Angel Arroyo Garcia
Mean value properties in metric measure spaces
Henri Ylinen
Type and cotype of anisotropic Besov spaces
Jarkko Siltakoski
On the normalized p(x)-Laplace equation
Joonas Heino
Continuous time tug-of-war with space and time dependent weights
• Thu Sept 7, 14:15-16:00 @MaA 203
Amal Attouchi
Regularity results for degenerate or singular parabolic equations in nondivergence form
Eero Ruosteenoja
Regularity properties of tug-of-war games
Eija Laukkarinen
On stochastic derivatives and integrals
Thuan Nguyen
A note on approximation for stochastic integrals in L\'evy setting

### Spring 2017:

• (Analysis Seminar) Thu May 18, 14:15-15:15 @MaD380
Alessio Porretta
PDE systems in mean field games theory
• Tue May 9, 10:15-12:00 @MaA 203
Thuan Nguyen
An embedding theorem for convex combination spaces and applications

Abstract: In this talk, we embed a metric space endowed with a convex combination operation, named convex combination space, into a Banach space, where the embedding preserves the structures of the metric and convex combination. Applications of this embedding are also established for random elements taking values in this kind of space. On the one hand, some properties of expectation such as the representation of expectation through continuous affine mappings and the linearity of expectation will be given. On the other hand, the notion of conditional expectation will be also introduced and discussed. Thanks to the embedding theorem, we establish some basic properties of conditional expectation, Jensen's inequality, convergences of martingales and an ergodic theorem.

• Tue May 2, 10:15-12:00 @MaA 203
Amal Attouchi
Boundary regularity for parabolic equations II
• Tue April 25, 10:15-12:00 @MaA 203
Amal Attouchi
Boundary regularity for parabolic equations I

Abstract: In this talk we will review and discuss some results about the boundary regularity of solutions of parabolic differential equations. The results depend on the different geometric assumptions on the parabolic boundary. My talk is based on some papers of Lihe Wang and the works of Krylov.

• Tue April 18, 10:15-12:00 @MaA 203
Joonas Heino
Approximation of viscosity solutions II
Stefan Geiss
Sobolev spaces on infinite dimensional spaces III
• Tue April 04, 10:15-12:00 @MaA 203
Joonas Heino
Approximation of viscosity solutions

Abstract: It is well known that the sup-convolutions and inf-convolutions yield good approximations of viscosity subsolutions and supersolutions, respectively. In this talk, I will show that the sup-convolution of a continuous viscosity subsolution to the normalized p(x)-Laplacian is also a viscosity subsolution to the equation up to a small error. The talk is based on the work of H. Ishii from 1995.

• Tue March 28, 10:15-12:00 @MaA 203
Stefan Geiss
Sobolev spaces on infinite dimensional spaces II
• Tue March 21, 10:15-12:00 @MaA 203
Mikko Parviainen
Boundary regularity for viscosity solutions II
• Tue March 14, 10:15-12:00 @MaA 203
Christophe Andrieu (University of Bristol)
Some L2 analysis results for a class of Monte Carlo Markov chains & processes
• Tue March 7, 10:15-12:00 @MaA 203
Mikko Parviainen
Boundary regularity for viscosity solutions I
• (Analysis Seminar) Wed March 1, 14:15-15:00 @MaD 380
Christopher Hopper (Aalto University)
Partial Regularity for Holonomic Minimisers of Quasiconvex Functionals

Abstract: We prove partial regularity for local minimisers of certain strictly quasiconvex integral functionals, over a class of Sobolev mappings into a compact Riemannian manifold, to which such mappings are said to be holonomically constrained. Several applications to variational problems in condensed matter physics with broken symmetries are also discussed, in particular those concerning the superfluidity of liquid helium-3 and nematic liquid crystals.

• Tue February 21, 10:15-12:00 @MaA 203
Stefan Geiss
Sobolev spaces on infinite dimensional spaces I
• Tue February 14, 10:15-12:00 @MaA 203
Steffen Dereich
Probabilistic analysis of complex networks with fitness

Abstract: A popular model for complex networks is the preferential attachment model which gained popularity in the end of the 90's since it gives a simple explanation for the appearance of power laws in real world networks. Mathematically, one considers a sequence of random graphs that is built dynamically according to a simple rule. In each step a new vertex is added and linked randomly by a random or deterministic number of edges to the vertices already present in the system. In this process, links to vertices with high degree are preferred. A variant of the model, additionally, assigns each vertex a random positive fitness (say a μ-distributed value) which has a linear impact on its attractivity in the network formation.
Such network models show a phase transition for compactly supported μ. In the condensation phase, in the limit, there is a comparably small set of vertices (the condensate) that attracts a constant fraction of new links established by new vertices. This condensation effect was observed for the first time by Bianconi and Barási in 2001, where it was coined Bose-Einstein phase due to similarities to Bose-Einstein condensation. The fitness of the vertices in the condensate gradually converges to the essential supremum of μ and in the talk we discuss the dynamics of this process.

• Tue February 07, 10:15-12:00 @MaA 203
Jarkko Siltakoski
Connections between weak and viscosity solutions II
• Tue January 31, 10:15-12:00 @MaA 203
Henri Ylinen
Integrability and tail behavior of Radon Gaussian random variables II
• Tue January 24, 10:15-12:00 @MaA 203
Jarkko Siltakoski
Connections between weak and viscosity solutions I
• Tue January 17, 10:15-12:00 @MaA 203
Henri Ylinen
Integrability and tail behavior of Radon Gaussian random variables I

### Autumn 2016:

• Thu December 1, 12:15-14:00 @ MaD381
Francesco Russo (ENSTA-ParisTech)
BSDEs, càdlàg martingale problems and mean-variance hedging under basis risk

The talk will be based on joint work with Ismail Laachir (ZELIADE and ENSTA ParisTech)
SIAM Journal on Financial Mathematics (SIFIN), vol. 7, pp. 308-356, 2016.

Abstract: The aim of this talk consists in introducing a new formalism for the deterministic analysis associated with backward stochastic differential equations driven by general càdlàg martingales. When the martingale is a standard Brownian motion, the natural deterministic analysis is provided by the solution of a semilinear PDE of parabolic type. A significant application concerns the hedging problem under basis risk of a contingent claim g(XT,ST), where S (resp. X) is an underlying price of a traded (resp. non-traded but observable) asset, via the celebrated Föllmer-Schweizer decomposition. We revisit the case when the couple of price processes $(X,S)$ is a diffusion and we provide explicit expressions when (X,S) is an exponential of additive processes.

• (Analysis seminar) Wed November 30, 14:15-15:15 @ MaD380
Espen Jakobsen (Norwegian University of science and Technology, Trondheim)
On error estimates for monotone approximations for Bellman and Isaacs equations

Abstract: In this talk I will discuss different analytical approaches to obtaining error estimates for monotone numerical approximations of Bellman and Isaacs equations from deterministic and stochastic control and game theory. I will focus on local equations and try to explain why we can have general results for first order equations, but only partial results for second order equations, especially for non-convex Isaacs equations.
Towards the end of the talk, I will discuss some recent results for nonlocal Isaacs equations, where a 'general' error estimate is possible even if the order of the equation is greater than one. This latter result is joint work with Imran H. Biswas and Indranil Chowdhury, both at TIFR, Bangalore, India.

• Fr November 25, 12:15–14:00 @ MaD381
Antti Luoto
Reading Seminar: A stochastic version of Pontryagin's maximum principle II
• Fr November 18, 12:15–14:00 @ MaD381
Paolo di Tella (Technical University Dresden)
About a hedging method in incomplete financial markets

Abstract: In this talk we present a hedging method based on Fourier transform for contingent claims in incomplete market models. First we give a very basic introduction to the problem of hedging in complete and incomplete markets and briefly discuss the well-known Black&Scholes; model. Then we come to more general market models, as stochastic volatility models, which are, in general, incomplete. In this setting we motivate and explain the role of "semi-static hedging" and show how to reduce it to a quadratic minimization problem. We apply the general method to the special case of the Heston model, and in this context we also present some numerical results. This is a joint work with M. Keller-Ressel and M. Haubold from University of Technology, Dresden.

• Fr November 11, 12:15–14:00 @ MaD381
Antti Luoto
Reading Seminar: A stochastic version of Pontryagin's maximum principle I
• Fr November 4, 12:15–14:00 @ MaD381
Stefan Geiss
Reading Seminar (RS) on Besov spaces in probability: Suslin spaces, analytic sets an probability: and overview II
Henri Ylinen
• Fr October 28, 12:15–14:00 @ MaD381
Eero Ruosteenoja

Abstract: Pontryagin's maximum principle was formulated in 1956 by Pontryagin and his students Boltyansky and Gamkrelidze. It provides a general set of necessary conditions for an extremum in an optimal control problem. My presentation is based on a monograph by Fleming and Soner.

• Fr October 21, 12:15–14:00 @ MaD245
Stefan Geiss
Reading Seminar (RS) on Besov spaces in probability: Suslin spaces, analytic sets an probability: and overview I
Henri Ylinen
• Fr October 14, 12:15–14:00 @ MaD245
Joonas Heino
Stochastic estimates for a random cylinder walk related to the normalized p(x)-Laplacian

Abstract: I will consider a random cylinder walk that is related to the normalized p(x)-Laplacian via a cancellation strategy for a player in a stochastic 'tug-of-war' game. This connection can be used to prove existence of a continuous viscosity solution to a boundary value problem with the normalized p(x)-Laplacian. I briefly explain how this relation can be used and then, concentrate on key hitting time estimates for the cylinder walk.

• Fr October 7, 12:15–14:00 @ MaD245
Juha Ylinen
How to use regularity structures for solving SPDEs, a sketch
(based on Martin Hairer's course at Barcelona 2016)
• (Analysis seminar) We October 5, 14:15-15:15 @ MaD380
Lasse Leskelä
Diclique clustering in a directed random intersection graph

Abstract: I will discuss a notion of clustering in directed graphs, which describes how likely two followers of a node are to follow a common target. The associated network motifs, called dicliques or bi-fans, have been found to be key structural components in various real-world networks. A two-mode statistical network model consisting of actors and auxiliary attributes is introduced, where an actor i decides to follow an actor j whenever i demands an attribute supplied by j. This directed random graph model admits nontrivial clustering properties of the aforementioned type, as well as power-law indegree and outdegree distributions. The talk is based on joint work with Mindaugas Bloznelis (U Vilnius), available at arXiv:1607.02278.

• Fr September 30, 12:15–14:00 @ MaD245
Pekka Lehtelä
A weak Harnack estimate for supersolutions to the porous medium equation
• Fr September 23, 12:15–14:00 @ MaD381
Bruno Bouchard
General a-priori estimates for super-solutions of BSDEs and Doob-Meyer-Mertens decomposition of g-supermatingales

Abstract:In this talk, we will show how two classical tools of the general theory of stochastic processes can be adapted to the framework of BSDEs to obtain new estimates and Doob-Meyer type decompositions. More precisely, we shall first explain how a-priori estimates can be obtained for general super-solutions of BSDEs in general filtrations by using controls on the Doob-Meyer decomposition of a supermartingale du to Meyer. This allows one to prove for instance the well-posedness of Lp-solutions of reflected BSDEs in a filtration that is only assumed to satisfy the usual conditions. Second, we shall explain how the proof of the Doob-Meyer decomposition for ladlag supermartingales du to Mertens can be adapted to the setting of g-supermatingales. This provides a representation in terms of supersolutions of BSDEs without any a-priori regularity on the path. Non-trivial examples of application will be discussed.

• Fr September 16, 12:15–14:00 @ MaD245
Juha Ylinen
A glimpse of regularity structures
(based on Martin Hairer's course at Barcelona 2016)

### Spring 2016:

• Mo May 16, 10:15–12:00 @ MaD380
Mikko Parviainen
• Mo April 18, 10:15–12:00 @ MaD380
Stefan Geiss
On uniqueness for sub-quadratic stochastic backwards equations
• Mo April 11, 10:15–12:00 @ MaD380
Eero Ruosteenoja
Uniform gradient estimates for p-Laplace type equations

Abstract: I will present techniques of nonlinear potential theory to obtain uniform gradient estimates for nonlinear PDEs. The talk is based on the work of Frank Duzaar and Giuseppe Mingione. The techniques can be used to show local C1,α regularity for the normalized p-Poisson equation.

• Mo April 4, 10:15–12:00 @ MaD380
Petteri Piiroinen (Tampere University of Technology)
Feynman-Kac Formulae and Stochastic Homogenization
• Mo March 14, 10:15–12:00 @ MaD380
Anni Laitinen
Convergence rate for the hedging error of a path-dependent example (joint work with Dario Gasbarra)
• Thu February 25, 10:15–12:00 @ MaA203
Vesa Julin
Small Perturbation Solutions for Elliptic Equations
(based on a paper by O. Savin with the same title)
• Thu February 18, 10:15–12:00 @ MaA203
Stefan Geiss
On decoupling inequalities in Banach spaces II (joint work with S. Cox, Amsterdam)
• Thu February 11, 10:15–12:00 @ MaA203
Amal Attouchi
Potential theory to derive gradient estimates

In this talk we show how different gradient estimates can be obtained via suitable non-linear potentials. We start by presenting some introductory aspects of potential theory and its various applications. In a second part we discuss a recent local Lipschitz regularity result for degenerate elliptic equations obtained by Duzaar and Mingione via gradient potential estimates. In the last part we show how to use this result to drive uniform C1,α estimate for viscosity solutions of the Poisson problem of the normalized p-Laplacian when the source term is not bounded but belongs to some Lebesgue space.

• Thu February 04, 10:15–12:00 @ MaA203
Stefan Geiss
On decoupling inequalities in Banach spaces I (joint work with S. Cox, Amsterdam)
• Thu January 28, 10:15–12:00 @ MaA203
Christel Geiss
Stroock and Varadhan's convergence of Markov chains II
• Thu January 21, 10:15–12:00 @ MaA203
Christel Geiss
Stroock and Varadhan's convergence of Markov chains I

### Autumn 2015:

• Fr December 11, 12:15–14:00 @ MaD381
Antti Luoto
On the mean first exit time of a Brownian bridge with drift
• Fr December 4, 12:15–14:00 @ MaD381
Eero Ruosteenoja
Regularity for parabolic PDEs via stochastic games
• Fr November 20, 12:15–14:00 @ MaD381
Juha Ylinen
Split solutions to multidimensional quadratic BSDEs and related local equilibria
• Fr November 13, 13:15–15:00 @ MaD381
Mikko Parviainen
C1,α regularity for the normalized p-parabolic equation
We discuss the recent proof of Jin and Silvestre for C1,α regularity for the normalized or game theoretic p-parabolic equation.
• Fr October 23, 12:15–14:00 @ MaA204
Lauri Viitasaari (Aalto University)
Linear BSDEs driven by Gaussian processes
• Fr October 16, 12:15–14:00 @ MaA204
Pekka Matomäki
On an optimal variance stopping problem

I will consider quite comprehensively the question of optimally stopping a variance of an unkilled linear diffusion. Especially, I will show its close connection to game theory. Also, unlike in the usual linear optimal stopping problems, I will demonstrate that the optimal solution might follow a genuine mixed strategy policy in this non-linear variance scheme.

• Fr October 9, 12:15–14:00 @ MaA204
Jonas Tölle (University of Hamburg)
Stability of solutions and ergodicity for stochastic local and nonlocal p-Laplace equations joint work with Benjamin Gess (University of Bielefeld)
We provide a general framework for the stability of solutions to stochastic partial differential equations with respect to perturbations of the drift. More precisely, we consider stochastic partial differential equations with drift given as the subdifferential of a convex function and prove continuous dependence of the solutions with regard to random Mosco convergence of the convex potentials. To this aim, we identify the concept of stochastic variational inequalities (SVI) as a well-suited framework to study such stability properties. In particular, we provide an SVI treatment for stochastic nonlocal p-Laplace equations and prove their convergence to the respective local models. Furthermore, ergodicity for local and nonlocal stochastic singular p-Laplace equations is proven, without restriction on the spatial dimension and for all p ∈ [1, 2). This generalizes previous results from [Gess, Tölle; JMPA, 2014], [Liu, Tölle; ECP, 2011], [Liu; JEE, 2009]. In particular, the results include the multivalued case of the stochastic (nonlocal) total variation flow. Under appropriate rescaling, the convergence of the unique invariant measure for the nonlocal stochastic p-Laplace equation to the unique invariant measure of the local stochastic p-Laplace equation is proven.
• Fr October 2, 12:15–14:00 @ MaA204
Matti Vihola
Unbiasing Monte Carlo estimates of SDEs
• Fr September 25, 12:15–14:00 @ MaA204
Joonas Heino
On estimating the density of a sum of i.i.d. random vectors distributed uniformly in a ball
We show a certain density estimate needed to prove that a uniform measure density condition implies game regularity in a stochastic game called 'tug-of-war with noise'. A classical theorem due to B. V. Gnedenko states that if Z1,...,Zn are i.i.d. random variables having a bounded density function f, expectation zero and finite variance, the density of a random variable (Z1 + ...+ Zn) divided by the scaling factor √ n*Var(Z) tends uniformly to the density of the normal distribution as n increases. However, the scaling in our setting is different and we are interested in the density for all n. Thus, we cannot use the classical limit theorem. We calculate the characteristic function and use some integral estimates to the inversion formula. In addition, we use Hoeffdings's (or Azuma's or Bernstein's) inequality.
• Fr September 18, 12:15–14:00 @ MaA204
Short presentations on research topics
by Joonas Heino, Hannes Luiro and Angel Arroyo
• Fr September 11, 12:15–14:00 @ MaA204
Short presentations on research topics
by Mikko Parviainen, Eero Ruosteenoja, Juhana Siljander, and Amal Attouchi

### Spring 2015:

• Thu 28 May 14:15–16:00 @ MaA211
David Nualart
Fractional Brownian Motion: Stochastic Calculus and Applications
• The fractional Brownian motion is a centered Gaussian process with stationary increments, which depends on a parameter H in (0,1) called the Hurst index. In this talk we will first describe some basic properties of this process such as self-similarity, long-range dependence and finite p-variation. The fractional Brownian motion has become a plausible model in a wide range of physical phenomena including financial time series, internet traffic and turbulence. The applications of the fractional Brownian motion require the construction of a suitable stochastic calculus, similar to the classical Ito calculus. We will present several approaches to the stochastic calculus with respect to the fractional Brownian motion using path-wise techniques, Riemann sums and Malliavin calculus. In the last part of the talk we will discuss efficient numerical schemes for stochastic differential equations driven by a fractional Brownian motion.
• Mo 4 May 14:15–16:00 @ MaD355
Eija Laukkarinen
Differentiability in Malliavin calculus for Lévy processes
Given a Lévy process X, we investigate classes of real functions f such that f(X<pT) is differentiable in the sense of Malliavin calculus for Lévy processes. If X is the Brownian motion, then Malliavin differentiability is equivalent to f being in a weighted Sobolev space. For pure jump Lévy processes we examine conditions on f which yield Malliavin differentiability. Such conditions we find for example in terms of the real interpolation between Lipschitz-continuous functions and bounded functions, where the main interpolation parameter is determined by the Blumenthal-Getoor-index of the Lévy measure.
• Thu 23 April 14:15–16:00 @ MaD302
Rainer Buckdahn
Nonlinear stochastic differential games involving a major player and a large number of collectively acting minor players
In the talk we consider a 2-person zero-sum non linear stochastic differential game, in which the one player is a major one and the other player is formed by N collectively acting minor players, whose dynamics are driven by independent Brownian motions but who intervene with their control in a same manner. This leads to a pay-off/cost functional, defined through a backward SDE, which averages over the minor players. For the game with the N minor players we consider a weak solution, which makes it possible to study the game by using controls. Under suitable assumptions the saddle-point controls of the game are determined. The main objective on which the talk focuses is the limit behavior of the stochastic differential game and of the saddle-point controls, as the number N of minor players tends to infinity. The limit stochastic differential game -a mean-field game- is discussed and its saddle-point controls are characterised as the limit of the saddle-point controls of the game with N minor players.
The talk is based on a common work by Shige Peng and Juan Li (Shandong University, Jinan and Weihai) together with the speaker
• Thu 16 April 14:15–16:00 @ MaD302
Stefan Geiss

Conference report 'Stochastic Analysis, Controlled Dynamical Systems
Christel Geiss On BSDEs with bounded solutions

• Thu 09 April 14:15–16:00 @ MaD259
Paavo Salminen
Optimal stopping via expected suprema
• In this talk we consider optimal stopping problems (OSP) for general strong Markov processes. Since the value function of OSP is excessive it is natural to study diﬀerent representations of excessive functions in the context of optimal stopping. In this talk we focus on the representation via expected suprema. The main body of the talk consists of a veriﬁcation theorem for the value function. The result generalizes ﬁndings for Lévy processes obtained essentially via the Wiener-Hopf factorization. Some examples are discussed. The talk is mainly based onSören Christensen, Paavo Salminen, Bao Ta: Optimal stopping of strong Markov processes. SPA 123: 1138-1159, 2013.
• Thu 26 March 14:15–16:00 @ MaA105
Antti Luoto
Brownian bridge and certain mean exit times
• Thu 5 March 14:15–16:00 @ MaD302
Hans Hartikainen
A Dynamic Programming Principle for the p-Laplacian, 1  ≤ p <  ∞ III
• Thu 26 Feb 14:15–16:00 @ MaD302
Hans Hartikainen
A Dynamic Programming Principle for the p-Laplacian, 1  ≤ p <  ∞ II
• Thu 19 Feb 14:15–15:00 @ MaD302
Christel Geiss
Malliavin derivative of random functions III
Hans Hartikainen
A Dynamic Programming Principle for the p-Laplacian, 1  ≤ p <  ∞ I
• Thu 12 Feb 14:15–16:00 @ MaD302
Part 7 (Mikko Parviainen)
• Thu 5 Feb 14:15–16:00 @ MaD302
Christel Geiss
Malliavin derivative of random functions II
• Thu 29 Jan 14:15–16:00 @ MaD302
Part 6 (Mikko Parviainen)

### Autumn 2014:

• Mon 15 Dec 10:15–12:00 @ MaD380
Part 5 (Stefan Geiss)
• Mon 8 Dec 10:15–12:00 @ MaD380
Stefan Geiss
BSDEs and reverse Hölder inequalities III+1/2
Christel Geiss
Malliavin derivative of random functions
• Mon 1 Dec 10:15–12:00 @ MaD380
Part 4 (Stefan Geiss)
• Mon 24 Nov 10:15–12:00 @ MaD380
Stefan Geiss
BSDEs and reverse Hölder inequalities III
• Mon 17 Nov 10:15–12:00 @ MaD380
Part 3
• Mon 10 Nov 10:15–12:00 @ MaD380
Stefan Geiss
BSDEs and reverse Hölder inequalities II
• Mon 3 Nov 10:15–12:00 @ MaD380
Part 2 (Christel Geiss)
• Mon 27 Oct 10:15–12:00 @ MaD380
Antti Luoto
Short presentation on research topic
Stefan Geiss
BSDEs and reverse Hölder inequalities I
• Mon 20 Oct 10:15–12:00 @ MaD381
Stochastic differential games and viscosity solutions... by Buckdahn and Li
• Thu 16 Oct 14:15–16:00 @ MaD245
Juha Ylinen, Heikki Seppälä
Short presentations on research topics (30 min each)
• Thu 9 Oct 14:15–16:00 @ MaD245
Christel Geiss, Anni Laitinen and Eija Laukkarinen
Short presentations on research topics (30 min each)

### Spring 2014

• Tue 20 May 14:15–16:00 @ MaD380
Antti Luoto (U Jyväskylä)
Lévy processes and stochastic integration
• Tue 29 Apr 14:15–16:00 @ MaD380
Petteri Piiroinen (U Helsinki)
Probabilistic interpretation of electrical impedance tomography
• Tue 15 Apr 14:15–16:00 @ MaD380
Antti Luoto (U Jyväskylä)
Introduction to Lévy processes
• Tue 11 Mar 14:15–16:00 @ MaD381
Matti Vihola (U Jyväskylä)
On Markov chain convergence rates
• Tue 25 Feb 14:15–16:00 @ MaD381
Lasse Leskelä (U Jyväskylä)
Selecting martingale couplings of Markov kernels the Borel way
• Tue 18 Feb 14:15–16:00 @ MaD381
Heikki Seppälä (U Jyväskylä)
Approximating stochastic integrals: a characterization for the optimal approximation rate
• Tue 4 Feb 14:15–16:00 @ MaD381
Mikko Kuronen (U Jyväskylä)
The giant component in the binomial random intersection digraph
• Tue 21 Jan 14:15–16:00 @ MaD381
Juha Ylinen (U Jyväskylä)
Quantitative BMO in Wiener space, and Martingale Representation Theorem with a bounded integrand

### Fall 2013

• Thu 21 Nov 14:15–16:00 @ MaA103
Mikko Stenlund (U Helsinki)
An adiabatic dynamical system as a stochastic process

Abstract: The statistical properties of dynamical systems are traditionally studied in the context of their invariant measures. Motivated by non-equilibrium phenomena in nature, we wish to step out of the above setup. To this end, we introduce a model whose characteristics change slowly with time; hence the word adiabatic in the title. Solving a martingale problem in the spirit of Stroock and Varadhan, we show that repeated observations of the state of the system yield a certain stochastic diffusion process.

• Thu 14 Nov 14:15–16:00 @ MaD380
Alexander Steinicke (U Innsbruck)
The functional representation of an adapted stochastic process
• Thu 31 Oct 12:15–14:00 @ MaD248
Heikki Seppälä (U Jyväskylä)
Approximation of loan portfolio credit risk
• Wed 16 Oct 12:30–14:00 @ MaD302
Matti Vihola (U Jyväskylä)
Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms
• Wed 9 Oct 12:15–14:00 @ MaD302
Lasse Leskelä
(U Jyväskylä)
Rumor spreading and first-passage times in a large population of active and passive individuals
• Fri 23 Aug 13:15-14:00 @ MaD380
Xue-Mei Li (U Warwick)
McKean-Vlasov equation with singular potential
• Fri 23 Aug 14:15-15:00 @ MaD380
Martin Hairer (U Warwick)
Dynamics near criticality

### Spring 2013

• Existence and uniqueness techniques to the tug-of-war with noise
Mikko Parviainen (U Jyväskylä)
• Sonja Cox (ETH Zürich).

Abstract: Let X be the solution process of an SDE. Suppose one is interested in the function t -> IE( X(t) ) or in the distribution function of X(T). An approximation of these (infinite-dimensional!) quantities can be obtained by a Monte Carlo approximation based on a numerical simulation of the solution. In my talk I will discuss this procedure. Part of the talk is based on joint work with Martin Hutzenthaler, Arnulf Jentzen, and Jan van Neerven.

• Why adding a 'non-fixed jump' to a path of a Lévy process (with jumps) gives a path of the same process again II
Alexander Steinicke (U Innsbruck)
• Why adding a 'non-fixed jump' to a path of a Lévy process (with jumps) gives a path of the same process again
Alexander Steinicke (U Innsbruck)
• Concentration between Poincaré and Sobolev II
Stefan Geiss (U Jyväskylä)
• Concentration between Poincaré and Sobolev
Stefan Geiss (U Jyväskylä)
• Optimal transport from Lebesgue to Poisson
Karl-Theodor Sturm (University of Bonn)
• Concentration of measure in locally dependent spaces
Lasse Leskelä (U Jyväskylä)
• About Concentration of Measure Inequalities in Analysis
Tapio Rajala (U Jyväskylä)

### Fall 2012

• Chaos representation of Lévy driven BSDEs
Christel Geiss (University of Innsbruck)
• On the asymptotic analysis of the martingale estimating function approach for stochastic volatility models with jumps
Friedrich Hubalek (Technical University of Vienna) jointly with Petra Posedel (Zagreb)

Abstract: We provide explicit estimators for a class of continuous-time stochastic volatility models with jumps introduced by Barndorff-Nielsen and Shephard, that are observed in discrete time. We prove rigorously consistency and asymptotic normality based on the single assumption that all moments of the stationary distribution of the variance process are finite, and give explicit expressions for the asymptotic covariance matrix. We develop in detail the martingale estimating function approach for a bivariate model, that is not a diffusion, but admits jumps, and without using ergodicity arguments. We assume that both logarithmic returns and instantaneous variance are observed on a discrete grid of fixed width, and the observation horizon tends to infinity. As an illustration we provide a simulation study for daily increments, but the method applies unchanged for any time-scale, including high-frequency observations, without introducing any discretization error. The assumption that instantaneous variance is observable is a convenient simplification, that allows a rigorous and transparent theoretical analysis. Combining the estimating procedure with recent developments on volatility estimation or incorporating information from option prices and measures of trading intensity leads to methods that can also be applied in practice.

• On the approximation of stochastic integrals and almost sure convergence.
Emmanuel Gobet (Ecole Polytechnique, Paris).
Wed 28 Nov 16:15–18:00 @ MaD302.
• Pathwise estimates for the implicit Euler scheme for SDEs in Banach spaces.
Sonja Cox (ETH Zürich).
Tue 20 Nov 10:15–12:00 @ MaD302.

Abstract: In the first part of my talk I will explain what is meant by pathwise estimates for an approximation scheme of a stochastic differential equation (SDE), and why such estimates are of importance. In recent work by Jan van Neerven and myself, we obtained pathwise estimates for the implicit Euler scheme for SDEs in Banach spaces. In the second part of the talk I will sketch how we obtained these results and indicate what challenges arise when working in Banach spaces.

• Growth rate of breadth-first search trees in random intersection digraphs.
Mikko Kuronen (U Jyväskylä).
Thu 15 Nov 14:15–16:00 @ MaD380.
• Strong connectivity of random intersection digraphs.
Mikko Kuronen (U Jyväskylä).
Thu 1 Nov 14:15–16:00 @ MaD380.
• On the stochastic derivative of functionals of Lévy processes II.
Eija Laukkarinen (U Jyväskylä).
Thu 25 Oct 14:15–16:00 @ MaD380.
• On the stochastic derivative of functionals of Lévy processes.
Eija Laukkarinen (U Jyväskylä).
Thu 18 Oct 14:15–16:00 @ MaD380.
• Isonormal Gaussian representations of stochastic processes adapted to Brownian motion.
Juha Ylinen (U Jyväskylä).
Thu 11 Oct 14:15–16:00 @ MaD380.
• Exit times for Itō-processes (jointly with B. Bouchard and E. Gobet).
Stefan Geiss (U Jyväskylä).
Thu 4 Oct 14:15–16:00 @ MaD380.
• Lévy processes: Relating the Skorohod integral to a pathwise integral.
Eija Laukkarinen (U Jyväskylä).
Thu 27 Sep 14:15–16:00 @ MaD380.
• Chaos decomposition of Lévy processes in separable Banach spaces.
Florian Baumgartner (U Innsbruck)
Thu 20 Sep 14:15–16:00 @ MaD380.
• Stochastic Afternoon.Thu 13 Sep 15:00–17:00 @ MaD381.

In this informal kickoff meeting all researchers working on a stochastic-related research topic are invited to give a 5-15 min presentation, where they describe a research problem that they are planning to solve during the fall 2012 semester.

### Spring 2012

The seminar takes place on Fridays 12–14 at room MaA203, unless otherwise stated.

• Fri 4 May 12:15–14:00 @ MaA203. Stefan Geiss: Gradient and Hessian estimates for parabolic PDEs
• Fri 27 Apr 12:15–14:00 @ MaA203. Eija Laukkarinen: Distributional properties of a Lévy process influencing Malliavin fractional smoothness
• Fri 20 Apr 12:15–14:00 @ MaA203. Stefan Geiss: Fractional smoothness on the Wiener space under a change of measure with Muckenhoupt weights – Part II

Joint work with E. Gobet (École Polytechnique, Paris).

• Fri 13 Apr 12:15–14:00 @ MaA203. Stefan Geiss: Fractional smoothness on the Wiener space under a change of measure with Muckenhoupt weights – Part I

Joint work with E. Gobet (École Polytechnique, Paris).

• Fri 9 Mar 12:15–14:00 @ MaA203. Juha Ylinen: Conditional variance in probability space with Brownian motion
• Thu 23 Feb 14:15–16:00 @ MaA203. Antti Käenmäki: Ergodic methods in fractal geometry
• Fri 17 Feb 12:15–14:00 @ MaA203. Mikko Kuronen: Hard-core germ–grain models with power-law grain sizes
• Fri 27 Jan 12:15–14:00 @ MaA203. Matti Vihola: Markovian stochastic approximation

Abstract: Stochastic approximation is a framework unifying many random iterative algorithms occurring in a diverse range of applications. The stability of the process is often difficult to verify in practical applications and the process may even be unstable without additional stabilisation techniques. The talk covers some recent stability results in the case of a Markovian noise, which occurs when Markov chain Monte Carlo (MCMC) methods are used within stochastic approximation.

### Fall 2011

Students of the course MATS255 Markov processes present selected research articles on the theory and applications of Markov processes as part of their course requirements.

• Mon 28 Nov 14:15–16:00 @ MaD381. Heikki Seppälä: Interpolation spaces with parameter functions and approximations of stochastic integrals
• Mon 14 Nov 14:15–16:00 @ MaD381.  Janne Kujala: Necessary Conditions for Selectivity in the Dependence of Random Variables on External Factors

Abstract: Given a set of inputs to a system and a set of stochastic outputs depending on the inputs, we consider the question of which of the outputs depend on which inputs. We formulate conditions (on the joint distribution of the stochastic outputs as a function of the input values) for a given dependency structure to be satisfied. These conditions reduce to the problem of the existence of a joint distribution of random variables given the joint distributions of certain subsets of that set. We present necessary conditions for the existence of this joint distribution. The theory has many applications across sciences, ranging from psychology to quantum mechanics.
References:
http://dx.doi.org/10.3389/fpsyg.2010.00151
http://dx.doi.org/10.1016/j.jmp.2010.09.002
http://arxiv.org/abs/1108.3074
http://arxiv.org/abs/1110.1228
http://arxiv.org/abs/1110.2388

• Mon 17 Oct 14:15–15:00 @ MaD245.  Lasse Leskelä: Stochastic ordering of network throughputs using flow couplings

Abstract: A standard way to prove the stability of a queueing network is to show that the number of customers in the network is stochastically dominated by another network which is known to be stable. This approach is limited in practice because networks where the arrival and deparature rates depend on the queue lengths are usually not stochastically monotone. On the other hand, in most applications we care more about what goes through the network than what is inside it. In this talk I will describe a non-Markov coupling technique for Markov queueing networks that allows to stochastically order throughputs in networks where traditional couplings of customer populations would fail. This technique may be regarded as a probabilistic analogue of the Markov reward comparison developed by Nico van Dijk and others over the past two decades.

• Mon 10 Oct @ MaD245.  Matti Vihola: Coupling from the past

Abstract: Propp and Wilson (1996) introduced the coupling from the past (CFTP) algorithm to generate random samples exactly from a stationary distribution of a Markov chain. We will see why the algorithm works, go through some examples and discuss the limitations of the method.

• Mon 5 Sep @ MaD381.  Matti Vihola: Ergodicity of adaptive MCMC

Abstract: Markov chain Monte Carlo (MCMC) is a versatile method to approximate integrals in very general settings. The main difficulty in applications is to find a suitable proposal distribution for MCMC to make the method converge fast enough. Adaptive MCMC methods try to find a good proposal distribution automatically by learning continuously from the past samples. Such methods are easy to implement in practice, but because of the lost Markov property, the theoretical analysis needs more than classical Markov chain techniques.
This talk is an elementary introduction to the theory of adaptive MCMC. We will focus on algorithms and on the basic techniques to establish the correct ergodic properties.

• Thu 25 Aug 16:15–17:00 @ MaD381.  Anni Toivola: Approximating stochastic integrals in Lp

### Spring 2011

• 29 Jun Christophe Andrieu (University of Bristol, UK): Particle Markov chain Monte Carlo
• 25 May Teemu Pennanen: Reduced form models of corporate bond portfolios
• 27 Apr Yuri Kabanov (U.F.R. des Sciences et Technologie, France): No arbitrage under small transaction costs
• 13 Apr Mikko Parviainen: Random turn tug-of-war game with noise

Abstract: It was recently discovered that value functions for the random turn tug-of-war game with noise are related to p-harmonic functions. We utilize this connection and consider a situation in which a player has a strategy of forcing the game position to a chosen point

• 6 Apr Juha Ylinen: Backward Stochastic Differential Equations

Abstract: Linear Backward Stochastic Differential Equations (BSDEs) were first introduced by Bismut in 1973 in relation to stochastic optimal control theory. General uniqueness and existence results for nonlinear BSDEs were given by Pardoux and Peng in 1990. Formally the problem is to solve the stochastic differential equation
$$-dY_t = f(t,Y_t,Z_t)dt - Z_t dW_t,$$
where $Y_T = \xi$. What it means to solve this SDE will be clarified. It is important to notice that in contrast to Forward SDEs we have a condition on the final value instead of the starting value of the process $(Y_t, t \in [0,T])$. BSDEs come up naturally in the contingent claim valuation in complete markets, and an example of this is considered.

• 23 Mar Lasse Leskelä: Markov couplings, mass transportation, and stochastic relations

Abstract: Stochastic orders are powerful tools for approximating random processes whose distributions cannot be computed explicitly. For example, two stationary Markov processes can be ordered without explicit knowledge of the limiting distributions by means of an order-preserving Markov coupling. In this talk I will present a sharp characterization for the existence of order-preserving Markov couplings, which is based on a new notion of stochastic relations. I will also discuss how couplings of probability measures may be viewed as mass transportation problems. The theory is illustrated with applications to call centers and parallel queues with load balancing.

• 2 Mar Heikki Seppälä: Interpolation with general weight functions and approximation in L2(γ)

Abstract: Suppose we have a stochastic integral of form
$$f(W_1)=E(f(W_1)+\int_0^1 \lambda_t dW_t.$$
We want to approximate this integral discretely, which causes an approximation error. We investigate the connection between the smoothness properties of $f$ and the convergence speed of the quadratic approximation error rate of $f(W_1)$ as the number of discretization points increases. This problem is non-stochastic, since we consider the $L_2$-error.

• 16 Feb Teemu Pennanen: Convex duality in stochastic optimization and mathematical finance (continued)
• 9 Feb Teemu Pennanen: Convex duality in stochastic optimization and mathematical finance

Abstract: This paper proposes a general duality framework for the problem of minimizing a convex integral functional over a space of stochastic processes adapted to a given filtration. The framework unifies many well-known duality frameworks from operations research and mathematical finance. The unification allows the extension of some useful techniques from these two fields to a much wider class of problems. In particular, combining certain finite-dimensional techniques from convex analysis with measure theoretic techniques from mathematical finance, we are able to close the duality gap in some situations where traditional topological arguments fail. Applications to mathematical finance are given.

• 2 Feb Dario Gasbarra: Föllmer-Schweizer decomposition in discontinuous market models

Abstract: Let $X_t=X_0+ A_t + M_$ be the semimartingale decomposition of a discounted stock price under the objective measure $P$.
When the market is incomplete there are several equivalent martingale measures. Föllmer and Schweizer have proposed the minimal martingale measure $Q~P$ such that, when $Q$ is positive, $X$ is a local martingale under $Q$ and every $P$-martingale $L$ with $<L,M>$ is also a $Q$-martingale.If $H\in L^2(\Omega,F_T,Q)$ is a (discounted) contingent claim, the Kunita-Watanabe-Galtchouk martingale decomposition
$$H= E_Q(H)+\int_0^T H_s dX_s + L_T$$
where $L$ is a $Q$-local martingale with $<X,L>=0$ gives the hedging which minimizes the quadratic risk under $Q$.
On the other hand, it makes more sense to mimimize the quadratic risk under the objective measure $P$. The solution is the Föllmer-Schweizer decomposition
$$H=E_Q(H) + \int_0^T H_s' dX_s + L_T'$$
where $L$ is a local martingale under both $P$ and $Q$.
(FS) and (KWG) decompositions coincide only when $X$ is continuous. We discuss the discrete time case, a pure jump model and the general formula by Choulli et al. 2010 which relates (FS) and (KWG) decompositions.

Events