# Seminar on Stochastics and PDEs

### Next talks:

Thu Sept 28, 14:15-16:00 @MaA 203**Stefan Geiss**

On a generalization of a result of Klass about randomly stopped sums

### Autumn 2017:

- 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 formAbstract: 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}Δ^{N}_{p}u=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 applicationsAbstract: 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 IAbstract: 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 solutionsAbstract: 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 L^{2}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 FunctionalsAbstract: 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 fitnessAbstract: 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(X

_{T},S_{T}), 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 equationsAbstract: 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 marketsAbstract: 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**

(RS): Radon Gaussian measures - Fr October 28, 12:15–14:00 @ MaD381
**Eero Ruosteenoja**

Reading Seminar: Pontryagin's maximum principleAbstract: 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**

(RS): Radon Gaussian measures I - Fr October 14, 12:15–14:00 @ MaD245
**Joonas Heino**

Stochastic estimates for a random cylinder walk related to the normalized p(x)-LaplacianAbstract: 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 graphAbstract: 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-supermatingalesAbstract: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**

Gradient walk and p-harmonic functions - 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 equationsAbstract: 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 C

^{1,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 estimatesIn 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 C

^{1,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**

C^{1,α}regularity for the normalized p-parabolic equation

We discuss the recent proof of Jin and Silvestre for C^{1,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 problemI 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 Z_{1},...,Z_{n}are i.i.d. random variables having a bounded density function f, expectation zero and finite variance, the density of a random variable (Z_{1}+ ...+ Z_{n}) 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<p_{T}) 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

15:15–16:00 @ MaD302**Hans Hartikainen**

A Dynamic Programming Principle for the p-Laplacian, 1 ≤ p < ∞ I - Thu 12 Feb 14:15–16:00 @ MaD302
**Reading seminar**

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
**Reading seminar**

Part 6 (Mikko Parviainen)

### Autumn 2014:

- Mon 15 Dec 10:15–12:00 @ MaD380
**Reading seminar**

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
**Reading seminar**

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
**Reading seminar**

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
**Reading seminar**

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
**Reading seminar, part 1**

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