We prove the existence of (infinitesimally) invariant measures for a stochastic version of the porous medium equation (of exponent $m = 3$) with Dirichlet Laplacian on an open set in $\mathbb{R}^d$.

Let $(W, \mu, H)$ be an abstract Wiener space assume two $\nu_i, i =1, 2$ probabilities on $(W, \mathcal{B}(W))$. Assume that the Wasserstein distance between $\nu_1$ and $\nu_2$ with respect to the Cameron-Martin norm $$ d_H (\nu_1, \nu_2)=\left\{ \inf_{\beta} \int_{W \times W} |x-y|_{H}^{2} d \beta (x, y) \right\}^{1/2} $$ is finite, where the infimum is taken on the set of probability measures $\beta$ on $W \times W$ whose first and second marginals are $\nu_1$ and $\nu_2$ and that $\nu_1$ has regular disintegration along a sequence of finite dimensional projections. Then there exists a unique (cyclically monotone) map $T = I_W + \xi$, with $\xi : W \to H$, such that $T$ maps $\nu_1$ to $\nu_2$ and the measure $\gamma = (I \times T) \nu_1$ is the unique solution of the Monge-Kantorovitch problem. Besides, if $\nu_2 \ll \mu^{1)}$, then $T$ is stochastically invertible, i.e., there exists $S : W \to W$ such that $S \circ T = I_W$ $\nu_1$ a.s. and $T \circ S = I_W$ $\nu_2$ a.s. If $\nu_1 = \mu_2$, then there exists a 1-convex function $\phi$ in the Gaussian Sobolev space $\mathbb{D}_{2,1}$, such that $\xi = \nabla \phi$. These results imply that the quasi-invariant transformations of the Wiener space with finite Wasserstein distance from $\mu$ can be written as the composition of a transport map $T$ and a rotation, i.e., a measure preserving map. We give also 1-convex sub-solutions using by calculating the Gaussian jacobian. Finally the full solutions of the Monge-Ampère equation on $W$ are given with the help of the Itô calculus.

We exhibit some mutual interactions between potential theory for concrete function spaces on $\mathbb{R}^n$ and the Dirichlet space theory associated with symmetric Markov processes. Our first concern is the role of the Dirichlet form version of the capacitary strong type inequality in the study of the ultracontractivity of the transition semigroup of time changed symmetric Markov processes. In particular, we study time changes of symmetric stable processes in relation to $d$-bounds of measures. We next show how the theory on capacity and the spectral synthesis for the Dirichlet space can be well inherited to a general function space with contractive $p$-norm. A link connecting those two topics is a contractive Besov space over a $d$-set of $\mathbb{R}^n$.

A gauge theorem for the Stieltjes exponential of a right continuous additive functional satisfying a general Kato type condition is established. Results for the ordinary exponential are then obtained as corollaries.

This note discusses two topics; one is the notion of the multiple Wiener integral and the other is the Lévy-Itô decomposition of a Lévy process.

Both have been taken up by Professor K. Itô showing the significance in stochastic analysis. The extensive development of the stochastic analysis at present largely depends on these discoveries by him.

It seems to be a good time to remind his profound ideas and to discuss some of future directions in probability theory.

We consider vector lattices $\mathbb{D}$ generalizing quasi-regular Dirichlet spaces and give a characterization for bounded linear functionals on $\mathbb{D}$ to have a representation by an integral with respect to smooth measures. Applications to BV functions on Wiener space are also discussed.

This paper addresses the problem of approximating random variables in terms of sums consisting of a real constant and of a stochastic integral with respect to a given semimartingale $X$. The criterion is minimization of $\mathbf{L}^2$–distance, or "least-squares". This problem has a straightforward and well-known solution when $X$ is a Brownian motion or, more generally, a square-integrable martingale, with respect to the underlying probability measure $P$. We address the general, semimartingale case by means of a duality approach; the adjoint variables in this duality are signed measures, absolutely continuous with respect to $P$, under which $X$ behaves like a martingale. It is shown that this duality is useful, in that the value of an appropriately formulated dual problem can be computed fairly easily; that it "has no gap" (i.e., the values of the primal and dual problems coincide); that the signed measure which is optimal for the dual problem can be easily identified whenever it exists; and that the duality is also "strong", in the sense that one can then identify the optimal stochastic integral for the primal problem. In so doing, the theory presented here both simplifies and extends the extant work on the subject. It has also natural connections and interpretations in terms of the theory of "variance-optimal" and "mean-variance efficient" portfolios in Mathematical Finance, pioneered by H. Markowitz and then greatly extended by H. Föllmer, D. Sondermann and most notably M. Schweizer.

Soliton solutions and the tau function of the KdV equation are studied within the stochastic analytic framework. A key role is played by the Itô formula and the Cameron-Martin transformation.

Let $Xt$ be a continuous time Markov chain on some finitely ramified fractal graph given by putting i.i.d. random resistors on each cell. We prove that under an assumption that a renormalization map of resistors has a non-degenerate fixed point, $\alpha^{-n} X_{\tau^n t}$ converges in law to a non-degenerate diffusion process on the fractal as $n \to \infty$, where $\alpha$ is a spatial scale and $\tau$ is a time scale of the fractal. Especially, when the fixed point of the renormalization map is unique, the diffusion is a constant time change of Brownian motion on the fractal. These results improve and extend our former results in [10].

We study representations of martingales with jumps based on the filtration generated by a Lévy process. Two types of representation theorem are obtained. The first formula is valid for any martingale and written as the sum of the stochastic integral based on the Brownian motion and that based on the compensated Poisson random measure. See (0.1). The second formula is valid only for a process which is a martingale for any equivalent martingale measure. See (0.2). The latter representation formula is then applied to a problem in mathematical finance. The upper hedging strategy and the lower hedging strategy of a contingent claim is obtained through the representation kernel.

We discuss an application of risk-sensitive control to portfolio optimization problems for a general factor model, which is considered a variation of Merton's intertemporal capital asset pricing model ([18]). In the model the instantaneous mean returns as well as volatilities of the security prices are affected by economic factors and the security prices. The economic factors are assumed to satisfy stocahstic differential equations whose coefficients depend on the security prices as well as themselves. In such general incomplete market models under Markovian setting we consider constructing optimal strategies for risk-sensitive portfolio optimization problems on a finite time horizon. We study the Bellman equations of parabolic type corresponding to the optimization problems. Through analysis of the Bellman equations we construct optimal strategies from the solution of the equation. We further discuss the problem with partial information. We shall obtain a necessary condition for optimality using backward stochastic partial differential equations.

An optimization problem in mathematical finance, called the exponential hedging problem is addressed. First, the relations between the problem and the backward stochastic differential equation (abbreviated to BSDE) having a quadratic growth term in the drift are reviewed. Next, the asymptotic analysis by Davis (2000) for the problem and the motivation of this paper are stated. Further, with some extensions, his analysis is reinterpreted by using the asymptotic expansion of the BSDE with respect to a small parameter, which suggests an alternative approach to the analysis, and the result on an approximated optimizer is obtained.

The Meyer equivalence on an abstract Wiener space states that the $L^p$-norm of square root of the Ornstein-Uhlenbeck operator is equivalent to $L^p$-norm of the Malliavin derivative. We prove the equivalence in the framework of Orlicz space. We also discuss the logarithmic Sobolev inequality in $L^p$ setting and higher order logarithmic Sobolev inequality.

This article reviews Itô's procedure for constructing the Markov process generated by variable coefficient Lévy–Khinchine operators. In particular, it examines conditions under which Itô's procedure succeeds but more analytic procedures appear to fail.

Let $\mu$ be a positive Radon measure in the Kato class. We consider the spectral bound $C(\lambda) = -\inf \sigma(\mathcal{H}^{\lambda \mu})\ (\lambda \in \mathbb{R}^1)$ of a generalized Schrödinger operator $\mathcal{H}^{\lambda \mu} = -\frac{1}{2} \Delta - \lambda \mu$ on $\mathbb{R}^d$, and show that the spectral bound is differentiable if $d \le 4$ and $\mu$ is Green-tight.

A Harris flow is a stochastic flow on the real line given by SDE (2.1) below. We study the noise generated by Harris flows, particularly spectra of the noise. Our aim is to understand what lies beyond the finite order terms in the chaos expansion (the Wiener-Itô expansion) for nonstrong solutions of SDE (2.1).