arXiv subjects

Paavo Salminen

Publications and source records attributed to Paavo Salminen.

At least 19 recordsLinked to original sources

arXiv2024

Drawdowns of diffusions

In this paper we give excursion theoretical proofs of Lehoczky's formula (in an extended form allowing a lower bound for the underlying diffusion) for the joint distribution of the first drawdown time and the maximum before this time, and of Malyutin's formula for the joint distribution of the first hitting time and the maximum drawdown before this time. It is remarkable -- but there is a clean explanation -- that the excursion theoretical approach which we developed first for Lehoczky's formula provides also a proof for Malyutin's formula. Moreover, we discuss some generalizations and analyze the pure jump process describing the maximum before the first drawdown time when the size of the drawdown is varying

math.PR↗

arXiv2024

Occupation times on the legs of a diffusion spider

We study the joint moments of occupation times on the legs of a diffusion spider. Specifically, we give a recursive formula for the Laplace transform of the joint moments, which extends earlier results for a one-dimensional diffusion. For a Bessel spider, of which the Brownian spider is a special case, our approach yields an explicit formula for the joint moments of the occupation times.

math.PR↗

arXiv2023

Probabilistic aspects of Jacobi theta functions

In this note we deduce well known modular identities for Jacobi theta functions using the spectral representations associated with the real valued Brownian motion taking values on $[-1,+1]$. We consider two cases: (i) reflection at $-1$ and $+1$, (ii) killing at $-1$ and $+1$. It is seen that these two representations give, in a sense, most compact forms of the modular theta-function identities. We study also discrete Gaussian distributions generated by theta functions, and derive, in particular, addition formulas for discrete Gaussian variables.

math.PR↗

arXiv2022

Diffusion spiders: Green kernel, excessive functions and optimal stopping

A diffusion spider is a strong Markov process with continuous paths taking values on a graph with one vertex and a finite number of edges (of infinite length). An example is Walsh's Brownian spider where the process on each edge behaves as Brownian motion. We calculate the density of the resolvent kernel in terms of the characteristics of the underlying diffusion. Excessive functions are studied via the Martin boundary theory. The main result is an explicit expression for the representing measure of a given excessive function. These results are used to solve optimal stopping problems for diffusion spiders.

math.PR↗

arXiv2021

On a first hit distribution of the running maximum of Brownian motion

Let $(S_t)_{t\geq 0}$ be the running maximum of a standard Brownian motion $(B_t)_{t\geq 0}$ and $T_m:=\inf\{t; \, mS_t 0$. In this note we calculate the joint distribution of $T_m$ and $B_{T_m}$. The motivation for our work comes from a mathematical model for animal foraging. We also present results for Brownian motion with drift.

math.PR↗

arXiv2019

On occupation times of one-dimensional diffusions

In this paper we study the moment generating function and the moments of occupation time functionals of one-dimensional diffusions. Assuming, specifically, that the process lives on $\mathbb{R}$ and starts at~0, we apply Kac's moment formula and the strong Markov property to derive an expression for the moment generating function in terms of the Green kernel of the underlying diffusion. Moreover, the approach allows us to derive a recursive equation for the Laplace transforms of the moments of the occupation time on $\mathbb{R}_+$. If the diffusion has a scaling property, the recursive equation simplifies to an equation for the moments of the occupation time up to time 1. As examples of diffusions with scaling property we study in detail skew two-sided Bessel processes and, as a special case, skew Brownian motion. It is seen that for these processes our approach leads to simple explicit formulas. The recursive equation for a sticky Brownian motion is also discussed.

math.PR↗

arXiv2019

Optimal stopping of oscillating Brownian motion

We solve optimal stopping problems for an oscillating Brownian motion, i.e. a diffusion with positive piecewise constant volatility changing at the point $x=0$. Let $σ_1$ and $σ_2$ denote the volatilities on the negative and positive half-lines, respectively. Our main result is that continuation region of the optimal stopping problem with reward $((1+x)^+)^2$ is disconnected, if and only if $σ_1^2<σ_2^2<2σ_1^2$. Based on the fact that the skew Brownian motion in natural scale is an oscillating Brownian motion, the obtained results are translated into corresponding results for the skew Brownian motion.

math.PR↗

arXiv2018

On the local time process of a skew Brownian motion

We derive a Ray-Knight type theorem for the local time process (in the space variable) of a skew Brownian motion up to an independent exponential time. It is known that the local time seen as a density of the occupation measure and taken with respect to the Lebesgue measure has a discontinuity at the skew point (in our case at zero), but the local time taken with respect to the speed measure is continuous. In this paper we discuss this discrepancy by characterizing the dynamics of the local time process in both these cases. The Ray-Knight type theorem is applied to study integral functionals of the local time process of the skew Brownian motion. In particular, we determine the distribution of the maximum of the local time process up to a fixed time, which can be seen as the main new result of the paper.

math.PR↗

arXiv2018

Optimal stopping of Brownian motion with broken drift

We solve an optimal stopping problem where the underlying diffusion is Brownian motion on $\bf R$ with a positive drift changing at zero. It is assumed that the drift $μ_1$ on the negative side is smaller than the drift $μ_2$ on the positive side. The main observation is that if $μ_2-μ_1>1/2$ then there exists values of the discounting parameter for which it is not optimal to stop in the vicinity of zero where the drift changes. However, when the discounting gets bigger the stopping region becomes connected and contains zero. This is in contrast with results concerning optimal stopping of skew Brownian motion where the skew point is for all values of the discounting parameter in the continuation region.

math.PR↗

arXiv2018

On moments of integral exponential functionals of additive processes

For real-valued additive process $(X\_t)\_{t\geq 0}$ a recursive equation is derived for the entire positive moments of functionals $$I\_{s,t}= \int \_s^t\exp(-X\_u)du, \quad 0\leq s<t\leq\infty, $$ in case the Laplace exponent of $X\_t$ exists for positive values of the parameter. From the equation emergesan easy-to-apply sufficient condition for the finiteness of the moments. As an application we study first hitprocesses of diffusions.

math.PR↗

arXiv2017

On first exit times and their means for Brownian bridges

For a Brownian bridge from $0$ to $y$ we prove that the mean of the first exit time from interval $(-h,h), \,\, h>0,$ behaves as $O(h^2)$ when $h \downarrow 0.$ Similar behavior is seen to hold also for the 3-dimensional Bessel bridge. For Brownian bridge and 3-dimensional Bessel bridge this mean of the first exit time has a puzzling representation in terms of the Kolmogorov distribution. The result regarding the Brownian bridge is applied to prove in detail an estimate needed by Walsh to determine the convergence of the binomial tree scheme for European options.

math.PR↗

arXiv2016

On optimal stopping of multidimensional diffusions

This paper develops an approach for solving perpetual discounted optimal stopping problems for multidimensional diffusions, with special emphasis on the $d$-dimensional Wiener process. We first obtain some verification theorems for diffusions, based on the Green kernel representation of the value function associated with the problem. Specializing to the multidimensional Wiener process, we apply the Martin boundary theory to obtain a set of tractable integral equations involving only harmonic functions that characterize the stopping region of the problem. These equations allow to formulate a discretization scheme to obtain an approximate solution. The approach is illustrated through the optimal stopping problem of a $d$-dimensional Wiener process with a positive definite quadratic form reward function.

math.PR↗

arXiv2016

Timing in the Presence of Directional Predictability: Optimal Stopping of Skew Brownian Motion

We investigate a class of optimal stopping problems arising in, for example, studies considering the timing of an irreversible investment when the underlying follows a skew Brownian motion. Our results indicate that the local directional predictability modeled by the presence of a skew point for the underlying has a nontrivial and somewhat surprising impact on the timing incentives of the decision maker. We prove that waiting is always optimal at the skew point for a large class of exercise payoffs. An interesting consequence of this finding, which is in sharp contrast with studies relying on ordinary Brownian motion, is that the exercise region for the problem can become unconnected even when the payoff is linear. We also establish that higher skewness increases the incentives to wait and postpones the optimal timing of an investment opportunity. Our general results are explicitly illustrated for a piecewise linear payoff.

math.PR↗

arXiv2015

Impulse control and expected suprema

We consider a class of impulse control problems for general underlying strong Markov processes on the real line, which allows for an explicit solution. The optimal impulse times are shown to be of threshold type and the optimal threshold is characterized as a solution of a (typically nonlinear) equation. The main ingredient we use is a representation result for excessive functions in terms of expected suprema.

math.PR↗

arXiv2015

Riesz representation and optimal stopping with two case studies

In this paper we demonstrate that the Riesz representation of excessive functions is a useful and enlightening tool to study optimal stopping problems. After a short general discussion of the Riesz representation we concretize, firstly, on a d-dimensional and, secondly, a space-time one-dimensional geometric Brownian motion. After this, two classical optimal stopping problems are discussed: 1) the optimal investment problem and 2) the valuation of the American put option. It is seen in both of these problems that the boundary of the stopping region can be characterized as a unique solution of an integral equation arising immediately from the Riesz representation of the value function. In Problem 2 the derived equation coincides with the standard well-known equation found in the literature.

math.PR↗

arXiv2014

Irreversible Investment under Lévy Uncertainty: an Equation for the Optimal Boundary

We derive a new equation for the optimal investment boundary of a general irreversible investment problem under exponential Lévy uncertainty. The problem is set as an infinite time-horizon, two-dimensional degenerate singular stochastic control problem. In line with the results recently obtained in a diffusive setting, we show that the optimal boundary is intimately linked to the unique optional solution of an appropriate Bank-El Karoui representation problem. Such a relation and the Wiener Hopf factorization allow us to derive an integral equation for the optimal investment boundary. In case the underlying Lévy process hits any real point with positive probability we show that the integral equation for the investment boundary is uniquely satisfied by the unique solution of another equation which is easier to handle. As a remarkable by-product we prove the continuity of the optimal investment boundary. The paper is concluded with explicit results for profit functions of (i) Cobb-Douglas type and (ii) CES type. In the first case the function is separable and in the second case non-separable.

q-fin.PM↗

arXiv2014

Differentiablity of excessive functions of one-dimensional diffusions and the principle of smooth fit

The principle of smooth fit is probably the most used tool to find solutions to optimal stopping problems of one-dimensional diffusions. It is important, e.g., in financial mathematical applications to understand in which kind of models and problems smooth fit can fail. In this paper we connect - in case of one-dimensional diffusions - the validity of smooth fit and the differentiability of excessive functions. The basic tool to derive the results is the representation theory of excessive functions; in particular, the Riesz and Martin representations. It is seen that the differentiability may not hold in case the speed measure of the diffusion or the representing measure of the excessive function has atoms. As an example, we study optimal stopping of sticky Brownian motion. It is known that the validity of the smooth fit in this case depends on the value of the discounting parameter (when the other parameters are fixed). We decompose the size of the jump in the derivative of the value function in two factors. The first one is due to the atom of the representing measure and the second one due to the atom of the speed measure.

math.PR↗

arXiv2012

Optimal stopping of strong Markov processes

We characterize the value function and the optimal stopping time for a large class of optimal stopping problems where the underlying process to be stopped is a fairly general Markov process. The main result is inspired by recent findings for Lévy processes obtained essentially via the Wiener-Hopf factorization. The main ingredient in our approach is the representation of the $β$-excessive functions as expected suprema. A variety of examples is given.

math.PR↗