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For a L\'evy process $X$ on a finite time interval consider the probability that it exceeds some fixed threshold $x>0$ while staying below $x$ at the points of a regular grid. We establish exact asymptotic behavior of this probability as…
We study the risk criterion for investments based on the drawdown from the maximal value of the capital in the past. Depending on investor's risk attitude, thus his risk exposure, we find that the distribution of these drawdowns follows a…
The optimization criterion for dividends from a risky business is most often formalized in terms of the expected present value of future dividends. That criterion disregards a potential, explicit demand for stability of dividends. In…
We study the problem of optimizing the betting frequency in a dynamic game setting using Kelly's celebrated expected logarithmic growth criterion as the performance metric. The game is defined by a sequence of bets with independent and…
We establish an explicit pricing formula for the class of L\'evy-stable models with maximal negative asymmetry (Log-L\'evy model with finite moments and stability parameter $1<\alpha\leq 2$) in the form of rapidly converging series. The…
Control of drawdown, that is, the control of the drops in wealth over time from peaks to subsequent lows, is of great concern from a risk management perspective. With this motivation in mind, the focal point of this paper is to address the…
Stability of linear systems with uncertain bounded time-varying delays is studied under assumption that the nominal delay values are not equal to zero. An input-output approach to stability of such systems is known to be based on the bound…
When trading incurs proportional costs, leverage can scale an asset's return only up to a maximum multiple, which is sensitive to its volatility and liquidity. In a model with one safe and one risky asset, with constant investment…
We elaborate a scheme of trapping-expulsion management (TEM), in the form of the quadratic potential periodically switching between confinement and expulsion, as a means of stabilization of two-dimensional dynamical states against the…
We study the excess minimum risk in statistical inference, defined as the difference between the minimum expected loss in estimating a random variable from an observed feature vector and the minimum expected loss in estimating the same…
In this paper, we consider a simple discrete-time optimal betting problem using the celebrated Kelly criterion, which calls for maximization of the expected logarithmic growth of wealth. While the classical Kelly betting problem can be…
L\'evy's Upward Theorem says that the conditional expectation of an integrable random variable converges with probability one to its true value with increasing information. In this paper, we use methods from effective probability theory to…
In recent studies the truncated Levy process (TLP) has been shown to be very promising for the modeling of financial dynamics. In contrast to the Levy process, the TLP has finite moments and can account for both the previously observed…
We investigate two coupled properties of Levy stable random motions: The first passage times (FPTs) and the first passage leapovers (FPLs). While, in general, the FPT problem has been studied quite extensively, the FPL problem has hardly…
Using elementary arguments, we show how to derive $\mathbf{L}_p$-error bounds for the approximation of frictionless wealth process in markets with proportional transaction costs. For utilities with bounded risk aversion, these estimates…
We determine the variance-optimal hedge when the logarithm of the underlying price follows a process with stationary independent increments in discrete or continuous time. Although the general solution to this problem is known as backward…
The classical notion of L\'evy process is generalized to one that takes as its values probabilities on a first order model equipped with a commutative semigroup. This is achieved by applying a convolution product on definable probabilities…
Constraint tightening to non-conservatively guarantee recursive feasibility and stability in Stochastic Model Predictive Control is addressed. Stability and feasibility requirements are considered separately, highlighting the difference…
We generalize the classic Shiller cyclically adjusted price-earnings ratio (CAPE) used for prediction of future total returns of the stock market. We treat earnings growth as exogenous. The difference between log wealth and log earnings is…
We develop a rigorous walk-forward validation framework for algorithmic trading designed to mitigate overfitting and lookahead bias. Our methodology combines interpretable hypothesis-driven signal generation with reinforcement learning and…