We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

We consider a very general diffusion model for asset prices which allows the description of stochastic and past-dependent volatilities. Since this model typically yields an incomplete market, we show that for the purpose of pricing options, a small investor should use the minimal equivalent martingale measure associated to the underlying stock price process. Then we present stochastic numerical methods permitting the explicit computation of option prices and hedging strategies, and we illustrate our approach by specific examples.

Option pricing under incompleteness and stochastic volatility.

For a Brownian motion $B=(B_t)_{t\le 1}$ with $B_0=0$, {\bf E}$B_t=0$, {\bf E}$B_t^2=t$ problems of probability distributions and their characteristics are considered for the variables $$ \begin{array}{c} {\mathbb D} =\displaystyle\sup_{0\le t\le t'\le 1}(B_t-B_{t'}),\qquad {\mathbb D}_1=B_\sigma-\inf_{\sigma\le t'\le 1}B_{t'}, \\ {\mathbb D}_2=\displaystyle\sup_{0\le t\le\sigma'}B_{t}-B_{\sigma'}, \end{array} $$ where $\sigma$ and $\sigma'$ are times (non-Markov) of the absolute maximum and absolute minimum of the Brownian motion on $[0,1]$ (i.e., $B_\sigma=\sup_{0\le t\le 1}B_t$, $B_{\sigma'}=\inf_{0\le t'\le 1}B_{t'}$).

On Probability Characteristics of "Downfalls" in a Standard Brownian Motion

We consider a dynamic system whose state is governed by a linear stochastic differential equation with time-dependent coefficients. The control acts additively on the state of the system. Our objective is to minimize an integral cost which depends upon the evolution of the state and the total variation of the control process. It is proved that the optimal cost is the unique solution of an appropriate free boundary problem in a space-time domain. By using some decomposition arguments, the problems of a two-sided control, i.e. optimal corrections, and the case with constraints on the resources, i.e. finite fuel, can be reduced to a simpler case of only one-sided control, i.e. a monotone follower. These results are applied to solving some examples by the so-called method of similarity solutions.

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Additive control of stochastic linear systems with finite horizon

We start by briefly surveying a research on optimal stopping games since their introduction by Dynkin more than 40 years ago. Recent renewed interest to Dynkin’s games is due, in particular, to the study of Israeli (game) options introduced in 2000. We discuss the work on these options and related derivative securities for the last decade. Among various results on game options we consider error estimates for their discrete approximations, swing game options, game options in markets with transaction costs, and other questions.

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Dynkin's Games and Israeli Options

We consider the optimal stopping of a class of spectrally negative jump diffusions. We state a set of conditions under which the value is shown to have a representation in terms of an ordinary nonlinear programming problem. We establish a connection between the considered problem and a stopping problem of an associated continuous diffusion process and demonstrate how this connection may be applied for characterizing the stopping policy and its value. We also establish a set of typically satisfied conditions under which increased volatility as well as higher jump intensity decelerates rational exercise by increasing the value and expanding the continuation region.

A Class of Solvable Optimal Stopping Problems of Spectrally Negative Jump Diffusions

Introduction: This is one of the first intervention studies to examine how low- (LIT) and high-intensity endurance training (HIT) affect durability, defined as ‘time of onset and magnitude of deterioration in physiological-profiling characteristics over time during prolonged exercise’. Methods: Sedentary and recreationally active men (n = 16) and women (n = 19) completed either LIT (average weekly training time 6.8 ± 0.7 h) or HIT (1.6 ± 0.2 h) cycling for 10 weeks. Durability was analyzed before and after the training period from three factors during 3-h cycling at 48% of pretraining maximal oxygen uptake (VO2max): 1) by the magnitude and 2) onset of drifts (i.e. gradual change in energy expenditure, heart rate, rate of perceived exertion, ventilation, left ventricular ejection time, and stroke volume), 3) by the ‘physiological strain’, defined to be the absolute responses of heart rate and its variability, lactate, and rate of perceived exertion. Results: When all three factors were averaged the durability was improved similarly (time x group p = 0.42) in both groups (LIT: p = 0.03, g = 0.49; HIT: p = 0.01, g = 0.62). In the LIT group, magnitude of average of drifts and their onset did not reach statistically significance level of p < 0.05 (magnitude: 7.7 ± 6.8% vs. 6.3 ± 6.0%, p = 0.09, g = 0.27; onset: 106 ± 57 min vs. 131 ± 59 min, p = 0.08, g = 0.58), while averaged physiological strain improved (p = 0.01, g = 0.60). In HIT, both magnitude and onset decreased (magnitude: 8.8 ± 7.9% vs. 5.4 ± 6.7%, p = 0.03, g = 0.49; onset: 108 ± 54 min vs. 137 ± 57 min, p = 0.03, g = 0.61), and physiological strain improved (p = 0.005, g = 0.78). VO2max increased only after HIT (time x group p < 0.001, g = 1.51). Conclusion: Durability improved similarly by both LIT and HIT based on reduced physiological drifts, their postponed onsets, and changes in physiological strain. Despite durability enhanced among untrained people, a 10-week intervention did not alter drifts and their onsets in a large amount, even though it attenuated physiological strain.

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Durability is improved by both low and high intensity endurance training

We consider the representation of the value of a class of optimal stopping problems of linear diffusions in a linearized form as an expected supremum of a known function. We establish an explicit integral representation of this representing function by utilizing the explicitly known marginals of the joint probability distribution of the extremal processes. We also delineate circumstances under which the value of a stopping problem induces directly this representation and show how it is connected with the monotonicity of the generator. We compare our findings with existing literature and show, for example, how our representation is linked to the smooth fit principle and how it coincides with the optimal stopping signal representation. The intricacies of the developed integral representation are explicitly illustrated in various examples arising in financial applications of optimal stopping.

Pekka Matomäki

Papers

A Class of Solvable Optimal Stopping Problems of Spectrally Negative Jump Diffusions

Durability is improved by both low and high intensity endurance training

Expected Supremum Representation of a Class of Single Boundary Stopping Problems