Stochastic optimal control with random coefficients and associated stochastic Hamilton–Jacobi–Bellman equations

We consider the optimal control problem for stochastic differential equations (SDEs) with random coefficients under the recursive-type objective functional captured by the backward SDE (BSDE). Due to the random coefficients, the associated Hamilton–Jacobi–Bellman (HJB) equation is a class of second-order stochastic PDEs (SPDEs) driven by Brownian motion, which we call the stochastic HJB (SHJB) equation. In addition, as we adopt the recursive-type objective functional, the drift term of the SHJB equation depends on the second component of its solution. These two generalizations cause several technical intricacies, which do not appear in the existing literature. We prove the dynamic programming principle (DPP) for the value function, for which unlike the existing literature we have to use the backward semigroup associated with the recursive-type objective functional. By the DPP, we are able to show the continuity of the value function. Using the Itô–Kunita’s formula, we prove the verification theorem, which constitutes a sufficient condition for optimality and characterizes the value function, provided that the smooth (classical) solution of the SHJB equation exists. In general, the smooth solution of the SHJB equation may not exist. Hence, we study the existence and uniqueness of the solution to the SHJB equation under two different weak solution concepts. First, we show, under appropriate assumptions, the existence and uniqueness of the weak solution via the Sobolev space technique, which requires converting the SHJB equation to a class of backward stochastic evolution equations. The second result is obtained under the notion of viscosity solutions, which is an extension of the classical one to the case for SPDEs. Using the DPP and the estimates of BSDEs, we prove that the value function is the viscosity solution to the SHJB equation. For applications, we consider the linear-quadratic problem, the utility maximization problem, and the European option pricing problem. Specifically, different from the existing literature, each problem is formulated by the generalized recursive-type objective functional and is subject to random coefficients. By applying the theoretical results of this paper, we obtain the explicit optimal solution for each problem in terms of the solution of the corresponding SHJB equation.

Indefinite Linear-Quadratic Stochastic Control Problem for Jump-Diffusion Models with Random Coefficients: A Completion of Squares Approach

In this paper, we study the indefinite linear-quadratic (LQ) stochastic optimal control problem for stochastic differential equations (SDEs) with jump diffusions and random coefficients driven by both the Brownian motion and the (compensated) Poisson process. In our problem setup, the coefficients in the SDE and the objective functional are allowed to be random, and the jump-diffusion part of the SDE depends on the state and control variables. Moreover, the cost parameters in the objective functional need not be (positive) definite matrices. Although the solution to this problem can also be obtained through the stochastic maximum principle or the dynamic programming principle, our approach is simple and direct. In particular, by using the Itô-Wentzell’s formula, together with the integro-type stochastic Riccati differential equation (ISRDE) and the backward SDE (BSDE) with jump diffusions, we obtain the equivalent objective functional that is quadratic in control u under the positive definiteness condition, where the approach is known as the completion of squares method. Then the explicit optimal solution, which is linear in state characterized by the ISRDE and the BSDE jump diffusions, and the associated optimal cost are derived by eliminating the quadratic term of u in the equivalent objective functional. We also verify the optimality of the proposed solution via the verification theorem, which requires solving the stochastic HJB equation, a class of stochastic partial differential equations with jump diffusions.

Machine learning–augmented objective functional testing in the degenerative spine: quantifying impairment using patient-specific five-repetition sit-to-stand assessment

OBJECTIVE What is considered “abnormal” in clinical testing is typically defined by simple thresholds derived from normative data. For instance, when testing using the five-repetition sit-to-stand (5R-STS) test, the upper limit of normal (ULN) from a population of spine-healthy volunteers (10.5 seconds) is used to identify objective functional impairment (OFI), but this fails to consider different properties of individuals (e.g., taller and shorter, older and younger). Therefore, the authors developed a personalized testing strategy to quantify patient-specific OFI using machine learning. METHODS Patients with disc herniation, spinal stenosis, spondylolisthesis, or discogenic chronic low-back pain and a population of spine-healthy volunteers, from two prospective studies, were included. A machine learning model was trained on normative data to predict personalized “expected” test times and their confidence intervals and ULNs (99th percentiles) based on simple demographics. OFI was defined as a test time greater than the personalized ULN. OFI was categorized into types 1 to 3 based on a clustering algorithm. A web app was developed to deploy the model clinically. RESULTS Overall, 288 patients and 129 spine-healthy individuals were included. The model predicted “expected” test times with a mean absolute error of 1.18 (95% CI 1.13–1.21) seconds and R2 of 0.37 (95% CI 0.34–0.41). Based on the implemented personalized testing strategy, 191 patients (66.3%) exhibited OFI. Type 1, 2, and 3 impairments were seen in 64 (33.5%), 91 (47.6%), and 36 (18.8%) patients, respectively. Increasing detected levels of OFI were associated with statistically significant increases in subjective functional impairment, extreme anxiety and depression symptoms, being bedridden, extreme pain or discomfort, inability to carry out activities of daily living, and a limited ability to work. CONCLUSIONS In the era of “precision medicine,” simple population-based thresholds may eventually not be adequate to monitor quality and safety in neurosurgery. Individualized assessment integrating machine learning techniques provides more detailed and objective clinical assessment. The personalized testing strategy demonstrated concurrent validity with quality-of-life measures, and the freely accessible web app (https://neurosurgery.shinyapps.io/5RSTS/) enabled clinical application.

An Investgation About the Relationship Between Vasopressin and Oxytocin in Persistent Type Functional Neurological Symptom Disorder

Objective Functional neurogical symptom disorder (FNSD) is a somatic symptom disorder with loss of voluntary motor or sensory functions, which cannot be explained by another medical condition. The study aimed to examine the relationship of vasopressin and oxytocin in persistent type FNSD.Methods This study included 27 female patients between the ages of 20–57 who were diagnosed with FNSD according to DSM-5 and 27 healthy controls matched in terms of age and gender. Serum vasopressin and oxytocin levels were measured twice on the same day in fasting blood samples and the results were compared statistically.Results Vasopressin were lower in patients compared to controls while there was no difference between oxytocin levels. Childhood traumas were more common in patient group, and mean oxytocin level was lower in patients who exposed to childhood trauma, compared to controls. No significant difference was found between the groups in terms of vasopressin.Conclusion Changes in vasopressin and oxytocin balance in the pathogenesis of persistant FNSD, may likely to lead to physiological and behavioral consequences. Lower oxytocin levels may also be a marker of exposure to childhood trauma in FNSD. These neuropeptides plays important role in neuroendocrine balance of emotional behavior.

A method for choosing the regularization parameter of determining the right-hand side from integral observation

In this paper, the inverse problem of reconstructing the right-hand side from integral observation is studied by using variational method to minimizing objective functional combining with Tikhonov regularization. The L-curve method is suggested for choosing the regularization parameter.