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

Personalized cancer vaccines (PCVs) are reinvigorating vaccine strategies in cancer immunotherapy. In contrast to adoptive T-cell therapy and checkpoint blockade, the PCV strategy modulates the innate and adaptive immune systems with broader activation to redeploy antitumor immunity with individualized tumor-specific antigens (neoantigens). Following a sequential scheme of tumor biopsy, mutation analysis, and epitope prediction, the administration of neoantigens with synthetic long peptide (SLP) or mRNA formulations dramatically improves the population and activity of antigen-specific CD4+ and CD8+ T cells. Despite the promising prospect of PCVs, there is still great potential for optimizing prevaccination procedures and vaccine potency. In particular, the arduous development of tumor-associated antigen (TAA)-based vaccines provides valuable experience and rational principles for augmenting vaccine potency which is expected to advance PCV through the design of adjuvants, delivery systems, and immunosuppressive tumor microenvironment (TME) reversion since current personalized vaccination simply admixes antigens with adjuvants. Considering the broader application of TAA-based vaccine design, these two strategies complement each other and can lead to both personalized and universal therapeutic methods. Chemical strategies provide vast opportunities for (1) exploring novel adjuvants, including synthetic molecules and materials with optimizable activity, (2) constructing efficient and precise delivery systems to avoid systemic diffusion, improve biosafety, target secondary lymphoid organs, and enhance antigen presentation, and (3) combining bioengineering methods to innovate improvements in conventional vaccination, "smartly" re-educate the TME, and modulate antitumor immunity. As chemical strategies have proven versatility, reliability, and universality in the design of T cell- and B cell-based antitumor vaccines, the union of such numerous chemical methods in vaccine construction is expected to provide new vigor and vitality in cancer treatment.

Chemical Strategies to Boost Cancer Vaccines.

Infinite horizon risk-sensitive control of diffusions without any blanket stability assumptions

This paper studies the eigenvalue problem on $\mathbb{R}^d$ for a class of second order, elliptic operators of the form $\mathscr{L} = a^{ij}\partial_{x_i}\partial_{x_j} + b^{i}\partial_{x_i} + f$, associated with non-degenerate diffusions. We show that strict monotonicity of the principal eigenvalue of the operator with respect to the potential function $f$ fully characterizes the ergodic properties of the associated ground state diffusion, and the unicity of the ground state, and we present a comprehensive study of the eigenvalue problem from this point of view. This allows us to extend or strengthen various results in the literature for a class of viscous Hamilton-Jacobi equations of ergodic type with smooth coefficients to equations with measurable drift and potential. In addition, we establish the strong duality for the equivalent infinite dimensional linear programming formulation of these ergodic control problems. We also apply these results to the study of the infinite horizon risk-sensitive control problem for diffusions, and establish existence of optimal Markov controls, verification of optimality results, and the continuity of the controlled principal eigenvalue with respect to stationary Markov controls.

Strict monotonicity of principal eigenvalues of elliptic operators in $\mathbb{R}^d$ and risk-sensitive control

Ergodic Control of Diffusion Processes: Preface

In this article, we study risk-sensitive control problem with controlled continuous time pure jump process on a countable space as state dynamics. We prove multiplicative dynamic programming principle, elliptic and parabolic Harnack’s inequalities. Using the multiplicative dynamic programing principle and the Harnack’s inequalities, we prove the existence and a characterization of optimal risk-sensitive control under the near monotone condition.

Risk-Sensitive Control of Pure Jump Process on Countable Space with Near Monotone Cost

In this article, we study risk-sensitive control problem with controlled continuous time Markov chain state dynamics. Using multiplicative dynamic programming principle along with the atomic structure of the state dynamics, we prove the existence and a characterization of optimal risk-sensitive control under geometric ergodicity of the state dynamics along with a smallness condition on the running cost.

Risk-Sensitive Ergodic Control of Continuous Time Markov Processes With Denumerable State Space

We study infinite horizon discounted-cost and ergodic-cost risk-sensitive zero-sum stochastic games for controlled continuous time Markov chains on a countable state space. For the discounted-cost game, we prove the existence of value and saddle-point equilibrium in the class of Markov strategies under nominal conditions. For the ergodic-cost game, we prove the existence of values and saddle point equilibrium by studying the corresponding Hamilton-Jacobi-Isaacs equation under a certain Lyapunov condition.

Zero-sum risk-sensitive stochastic games for continuous time Markov chains

We study stochastic control problem for pure jump processes on a general state space with risk sensitive discounted and ergodic cost criteria For the discounted cost criterion we prove the existen

Chandan Pal

Papers

Risk-Sensitive Control of Pure Jump Process on Countable Space with Near Monotone Cost

Risk-Sensitive Ergodic Control of Continuous Time Markov Processes With Denumerable State Space

Zero-sum risk-sensitive stochastic games for continuous time Markov chains

Risk sensitive control of pure jump processes on a general state space

Synthesis of Amide Linked Cyclic Dinucleotide Analogues with Pyrimidine Bases