Based on a deterministic and stochastic process hybrid model, we use white noises to account for patient variabilities in treatment outcomes, use a hyperparameter to represent patient heterogeneity ...in a cohort, and construct a stochastic model in terms of Ito stochastic differential equations for testing the efficacy of three different treatment protocols in CAR T cell therapy. The stochastic model has three ergodic invariant measures which correspond to three unstable equilibrium solutions of the deterministic system, while the ergodic invariant measures are attractors under some conditions for tumor growth. As the stable dynamics of the stochastic system reflects long-term outcomes of the therapy, the transient dynamics provide chances of cure in short-term. Two stopping times, the time to cure and time to progress, allow us to conduct numerical simulations with three different protocols of CAR T cell treatment through the transient dynamics of the stochastic model. The probability distributions of the time to cure and time to progress present outcome details of different protocols, which are significant for current clinical study of CAR T cell therapy.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Under the framework of
G
-expectation and
G
-Brownian motion, we introduce Itô’s integral for stochastic processes without assuming quasi-continuity. Then we can obtain Itô’s integral on stopping ...time interval. This new formulation permits us to obtain Itô’s formula for a general
C
1
,
2
-function, which essentially generalizes the previous results of Peng (2006, 2008, 2009, 2010, 2010)
21–25 as well as those of Gao (2009)
8 and Zhang et al. (2010)
27.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Stopping-time Banach spaces is a collective term for the class of spaces of eventually null integrable processes that are defined in terms of the behaviour of the stopping times with respect to some ...fixed filtration. From the point of view of Banach space theory, these spaces in many regards resemble the classical spaces such as L1 or C(Δ), but unlike these, they do have unconditional bases. In the present paper, we study the canonical bases in the stopping-time spaces in relation to factorising the identity operator thereon. Since we work exclusively with the dyadic-tree filtration, this setup enables us to work with tree-indexed bases rather than directly with stochastic processes. En route to the factorisation results, we develop general criteria that allow one to deduce the uniqueness of the maximal ideal in the algebra of operators on a Banach space. These criteria are applicable to many classical Banach spaces such as (mixed-norm) Lp-spaces, BMO, SL∞, and others.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In this paper, we introduce the definitions of stopping time, forward and backward solutions to interval-valued differential equations under generalized Hukuhara differentiability, which could be ...applied to discuss the evolution of dynamical systems with practical backgrounds. By using these definitions, we study stopping time problems for the Malthusian population model and the logistic model in details. Then, some general conclusions about stopping time problems for interval-valued differential equations are considered and the results are shown to be feasible by providing some examples.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Motivated by problems in behavioural finance, we provide two explicit constructions of a randomized stopping time which embeds a given centred distribution μ on integers into a simple symmetric ...random walk in a uniformly integrable manner. Our first construction has a simple Markovian structure: at each step, we stop if an independent coin with a state-dependent bias returns tails. Our second construction is a discrete analogue of the celebrated Azéma–Yor solution and requires independent coin tosses only when excursions away from maximum breach predefined levels. Further, this construction maximizes the distribution of the stopped running maximum among all uniformly integrable embeddings of μ.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Using the spectral measure μS of the stopping time S, we define the stopping element XS as a Daniell integral ∫XtdμS for an adapted stochastic process (Xt)t∈J that is a Daniell summable vector-valued ...function. This is an extension of the definition previously given for right-order-continuous sub martingales with the Doob-Meyer decomposition property. The more general definition of XS necessitates a new proof of Doob's optional sampling theorem, because the definition given earlier for sub martingales implicitly used Doob's theorem applied to martingales. We provide such a proof, thus removing the heretofore necessary assumption of the Doob-Meyer decomposition property in the result. Another advancement presented in this paper is our use of unbounded order continuity of a stochastic process, which properly characterizes the notion of continuity of sample paths almost everywhere, found in the classical theory.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
The main question we would like to address in this paper is as follows: Given a geometric Brownian motion (GBM) as the underlying stock price model, what is the cumulative distribution function (CDF) ...for the trading profit or loss, call it g(t), when an affine feedback control strategy with stop-loss order is considered? Moreover, is it possible to obtain a closed-form characterization for the desired CDF for g(t) so that a theoretician or practical trader might be benefited from it? The answers to these questions are affirmative. In this paper, we provide a closed-form expression for the cumulative distribution function for the trading profit or loss. In addition, we show that the affine feedback controller with stop-loss order indeed generalizes the result without stop order in the sense of distribution function. Some simulations and illustrative examples are also provided as supporting evidence of the theory.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We define and study order continuity, topological continuity, γ-Hölder-continuity and Kolmogorov–Čentsov-continuity of continuous-time stochastic processes in vector lattices and show that every such ...kind of continuous submartingale has a continuous compensator of the same kind. The notion of variation is introduced for continuous time stochastic processes and for a γ-Hölder-continuous martingale with finite variation, we prove that it is a constant martingale. The localization technique for not necessarily bounded martingales is introduced and used to prove our main result which states that the quadratic variation of a continuous-time γ-Hölder continuous martingale X is equal to its compensator 〈X〉.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
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