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## Probability, Uncertainty and Quantitative Risk

January 2019 , Volume 4

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2019, 4: 8
doi: 10.1186/s41546-019-0042-6

*+*[Abstract](725)*+*[HTML](333)*+*[PDF](775.72KB)**Abstract:**

In this paper, we consider a class of nonlinear regression problems without the assumption of being independent and identically distributed. We propose a correspondent mini-max problem for nonlinear regression and give a numerical algorithm. Such an algorithm can be applied in regression and machine learning problems, and yields better results than traditional least squares and machine learning methods.

2019, 4: 5
doi: 10.1186/s41546-019-0039-1

*+*[Abstract](540)*+*[HTML](247)*+*[PDF](969.17KB)**Abstract:**

We develop a one-dimensional notion of affine processes under parameter uncertainty, which we call

*nonlinear affine processes*. This is done as follows: given a set Θ of parameters for the process, we construct a corresponding nonlinear expectation on the path space of continuous processes. By a general dynamic programming principle, we link this nonlinear expectation to a variational form of the Kolmogorov equation, where the generator of a single affine process is replaced by the supremum over all corresponding generators of affine processes with parameters in Θ. This nonlinear affine process yields a tractable model for Knightian uncertainty, especially for modelling interest rates under ambiguity.

We then develop an appropriate Itô formula, the respective term-structure equations, and study the nonlinear versions of the Vasiček and the Cox-Ingersoll-Ross (CIR) model. Thereafter, we introduce the nonlinear Vasiček-CIR model. This model is particularly suitable for modelling interest rates when one does not want to restrict the state space a priori and hence this approach solves the modelling issue arising with negative interest rates.

2019, 4: 4
doi: 10.1186/s41546-019-0038-2

*+*[Abstract](808)*+*[HTML](400)*+*[PDF](358.37KB)**Abstract:**

The main achievement of this paper is the finding and proof of Central Limit Theorem (CLT, see Theorem 12) under the framework of sublinear expectation. Roughly speaking under some reasonable assumption, the random sequence $\left\{ {1/\sqrt n \left( {{X_1} + \cdots + {X_n}} \right)} \right\}_{i = 1}^\infty $ converges in law to a nonlinear normal distribution, called G-normal distribution, where $\left\{ {{X_i}} \right\}_{i = 1}^\infty $ is an i.i.d. sequence under the sublinear expectation. It's known that the framework of sublinear expectation provides a important role in situations that the probability measure itself has non-negligible uncertainties. Under such situation, this new CLT plays a similar role as the one of classical CLT. The classical CLT can be also directly obtained from this new CLT, since a linear expectation is a special case of sublinear expectations. A deep regularity estimate of 2nd order fully nonlinear parabolic PDE is applied to the proof of the CLT. This paper is originally exhibited in arXiv.(math.PR/0702358v1).

2019, 4: 3
doi: 10.1186/s41546-019-0037-3

*+*[Abstract](766)*+*[HTML](340)*+*[PDF](659.51KB)**Abstract:**

The paper is devoted to the Cauchy problem of backward stochastic superparabolic equations with quadratic growth. We prove two Itô formulas in the whole space. Furthermore, we prove the existence of weak solutions for the case of one-dimensional state space, and the uniqueness of weak solutions without constraint on the state space.

2019, 4: 2
doi: 10.1186/s41546-019-0036-4

*+*[Abstract](543)*+*[HTML](266)*+*[PDF](1001.9KB)**Abstract:**

Conditional expectations (like, e.g., discounted prices in financial applications) are martingales under an appropriate filtration and probability measure. When the information flow arrives in a punctual way, a reasonable assumption is to suppose the latter to have piecewise constant sample paths between the random times of information updates. Providing a way to find and construct

*piecewise constant martingales*evolving in a connected subset of $\mathbb{R}$ is the purpose of this paper. After a brief review of possible standard techniques, we propose a construction scheme based on the sampling of latent martingales $\tilde Z$ with

*lazy clocks θ*. These

*θ*are time-change processes staying in arrears of the true time but that can synchronize at random times to the real (calendar) clock. This specific choice makes the resulting time-changed process

*Z*

_{t}= $\tilde Z$

_{θt}a martingale (called a

*lazy martingale*) without any assumption on $\tilde Z$, and in most cases, the lazy clock

*θ*is adapted to the filtration of the lazy martingale

*Z*, so that sample paths of Z on [0,

*T*] only requires sample paths of (

*θ*, $\tilde Z$) up to

*T*. This would not be the case if the stochastic clock

*θ*could be ahead of the real clock, as is typically the case using standard time-change processes. The proposed approach yields an easy way to construct analytically tractable lazy martingales evolving on (interval of) $\mathbb{R}$.

2019, 4: 1
doi: 10.1186/s41546-018-0035-x

*+*[Abstract](661)*+*[HTML](310)*+*[PDF](498.32KB)**Abstract:**

In this paper, we consider the mixed optimal control of a linear stochastic system with a quadratic cost functional, with two controllers-one can choose only deterministic time functions, called the deterministic controller, while the other can choose adapted random processes, called the random controller. The optimal control is shown to exist under suitable assumptions. The optimal control is characterized via a system of fully coupled forward-backward stochastic differential equations (FBSDEs) of mean-field type. We solve the FBSDEs via solutions of two (but decoupled) Riccati equations, and give the respective optimal feedback law for both deterministic and random controllers, using solutions of both Riccati equations. The optimal state satisfies a linear stochastic differential equation (SDE) of mean-field type. Both the singular and infinite time-horizonal cases are also addressed.

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