ISSN:

2095-9672

## Probability, Uncertainty and Quantitative Risk

Open Access Articles

2020, 5(0): 6
doi: 10.1186/s41546-020-00048-9

*+*[Abstract](93)*+*[HTML](35)*+*[PDF](454.83KB)**Abstract:**

It is well known that the minimal superhedging price of a contingent claim is too high for practical use. In a continuous-time model uncertainty framework, we consider a relaxed hedging criterion based on acceptable shortfall risks. Combining existing aggregation and convex dual representation theorems, we derive duality results for the minimal price on the set of upper semicontinuous discounted claims.

2020, 5(0): 5
doi: 10.1186/s41546-020-00047-w

*+*[Abstract](94)*+*[HTML](35)*+*[PDF](734.36KB)**Abstract:**

The recently proposed numerical algorithm, deep BSDE method, has shown remarkable performance in solving high-dimensional forward-backward stochastic differential equations (FBSDEs) and parabolic partial differential equations (PDEs). This article lays a theoretical foundation for the deep BSDE method in the general case of coupled FBSDEs. In particular, a posteriori error estimation of the solution is provided and it is proved that the error converges to zero given the universal approximation capability of neural networks. Numerical results are presented to demonstrate the accuracy of the analyzed algorithm in solving high-dimensional coupled FBSDEs.

2020, 5(0): 4
doi: 10.1186/s41546-020-00046-x

*+*[Abstract](108)*+*[HTML](39)*+*[PDF](604.43KB)**Abstract:**

We consider the problem of filtering an unseen Markov chain from noisy observations, in the presence of uncertainty regarding the parameters of the processes involved. Using the theory of nonlinear expectations, we describe the uncertainty in terms of a penalty function, which can be propagated forward in time in the place of the filter. We also investigate a simple control problem in this context.

2020, 5(0): 2
doi: 10.1186/s41546-020-00044-z

*+*[Abstract](88)*+*[HTML](38)*+*[PDF](335.83KB)**Abstract:**

Consider a single server queueing model which is observed over a continuous time interval (0,

*T*], where

*T*is determined by a suitable stopping rule. Let

*θ*be the unknown parameter for the arrival process and $\hat {\theta }_{T}$ be the maximum likelihood estimator of

*θ*. The main goal of this paper is to obtain a moderate deviation result of the maximum likelihood estimator for the single server queueing model under certain regular conditions.

2020, 5(0): 3
doi: 10.1186/s41546-020-00045-y

*+*[Abstract](77)*+*[HTML](37)*+*[PDF](926.7KB)**Abstract:**

For the class of (partially specified) internal risk factor models we establish strongly simplified supermodular ordering results in comparison to the case of general risk factor models. This allows us to derive meaningful and improved risk bounds for the joint portfolio in risk factor models with dependence information given by constrained specification sets for the copulas of the risk components and the systemic risk factor. The proof of our main comparison result is not standard. It is based on grid copula approximation of upper products of copulas and on the theory of mass transfers. An application to real market data shows considerable improvement over the standard method.

2020, 5(0): 1
doi: 10.1186/s41546-020-0043-5

*+*[Abstract](100)*+*[HTML](33)*+*[PDF](526.71KB)**Abstract:**

We use the functional Itô calculus to prove that the solution of a BSDE with singular terminal condition verifies at the terminal time: lim inf

_{t→T}

*Y*(

*t*) =

*ξ*=

*Y*(

*T*). Hence, we extend known results for a non-Markovian terminal condition.

2018, 3(0): 5
doi: 10.1186/s41546-018-0031-1

*+*[Abstract](75)*+*[HTML](35)*+*[PDF](3636.41KB)**Abstract:**

Asset returns are modeled by locally bilateral gamma processes with zero covariations. Covariances are then observed to be consequences of randomness in variations. Support vector machine regressions on prices are employed to model the implied randomness. The contributions of support vector machine regressions are evaluated using reductions in the economic cost of exposure to prediction residuals. Both local and global mean reversion and momentum are represented by drift dependence on price levels. Optimal portfolios maximize conservative portfolio values calculated as distorted expectations of portfolio returns observed on simulated path spaces. They are also shown to outperform classical alternatives.

2019, 4(0): 3
doi: 10.1186/s41546-019-0037-3

*+*[Abstract](90)*+*[HTML](42)*+*[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.

2017, 2(0): 6
doi: 10.1186/s41546-017-0018-3

*+*[Abstract](74)*+*[HTML](36)*+*[PDF](599.16KB)**Abstract:**

In this introductory paper to the issue, I will travel through the history of how quantitative finance has developed and reached its current status, what problems it is called to address, and how they differ from those of the pre-crisis world.

2018, 3(0): 4
doi: 10.1186/s41546-018-0030-2

*+*[Abstract](81)*+*[HTML](39)*+*[PDF](783.0KB)**Abstract:**

We study the existence and uniqueness of a solution to path-dependent backward stochastic Volterra integral equations (BSVIEs) with jumps, where pathdependence means the dependence of the free term and generator of a path of a càdlàg process. Furthermore, we prove path-differentiability of such a solution and establish the duality principle between a linear path-dependent forward stochastic Volterra integral equation (FSVIE) with jumps and a linear path-dependent BSVIE with jumps. As a result of the duality principle we get a comparison theorem and derive a class of dynamic coherent risk measures based on path-dependent BSVIEs with jumps.

2016, 1(0): 8
doi: 10.1186/s41546-016-0009-9

*+*[Abstract](81)*+*[PDF](593.73KB)**Abstract:**

This paper studies the backward-forward linear-quadratic-Gaussian (LQG) games with major and minor agents (players). The state of major agent follows a linear

*backward*stochastic differential equation (BSDE) and the states of minor agents are governed by linear

*forward*stochastic differential equations (SDEs). The major agent is dominating as its state enters those of minor agents. On the other hand, all minor agents are individually negligible but their state-average affects the cost functional of major agent. The mean-field game in such

*backward-major*and

*forward-minor*setup is formulated to analyze the decentralized strategies. We first derive the consistency condition via an auxiliary mean-field SDEs and a 3×2 mixed backward-forward stochastic differential equation (BFSDE) system. Next, we discuss the wellposedness of such BFSDE system by virtue of the monotonicity method. Consequently, we obtain the decentralized strategies for major and minor agents which are proved to satisfy the

*∊*-Nash equilibrium property.

2019, 4(0): 8
doi: 10.1186/s41546-019-0042-6

*+*[Abstract](86)*+*[HTML](36)*+*[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.

2017, 2(0): 5
doi: 10.1186/s41546-017-0016-5

*+*[Abstract](94)*+*[HTML](17)*+*[PDF](470.3KB)**Abstract:**

We show the well-posedness of backward stochastic differential equations containing an additional drift driven by a path of finite

*q*-variation with

*q*∈[1, 2). In contrast to previous work, we apply a direct fixpoint argument and do not rely on any type of flow decomposition. The resulting object is an effective tool to study semilinear rough partial differential equations via a Feynman-Kac type representation.

2018, 3(0): 1
doi: 10.1186/s41546-017-0025-4

*+*[Abstract](47)*+*[HTML](16)*+*[PDF](1600.36KB)**Abstract:**

We consider the class of affine LIBOR models with multiple curves, which is an analytically tractable class of discrete tenor models that easily accommodates positive or negative interest rates and positive spreads. By introducing an interpolating function, we extend the affine LIBOR models to a continuous tenor and derive expressions for the instantaneous forward rate and the short rate. We show that the continuous tenor model is arbitrage-free, that the analytical tractability is retained under the spot martingale measure, and that under mild conditions an interpolating function can be found such that the extended model fits any initial forward curve. This allows us to compute value adjustments (i.e. XVAs) consistently, by solving the corresponding ‘pre-default’ BSDE. As an application, we compute the price and value adjustments for a basis swap, and study the model risk associated to different interpolating functions.

2017, 2(0): 1
doi: 10.1186/s41546-017-0014-7

*+*[Abstract](95)*+*[HTML](35)*+*[PDF](594.73KB)**Abstract:**

In this paper, we study the recursive stochastic optimal control problems. The control domain does not need to be convex, and the generator of the backward stochastic differential equation can contain

*z*. We obtain the variational equations for backward stochastic differential equations, and then obtain the maximum principle which solves completely Peng's open problem.

2019, 4(0): 5
doi: 10.1186/s41546-019-0039-1

*+*[Abstract](54)*+*[HTML](16)*+*[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.

2017, 2(0): 7
doi: 10.1186/s41546-017-0019-2

*+*[Abstract](68)*+*[HTML](15)*+*[PDF](939.57KB)**Abstract:**

We apply to the concrete setup of a bank engaged into bilateral trade portfolios the XVA theoretical framework of Albanese and Crépey (2017), whereby so-called contra-liabilities and cost of capital are charged by the bank to its clients, on top of the fair valuation of counterparty risk, in order to account for the incompleteness of this risk. The transfer of the residual reserve credit capital from shareholders to creditors at bank default results in a unilateral CVA, consistent with the regulatory requirement that capital should not diminish as an effect of the sole deterioration of the bank credit spread. Our funding cost for variation margin (FVA) is defined asymmetrically since there is no benefit in holding excess capital in the future. Capital is fungible as a source of funding for variation margin, causing a material FVA reduction. We introduce a specialist initial margin lending scheme that drastically reduces the funding cost for initial margin (MVA). Our capital valuation adjustment (KVA) is defined as a risk premium, i.e. the cost of remunerating shareholder capital at risk at some hurdle rate.

2016, 1(0): 4
doi: 10.1186/s41546-016-0005-0

*+*[Abstract](69)*+*[PDF](577.61KB)**Abstract:**

We consider the problem of approximation of the solution of the backward stochastic differential equations in Markovian case. We suppose that the forward equation depends on some unknown finite-dimensional parameter. This approximation is based on the solution of the partial differential equations and multi-step estimator-processes of the unknown parameter. As the model of observations of the forward equation we take a diffusion process with small volatility. First we establish a lower bound on the errors of all approximations and then we propose an approximation which is asymptotically efficient in the sense of this bound. The obtained results are illustrated on the example of the Black and Scholes model.

2019, 4(0): 2
doi: 10.1186/s41546-019-0036-4

*+*[Abstract](80)*+*[HTML](33)*+*[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) $\mathbb{R}$.

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