scholarly journals A NOTE ON REAL-WORLD AND RISK-NEUTRAL DYNAMICS FOR HEATH–JARROW–MORTON FRAMEWORKS

2020 ◽  
Vol 23 (03) ◽  
pp. 2050020
Author(s):  
DAVID CRIENS
Keyword(s):  
Brownian Motion ◽  
Real World ◽  
Random Measure ◽  
Jump Condition ◽  
Poisson Random Measure ◽  
Change Of Measure ◽  
The Real ◽  
Infinite Dimensional ◽  
Risk Neutral ◽  
Equivalent Change Of Measure

We show that for time-inhomogeneous Markovian Heath–Jarrow–Morton models driven by an infinite-dimensional Brownian motion and a Poisson random measure an equivalent change of measure exists whenever the real-world and the risk-neutral dynamics can be defined uniquely and are related via a drift and a jump condition.

FIXING RISK NEUTRAL RISK MEASURES

2016 ◽  
Vol 19 (03) ◽  
pp. 1650021 ◽  
Author(s):  
HARVEY J. STEIN
Keyword(s):  
Risk Management ◽  
Real World ◽  
Risk Measures ◽  
Management Practice ◽  
Regulatory Capital ◽  
Change Of Measure ◽  
The Real ◽  
Alternative Approach ◽  
Risk Neutral ◽  
Credit Valuation Adjustment

In line with regulations and common risk management practice, the credit risk of a portfolio is managed via its potential future exposures (PFEs), expected exposures (EEs), and related measures, the expected positive exposure (EPE), effective expected exposure (EEE), and the effective expected positive exposure (EEPE). Notably, firms use these exposures to set economic and regulatory capital levels. Their values have a big impact on the capital that firms need to hold to manage their risks. Due to the growth of credit valuation adjustment (CVA) computations, and the similarity of CVA computations to exposure computations, firms find it expedient to compute these exposures under the risk neutral measure. Here, we show that exposures computed under the risk neutral measure are essentially arbitrary. They depend on the choice of numéraire, and can be manipulated by choosing a different numéraire. The numéraire can even be chosen in such a way as to pass backtests. Even when restricting attention to commonly used numéraires, exposures can vary by a factor of two or more. As such, it is critical that these calculations be carried out under the real world measure, not the risk neutral measure. To help rectify the situation, we show how to exploit measure changes to efficiently compute real world exposures in a risk neutral framework, even when there is no change of measure from the risk neutral measure to the real world measure. We also develop a canonical risk neutral measure that can be used as an alternative approach to risk calculations.

scholarly journals BSDEs with Logarithmic Growth Driven by Brownian Motion and Poisson Random Measure and Connection to Stochastic Control Problem

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Khalid Oufdil
Keyword(s):  
Optimal Control ◽  
Brownian Motion ◽  
Control Problem ◽  
Stochastic Optimal Control ◽  
Random Measure ◽  
Poisson Random Measure ◽  
One Dimensional ◽  
Stochastic Optimal Control Problem ◽  
Uniqueness Of The Solution ◽  
Logarithmic Growth

Abstract In this paper, we study one-dimensional backward stochastic differential equations under logarithmic growth in the 𝑧-variable ( | z | ⁢ | ln ⁡ | z | | ) (\lvert z\rvert\sqrt{\lvert\ln\lvert z\rvert\rvert}) . We show the existence and the uniqueness of the solution when the noise is driven by a Brownian motion and an independent Poisson random measure. In addition, we highlight the connection of such BSDEs with stochastic optimal control problem, where we show the existence of an optimal strategy for the control problem.

Reflected stochastic partial differential equations with jumps

2021 ◽  
pp. 2250002
Author(s):  
Hongchao Qian ◽  
Jun Peng
Keyword(s):  
Brownian Motion ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
Random Measure ◽  
Poisson Random Measure ◽  
Uniqueness Of Solutions ◽  
Partial Differential ◽  
Penalization Method

In this paper, we establish the existence and uniqueness of solutions of reflected stochastic partial differential equations (SPDEs) driven both by Brownian motion and by Poisson random measure in a convex domain. Penalization method plays a crucial role.

scholarly journals Optimal strong convergence rates of some Euler-type timestepping schemes for the finite element discretization SPDEs driven by additive fractional Brownian motion and Poisson random measure

2020 ◽  
Author(s):  
Aurelien Junior Noupelah ◽  
Antoine Tambue
Keyword(s):  
Finite Element ◽  
Brownian Motion ◽  
Strong Convergence ◽  
Fractional Brownian Motion ◽  
Convergence Rates ◽  
Random Measure ◽  
Poisson Random Measure ◽  
Element Discretization ◽  
Exponential Integrator ◽  
Strong Convergence Rates

AbstractIn this paper, we study the numerical approximation of a general second order semilinear stochastic partial differential equation (SPDE) driven by a additive fractional Brownian motion (fBm) with Hurst parameter $H>\frac {1}{2}$ H > 1 2 and Poisson random measure. Such equations are more realistic in modelling real world phenomena. To the best of our knowledge, numerical schemes for such SPDE have been lacked in the scientific literature. The approximation is done with the standard finite element method in space and three Euler-type timestepping methods in time. More precisely the well-known linear implicit method, an exponential integrator and the exponential Rosenbrock scheme are used for time discretization. In contract to the current literature in the field, our linear operator is not necessary self-adjoint and we have achieved optimal strong convergence rates for SPDE driven by fBm and Poisson measure. The results examine how the convergence orders depend on the regularity of the noise and the initial data and reveal that the full discretization attains the optimal convergence rates of order $\mathcal {O}(h^{2}+\varDelta t)$ O ( h 2 + Δ t ) for the exponential integrator and implicit schemes. Numerical experiments are provided to illustrate our theoretical results for the case of SPDE driven by the fBm noise.

Predictable solution for reflected BSDEs when the obstacle is not right-continuous

2020 ◽  
Vol 28 (4) ◽  
pp. 269-279
Author(s):  
Mohamed Marzougue ◽  
Mohamed El Otmani
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Random Measure ◽  
Backward Stochastic Differential Equations ◽  
Poisson Random Measure ◽  
One Dimensional ◽  
General Filtration ◽  
Reflected Bsdes

AbstractIn the present paper, we consider reflected backward stochastic differential equations when the reflecting obstacle is not necessarily right-continuous in a general filtration that supports a one-dimensional Brownian motion and an independent Poisson random measure. We prove the existence and uniqueness of a predictable solution for such equations under the stochastic Lipschitz coefficient by using the predictable Mertens decomposition.

scholarly journals STOCHASTIC STABILIZATION OF DYNAMICAL SYSTEMS USING LÉVY NOISE

2010 ◽  
Vol 10 (04) ◽  
pp. 509-527 ◽  
Author(s):  
DAVID APPLEBAUM ◽  
MICHAILINA SIAKALLI
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Dynamical Systems ◽  
Nonlinear Differential Equation ◽  
Random Noise ◽  
Random Measure ◽  
Poisson Random Measure ◽  
Stochastic Stabilization ◽  
Perturbed System ◽  
Exponentially Stable

We investigate the perturbation of the nonlinear differential equation [Formula: see text] by random noise terms consisting of Brownian motion and an independent Poisson random measure. We find conditions under which the perturbed system is almost surely exponentially stable and estimate the corresponding Lyapunov exponents.

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