scholarly journals Pricing Defaultable Securities under Actual Probability Measure

2014 ◽  
Vol 2 (4) ◽  
pp. 313-334
Author(s):  
Jianfen Feng ◽  
Dianfa Chen ◽  
Mei Yu
Keyword(s):  
Differential Equation ◽  
Probability Measure ◽  
Stochastic Differential Equation ◽  
Credit Risk ◽  
Discrete Time ◽  
Continuous Time ◽  
Backward Stochastic Differential Equation ◽  
New Approach ◽  
Defaultable Securities

AbstractIn this paper, a new approach is developed to estimate the value of defaultable securities under the actual probability measure. This model gives the price framework by means of the method of backward stochastic differential equation. Such a method solves some problems in most of existing literatures with respect to pricing the credit risk and relaxes certain market limitations. We provide the price of defaultable securities in discrete time and in continuous time respectively, which is favorable to practice to manage real credit risk for finance institutes.

scholarly journals Homogenization for deterministic maps and multiplicative noise

2013 ◽  
Vol 469 (2156) ◽  
pp. 20130201 ◽  
Author(s):  
Georg A. Gottwald ◽  
Ian Melbourne
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Multiplicative Noise ◽  
Stochastic Integrals ◽  
Deterministic System ◽  
One Dimensional ◽  
Stable Lévy Process ◽  
Fast Flow

A recent paper of Melbourne & Stuart (2011 A note on diffusion limits of chaotic skew product flows. Nonlinearity 24 , 1361–1367 (doi:10.1088/0951-7715/24/4/018)) gives a rigorous proof of convergence of a fast–slow deterministic system to a stochastic differential equation with additive noise. In contrast to other approaches, the assumptions on the fast flow are very mild. In this paper, we extend this result from continuous time to discrete time. Moreover, we show how to deal with one-dimensional multiplicative noise. This raises the issue of how to interpret certain stochastic integrals; it is proved that the integrals are of Stratonovich type for continuous time and neither Stratonovich nor Itô for discrete time. We also provide a rigorous derivation of super-diffusive limits where the stochastic differential equation is driven by a stable Lévy process. In the case of one-dimensional multiplicative noise, the stochastic integrals are of Marcus type both in the discrete and continuous time contexts.

scholarly journals Dynkin game under g-expectation in continuous time

2020 ◽  
Vol 9 (2) ◽  
pp. 459-470
Author(s):  
Helin Wu ◽  
Yong Ren ◽  
Feng Hu
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Continuous Time ◽  
Value Function ◽  
Backward Stochastic Differential Equation ◽  
Saddle Points ◽  
Value Functions ◽  
Dynkin Game ◽  
The Value Function

Abstract In this paper, we investigate some kind of Dynkin game under g-expectation induced by backward stochastic differential equation (short for BSDE). The lower and upper value functions $$\underline{V}_t=ess\sup \nolimits _{\tau \in {\mathcal {T}_t}} ess\inf \nolimits _{\sigma \in {\mathcal {T}_t}}\mathcal {E}^g_t[R(\tau ,\sigma )]$$ V ̲ t = e s s sup τ ∈ T t e s s inf σ ∈ T t E t g [ R ( τ , σ ) ] and $$\overline{V}_t=ess\inf \nolimits _{\sigma \in {\mathcal {T}_t}} ess\sup \nolimits _{\tau \in {\mathcal {T}_t}}\mathcal {E}^g_t[R(\tau ,\sigma )]$$ V ¯ t = e s s inf σ ∈ T t e s s sup τ ∈ T t E t g [ R ( τ , σ ) ] are defined, respectively. Under some suitable assumptions, a pair of saddle points is obtained and the value function of Dynkin game $$V(t)=\underline{V}_t=\overline{V}_t$$ V ( t ) = V ̲ t = V ¯ t follows. Furthermore, we also consider the constrained case of Dynkin game.

scholarly journals What is the time value of a stream of investments?

2005 ◽  
Vol 42 (3) ◽  
pp. 861-866 ◽  
Author(s):  
Ragnar Norberg ◽  
Mogens Steffensen
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Stochastic Integral ◽  
Asset Price ◽  
Expected Value ◽  
Integral Expression ◽  
Time Value ◽  
Price Process

The titular question is investigated for fairly general semimartingale investment and asset price processes. A discrete-time consideration suggests a stochastic differential equation and an integral expression for the time value in the continuous-time framework. It is shown that the two are equivalent if the jump part of the price process converges. The integral expression, which is the answer to the titular question, is the sum of all investments accumulated with returns on the asset (a stochastic integral) plus a term that accounts for the possible covariation between the two processes. The arbitrage-free price of the time value is the expected value of the sum (i.e. integral) of all investments discounted with the locally risk-free asset.

The Hamiltonian Generating Quantum Stochastic Evolutions in the Limit from Repeated to Continuous Interactions

2015 ◽  
Vol 22 (04) ◽  
pp. 1550022
Author(s):  
Matteo Gregoratti
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Schrodinger Equation ◽  
Hamiltonian Operator ◽  
The Other ◽  
Stochastic Evolution ◽  
Quantum Stochastic Differential Equation ◽  
Proper Formulation

We consider a quantum stochastic evolution in continuous time defined by the quantum stochastic differential equation of Hudson and Parthasarathy. On one side, such an evolution can also be defined by a standard Schrödinger equation with a singular and unbounded Hamiltonian operator K. On the other side, such an evolution can also be obtained as a limit from Hamiltonian repeated interactions in discrete time. We study how the structure of the Hamiltonian K emerges in the limit from repeated to continuous interactions. We present results in the case of 1-dimensional multiplicity and system spaces, where calculations can be explicitly performed, and the proper formulation of the problem can be discussed.

scholarly journals What is the time value of a stream of investments?

2005 ◽  
Vol 42 (03) ◽  
pp. 861-866 ◽  
Author(s):  
Ragnar Norberg ◽  
Mogens Steffensen
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Stochastic Integral ◽  
Asset Price ◽  
Expected Value ◽  
Integral Expression ◽  
Time Value ◽  
Price Process

The titular question is investigated for fairly general semimartingale investment and asset price processes. A discrete-time consideration suggests a stochastic differential equation and an integral expression for the time value in the continuous-time framework. It is shown that the two are equivalent if the jump part of the price process converges. The integral expression, which is the answer to the titular question, is the sum of all investments accumulated with returns on the asset (a stochastic integral) plus a term that accounts for the possible covariation between the two processes. The arbitrage-free price of the time value is the expected value of the sum (i.e. integral) of all investments discounted with the locally risk-free asset.

scholarly journals A decomposition approach for the discrete-time approximation of FBSDEs with a jump

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Idris Kharroubi ◽  
Thomas Lim
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Backward Stochastic Differential Equation ◽  
Jump Process ◽  
Quadratic Growth ◽  
Decomposition Approach ◽  
Time Approximation ◽  
Discrete Time Approximation

AbstractWe are concerned with the discretization of a solution of a forward-backward stochastic differential equation (FBSDE) with a jump process depending on the Brownian motion. In this paper, we study the cases of Lipschitz generators and the generators with a quadratic growth with respect to the variable

scholarly journals Lp (p ≥ 1) solutions of multidimensional BSDEs with monotone generators in general time intervals

2014 ◽  
Vol 15 (01) ◽  
pp. 1550002 ◽  
Author(s):  
Li-Shun Xiao ◽  
Sheng-Jun Fan ◽  
Na Xu
Keyword(s):  
Differential Equation ◽  
Weak Convergence ◽  
Stochastic Differential Equation ◽  
Existence And Uniqueness ◽  
Additional Assumption ◽  
Backward Stochastic Differential Equation ◽  
Time Interval ◽  
Time Intervals ◽  
Lipschitz Continuous ◽  
General Time

In this paper, we are interested in solving general time interval multidimensional backward stochastic differential equation in Lp (p ≥ 1). We first study the existence and uniqueness for Lp (p > 1) solutions by the method of convolution and weak convergence when the generator is monotonic in y and Lipschitz continuous in z both non-uniformly with respect to t. Then we obtain the existence and uniqueness for L1 solutions with an additional assumption that the generator has a sublinear growth in z non-uniformly with respect to t.

Sign in / Sign up

Export Citation Format

Share Document