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

We consider the quantum stochastic differential equation introduced by Hudson and Parthasarathy to describe the stochastic evolution of an open quantum system together with its environment. We study the (unbounded) Hamiltonian operator generating the unitary group connected, as shown by Frigerio and Maassen, to the solution of the equation. We find a densely defined restriction of the Hamiltonian operator; in some special cases we prove that this restriction is essentially self-adjoint and in one particular case we get the whole Hamiltonian with its full domain.

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Homogenization for deterministic maps and multiplicative noise

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2013.0201 ◽

2013 ◽

Vol 469 (2156) ◽

pp. 20130201 ◽

Cited By ~ 31

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.

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Quantum Stochastic Differential Equation is Unitarily Equivalent to a Symmetric Boundary Value Problem in Fock Space

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025798000120 ◽

1998 ◽

Vol 01 (02) ◽

pp. 175-199 ◽

Cited By ~ 12

Author(s):

Alexander M. Chebotarev

Keyword(s):

Differential Equation ◽

Schrödinger Equation ◽

Boundary Value Problem ◽

Stochastic Differential Equation ◽

Schrodinger Equation ◽

Fock Space ◽

Boundary Value ◽

Weak Limit ◽

Quantum Stochastic Differential Equation ◽

Markov Evolution

We show a new remarkable connection between the symmetric form of a quantum stochastic differential equation (QSDE) and the strong resolvent limit of the Schrödinger equations in Fock space: the strong resolvent limit is unitarily equivalent to QSDE in the adapted (or Ito) form, and the weak limit is unitarily equivalent to the symmetric (or Stratonovich) form of QSDE. We also prove that QSDE is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation in Fock space. The boundary condition describes standard jumps in phase and amplitude of components of Fock vectors belonging to the range of the resolvent. The corresponding Markov evolution equation (the Lindblad or Markov master equation) is derived from the boundary value problem for the Schrödinger equation.

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What is the time value of a stream of investments?

Journal of Applied Probability ◽

10.1239/jap/1127322033 ◽

2005 ◽

Vol 42 (3) ◽

pp. 861-866 ◽

Cited By ~ 4

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.

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Pricing Defaultable Securities under Actual Probability Measure

Journal of Systems Science and Information ◽

10.1515/jssi-2014-0313 ◽

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.

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What is the time value of a stream of investments?

Journal of Applied Probability ◽

10.1017/s0021900200000838 ◽

2005 ◽

Vol 42 (03) ◽

pp. 861-866 ◽

Cited By ~ 1

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.

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A Stochastic Differential Equation Driven by Poisson Random Measure and Its Application in a Duopoly Market

Mathematical Problems in Engineering ◽

10.1155/2020/3256859 ◽

2020 ◽

Vol 2020 ◽

pp. 1-12

Author(s):

Tong Wang ◽

Hao Liang

Keyword(s):

Differential Equation ◽

Stochastic Differential Equation ◽

Finite Number ◽

Random Measure ◽

Approximate Solutions ◽

The Other ◽

Poisson Random Measure ◽

Verification Theorem ◽

Hamilton Jacobi Bellman ◽

Duopoly Market

We investigate a stochastic differential equation driven by Poisson random measure and its application in a duopoly market for a finite number of consumers with two unknown preferences. The scopes of pricing for two monopolistic vendors are illustrated when the prices of items are determined by the number of buyers in the market. The quantity of buyers is proved to obey a stochastic differential equation (SDE) driven by Poisson random measure which exists a unique solution. We derive the Hamilton-Jacobi-Bellman (HJB) about vendors’ profits and provide a verification theorem about the problem. When all consumers believe a vendor’s guidance about their preferences, the conditions that the other vendor’s profit is zero are obtained. We give an example of this problem and acquire approximate solutions about the profits of the two vendors.

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Discrete-Time First-Order Plus Dead-Time Model-Reference Trade-off PID Control Design

Applied Sciences ◽

10.3390/app9163220 ◽

2019 ◽

Vol 9 (16) ◽

pp. 3220 ◽

Cited By ~ 3

Author(s):

Ryo Kurokawa ◽

Takao Sato ◽

Ramon Vilanova ◽

Yasuo Konishi

Keyword(s):

Discrete Time ◽

Pid Control ◽

Continuous Time ◽

Dead Time ◽

Design Method ◽

Control Design ◽

The Other ◽

Time Model ◽

Trade Off ◽

First Order

The present study proposes a novel proportional-integral-derivative (PID) control design method in discrete time. In the proposed method, a PID controller is designed for first-order plus dead-time (FOPDT) systems so that the prescribed robust stability is accomplished. Furthermore, based on the control performance, the relationship between the servo performance and the regulator performance is a trade-off relationship, and hence, these items are not simultaneously optimized. Therefore, the proposed method provides an optimal design method of the PID parameters for optimizing the reference tracking and disturbance rejection performances, respectively. Even though such a trade-off design method is being actively researched for continuous time, few studies have examined such a method for discrete time. In conventional discrete time methods, the robust stability is not directly prescribed or available systems are restricted to systems for which the dead-time in the continuous time model is an integer multiple of the sampling interval. On the other hand, in the proposed method, even when a discrete time zero is included in the controlled plant, the optimal PID parameters are obtained. In the present study, as well as the other plant parameters, a zero in the FOPDT system is newly normalized, and then, a universal design method is obtained for the FOPDT system with the zero. Finally, the effectiveness of the proposed method is demonstrated through numerical examples.

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Dynkin game under g-expectation in continuous time

Arabian Journal of Mathematics ◽

10.1007/s40065-020-00281-2 ◽

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.

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Pure Gaussian state generation via dissipation: a quantum stochastic differential equation approach

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2011.0529 ◽

2012 ◽

Vol 370 (1979) ◽

pp. 5324-5337 ◽

Cited By ~ 22

Author(s):

Naoki Yamamoto

Keyword(s):

Differential Equation ◽

Steady State ◽

Stochastic Differential Equation ◽

Cluster State ◽

Dissipative System ◽

Complete Characterization ◽

Gaussian State ◽

Quantum State Transfer ◽

Quantum Stochastic Differential Equation ◽

Gaussian System

Recently, the complete characterization of a general Gaussian dissipative system having a unique pure steady state was obtained. This result provides a clear guideline for engineering an environment such that the dissipative system has a desired pure steady state such as a cluster state. In this paper, we describe the system in terms of a quantum stochastic differential equation (QSDE) so that the environment channels can be explicitly dealt with. Then, a physical meaning of that characterization, which cannot be seen without the QSDE representation, is clarified; more specifically, the nullifier dynamics of any Gaussian system generating a unique pure steady state is passive. In addition, again based on the QSDE framework, we provide a general and practical method to implement a desired dissipative Gaussian system, which has a structure of quantum state transfer.

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