Abstract

We introduce the concept of a mild solution for the right Hudson-Parthasarathy quantum stochastic differential equation, prove existence and uniqueness results, and show the correspondence between our definition and similar ideas in the theory of classical stochastic differential equations. The conditions that a process must satisfy in order for it to be a mild solution are shown to be strictly weaker than those for it to be a strong solution by exhibiting a class of coefficient matrices for which a mild unitary solution can be found, but for which no strong solution exists.

Highlights

  • One of the main analytical difficulties in the theory of stochastic differential equations arises whenever the coefficients driving the equation consist of unbounded operators — a requirement that is largely unavoidable in the pursuit of interesting models

  • We introduce the concept of a mild solution for the right Hudson-Parthasarathy quantum stochastic differential equation, prove existence and uniqueness results, and show the correspondence between our definition and similar ideas in the theory of classical stochastic differential equations

  • The conditions that a process must satisfy in order for it to be a mild solution are shown to be strictly weaker than those for it to be a strong solution by exhibiting a class of coefficient matrices for which a mild unitary solution can be found, but for which no strong solution exists

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Summary

Introduction

One of the main analytical difficulties in the theory of stochastic differential equations (both classical and quantum) arises whenever the coefficients driving the equation consist of unbounded operators — a requirement that is largely unavoidable in the pursuit of interesting models. Xt = ξ + AXs ds + BXs dWs, in particular the two integrals on the right hand side must be well-defined, and for this to be true we must have Xt ∈ Dom A ∩ Dom B almost surely. If both A and B are unbounded any study of (1.1) must incorporate an investigation of how well their domains match up. The coefficient matrix [Fβα] is made up of (unbounded) operators acting on another Hilbert space h, and the solution process U = (Ut)t≥0 consists of contraction operators on the tensor product Hilbert space h ⊗ F.

INTRODUCTION
Tensor product conventions
Quantum stochastic calculus
Existence results
A mild solution that cannot be a strong solution
Conclusion

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