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International Journal of Stochastic Analysis
Volume 2012 (2012), Article ID 858736, 9 pages
Relations between Stochastic and Partial Differential Equations in Hilbert Spaces
Institute of Mathematics and Computer Sciences (IMCS), Ural Federal University, Lenin Avenue 51, 620083 Ekaterinburg, Russia
Received 31 May 2012; Accepted 20 August 2012
Academic Editor: Yaozhong Hu
Copyright © 2012 I. V. Melnikova and V. S. Parfenenkova. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The aim of the paper is to introduce a generalization of the Feynman-Kac theorem in Hilbert spaces. Connection between solutions to the abstract stochastic differential equation and solutions to the deterministic partial differential (with derivatives in Hilbert spaces) equation for the probability characteristic is proved. Interpretation of objects in the equations is given.
The Feynman-Kac theorem in the numerical (vector) case relates solutions of the Cauchy problem for stochastic equations with a Brownian motion , : with solutions of the Cauchy problem for deterministic partial differential equations: for the probability characteristic with an arbitrary Borel function . Here means the mathematical expectation of a solution to (1.1) with initial value , .
Study of the relationship between the problems (1.1) and (1.2) was initially caused by the needs of physics. For example, the process describes the random motion of particles in a liquid or gas, and is a probability characteristic such as temperature, determined by the Kolmogorov equation. In the last years, the importance of the relationship between stochastic (1.1) and deterministic (1.2) problems has become more acute with the development of numerical methods and applications in financial mathematics. Here , for example, stock price at time , then is the value of stock options, determined by the famous Black-Scholes equation [1, 2]. Moreover, there exist recent applications of infinite-dimensional stochastic equations in financial mathematics . For example, consider function that is the price at time of the coupon bond with maturity date . Let be parametrized as for all and , be the forward curve; that is, . Then the Musiela reparametrization , , in the special case of zero HJM shift, satisfies the following equation in a Hilbert space of functions acting from to : where is the generator of the right-shifts semigroup in , is a -valued Wiener (in particular -Wiener) process, and is a random mapping from Hilbert space to . Here the value of bond options may be calculated, at least numerically, via defined for . Thereby it requires an infinite-dimensional analogue of the connection between problems (1.1) and (1.2), which the present paper is devoted to.
Generalization of the Feynman-Kac theorem to the infinite-dimensional case raises many questions related with the very formulation of the problem in infinite-dimensional spaces, the definition of relevant objects and a rigorous rationale for the relationship between mentioned problems.
The paper considers the stochastic Cauchy problem in Hilbert spaces, which is the infinite-dimensional generalization of the problem (1.1): We prove the infinite-dimensional case of the Feynman-Kac theorem in a standard basic conditions such as is the generator of a -semigroup in a Hilbert space , is a bounded operator from a Hilbert space to , and is a -valued -Wiener process. This is done for the clarity of proof despite the fact that the conditions are likely to be weakened, and the statement is true in more general assumptions. For example, the main result can be applied for (1.3) related to the bond price in more general case . The reduction of constraints to the Feynman-Kac theorem is also the subject of further research.
For the problem (1.4) we associate a problem for a deterministic partial differential equation which is a generalization of the problem (1.2) in the case of Hilbert spaces. The paper gives the rigorous interpretation of the objects included in the stochastic and deterministic equations and proves the connection between their solutions. The proof of the relation is obtained as a generalization of the finite-dimensional Feynman-Kac theorem . In contrast to the well-known works as , the proof is performed without using semigroup technique for some operator family defined as that gives a rigorous justification of the study only in certain cases. Particular attention is paid to the subtle issue of transition from zero expectation for a function of to equality for itself.
2. Definitions and Auxiliary Statements
We start with interpretation for the objects of the stochastic problem (1.4). Let the operator be the generator of the -semigroup in Hilbert space . This ensures uniform well-posedness of the Cauchy problem for the corresponding homogeneous equation , as well as existence and uniqueness of a weak solution to the stochastic problem (1.4) with -Wiener process , where is a trace class operator [4, 5].
Let a -valued stochastic process be -Wiener, that is, satisfy the following conditions:(i),(ii) has continuous trajectories,(iii) has independent increments,(iv) is normally distributed with and , . The -Wiener process can be expanded into the convergent series where are independent Brownian motions, is the sequence of eigenvalues of , and is the complete orthonormal system in corresponding to the : .
Define the function that transforms into . The is a measurable function from to . We show that satisfies the following infinite-dimensional deterministic problem: corresponding to the stochastic one (1.4).
At the beginning we make the sense to the terms of (2.2). The derivatives and are understood in the sense of Frechet, that means and . More precisely The term requires special attention. Expression is usually defined as the trace of an operator acting in the same Hilbert space. The operator under the trace sign in (2.2) maps Hilbert space to its adjoint .
Using the traditional definition of the trace, we can make sense to its expression, using the Riesz theorem on the isomorphism and , that is, identifying with . Note that the isomorphism allows us to consider operators , , and as mappings from to , from to , and to , respectively. Then operator transfers the Hilbert space to , and trace of this operator can be understood in the usual sense.
We give more rigorous interpretation to the concept of the trace for an operator acting from a Hilbert space to : for the purpose we write an arbitrary operator in the form Then can be understood as , , where is the basis in . Considering the proposed interpretation of the trace, the expression with has a clear and definite sense.
Now we prove necessary properties of the process that is a solution of (1.4), and function that determines the relationship between the solutions of problems (1.4) and (2.2). We obtain the required properties for the case of more general processes, the diffusion processes, to which the solution of (1.4) is a special case.
An -valued Ito process is called diffusion if it can be written in the form In the paper we consider the diffusion processes for which the existence and uniqueness of solution to the stochastic Cauchy problem for (2.5) are fulfilled. For example, such additional condition is guaranteed by the estimate to the coefficients and : , , , (see Theorem 2.1 , ch. VII). Note that in the particular case of the problem (1.4) the unique solution of (2.5) can be written as sum of the term depending on the initial value and the stochastic convolution term (see, e.g., [4, 5]): where the family is the -semigroup with the generator .
To prove the relationship under the study it is important to establish the Markov property for the solution of the Cauchy problem (1.4). The following statement is a generalization of the finite-dimensional case result (Theorem 7.1.2 ) to the case of Hilbert spaces.
Proposition 2.1. Let , be Borel-measurable and let be a diffusion Ito process. Then satisfies the Markov property with respect to a -algebra defined by -Wiener process :
Proof. Let , , be the solution of (2.5) with the condition . By the uniqueness of a solution to the Cauchy problem for (2.5) we have , , almost surely. Define the map . Then to prove the property (2.7) is sufficient to obtain the equality
Introduce the function . It is measurable as superposition of measurable functions. Fix some and consider some partition of the segment : , . Let where is a characteristic function of the semiopen interval . Then we have The first and the last equalities follow from definition of the . The second one is a consequence of the fact that the characteristic functions of intervals do not depend on the variable . Since , the third equality holds because is independent of the filter for all .
Now note that in . So, let tends to infinity in the established relation (2.10); then we have Thus, as we conclude that Using the diffusion property of Ito process we obtain The last two equalities imply the desired relation (2.7).
Note that if the Ito process is a solution to (1.4), it is diffusive and by the statement established previously has the Markov property.
Corollary 2.2. By the homogeneity in time of diffusion processes the following relation is fulfilled:
Corollary 2.3. Markov property can be written as follows:
The following statement generalizes Theorem 5.50 .
Proposition 2.4. Suppose a process satisfies the conditions of Proposition 2.1. Then the process is martingale, that is:
Proof. According to the Proposition 2.1, has the Markov property. Therefore and we obtain the following equalities: The first equality implies the obtained representation for the process via the conditional expectation. The second equality follows from the properties of conditional expectation. The third one is the direct consequence of the Markov property for . The last equality follows from the definition of the process and completes the proof.
3. Proof of the Main Result
Proof. Applying the Ito formula in Hilbert spaces  to as a function from the solution of the problem (1.4) we obtain
This equality is written in the form of differentials (increments). In the integral form it can be written as follows:
Apply the expectation to both sides of the equation. From the definition of Ito integral (via the approximation in the mean square by step processes) and the properties of the -Wiener process, we obtain . Further, since the process is martingale, we have
Hence, using the theorem of Tonelli-Fubini in Hilbert spaces and equalities mentioned previously we conclude The last equality is true for all . Therefore, Rewrite this equality at the origin point : that is Note that does not depend on ; thus . Using the Lebesgue dominated convergence theorem, the fact that the mappings , and are independent of the variable , and that the expectation is an integral of the variable , we conclude that all these operators commute with the operator . Furthermore, according to the interpretation of trace given in Section 2, it also commutes with the operator by the following arguments: Note Hence, we obtain
Until now we have considered the Cauchy problem (1.4) where . Consider this problem for the same equation with the initial condition at the moment : Then by the arguments similar to that conducted previously the equality holds. Varying we obtain (3.12) for . It remains to note That completes the proof.
In conclusion we note that the Feynman-Kac theorem in the numerical case establishes the interrelation between the stochastic and the deterministic problems on both sides [1, 2]. In numerical methods this relationship is indeed important to both sides: numerical methods obtained for stochastic equations are used for solving differential equations in partial derivatives, and basic methods for partial differential equations allow to obtain the characteristics of solutions to stochastic problems (see, e.g., ). Therefore, in addition to the previous result, it is important to establish the connection in the opposite direction and to construct the methods of solution for infinite-dimensional problem. Those are the subjects of the future studies related to the generalization of the Feynman-Kac theorem for the case of Hilbert spaces.
This research was carries out with the support of RFBR no. 10-01-96003p and the programme of Ministry of Education and Science no. 1.1016.2011.
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