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A framework for the optimal sparse-control of the probability density function of a jump-diffusion process is presented. This framework is based on the partial integro-differential Fokker-Planck (FP) equation that governs the time evolution of the probability density function of this process. In the stochastic process and, correspondingly, in the FP model the control function enters as a time-dependent coefficient. The objectives of the control are to minimize a discrete-in-time, resp. continuous-in-time, tracking functionals and its L2- and L1-costs, where the latter is considered to promote control sparsity. An efficient proximal scheme for solving these optimal control problems is considered. Results of numerical experiments are presented to validate the theoretical results and the computational effectiveness of the proposed control framework.

Recently, largely motivated by computational finance applications, there has been a growing interest in stochastic jump-diffusion processes. In fact, empirical facts suggest that a discontinuous path could be most appropriate for describing the dynamics of stock prices; see [

When one considers decision making issues involving random quantities, stochastic optimization problems must be solved. Such problems have largely been examined in the scientific literature, because of the numerous applications in, e.g., physics, biology, finance, and economy [

In this work, we tackle the issue of controlling a stochastic process by following an alternative approach already proposed in [

In the case of our JD process, the FP equation takes the form of a PIDE endowed with initial and boundary conditions. While the Cauchy data must be the initial distribution of the given random variable, the boundary conditions of a FP problem depend on the considered model. For the derivation of the FP equation and a discussion about boundary conditions, see [

Infinite-dimensional optimization is a very active research field, motivated by a broad range of applications ranging from, e.g., fluid flow, space technology, heat phenomena, and image reconstruction; see, e.g., [

Very recently, PDE-based optimal control problems with sparsity promoting L^{1}-cost functionals have been investigated starting with [^{1} cost functional and apply the proximal algorithm proposed in [

This paper is organized as follows. In the next section, we discuss the functional setting of the FP problem modeling the evolution of the PDF of a JD stochastic process. In Section 3, we formulate our optimal control problems. Section 4 is devoted to the formulation of the corresponding first-order optimality systems. In Section 5, we discuss the discretization of the state and adjoint equations of the optimality system. In Section 6, we illustrate a proximal method for solving our optimal control problems. Section 7 is devoted to presenting results of numerical tests, including a discussion on the robustness of the algorithm to the choice of the parameters of the optimization problem. A section of conclusions completes this work.

In this section, we introduce a JD process and the corresponding FP equation that models the time evolution of the PDF of this process. Further, we discuss well-posed- ness and regularity of solutions to our FP problem.

We consider a time interval

where

Define

tive definite, and hence there exists

In this work, we consider a stochastic process with reflecting barriers. This assumption determines the boundary conditions for the FP equation corresponding to (1), see below. Define

where the differential operator

and

respectively. The definition of g in (5) takes into account the presence of reflecting barriers and the dependence on the jump amplitude

Notice that the differential operator

where

and

for each

The PDF f of X in (1) in the bounded domain

where

Notice that the flux F corresponds to the differential part of the FP equation, that is, to the drift and diffusion components of the stochastic process. In order to take into account the action of a reflecting barrier on the jumps, we consider a suitable definition of the kernel g, which can be conveniently illustrated in the one-dimensional case as follows.

Consider

where H is the Heaviside step function defined by

We normalize g and

The next remark motivates the choice of the boundary conditions (8) and of the condition (10).

Remark 2.1. Assume (8) and (10). Provided that

That is, the total probability over the space domain

Our FP problem is stated as follows

Next, we recall some definitions concerning the functional spaces needed to state the existence and uniqueness of solutions to (11). The space

These spaces are endowed with the following norms

where

We assume that the coefficients a and b in (4) satisfy the following conditions

for each

Notice that a and b must be defined on the closure

with

We have the following theorem [

Theorem 2.1. Assume

Proof. See [

Remark 2.2. Provided that

Consider the following spaces

Given the interval

which are also Banach spaces [

respectively. We consider the following space

which is a Hilbert space [

With this preparation, we can recall the following theorem [

Theorem 2.2. The embedding

The following proposition provides a useful a priori estimate of the solution to (11).

Proposition 2.3. Let

Proof. Consider the H inner product of the equation in (11) with f. Exploiting the properties of the Gelf and triple, we have

We make use of the following fact [

First, we exploit the zero-flux boundary conditions in (11) and the coercivity of a as given in (2). Moreover, we make use of the following Cauchy inequality

which holds for each

We choose

We have

Therefore we have

Recalling the definition of I in (5) and defining

Since

Therefore,

Define

By applying the Gronwall inequality, we have

Next, we outline how to obtain an upper bound of

where we used the PIDE and the boundary condition of the FP problem in (11). Proceeding as above, we obtain

with

up to a redefinition of the constant

Proposition 2.4. Assume (13) and

Proof. The statement follows from the a priori estimates of Proposition 2.3 and Theorem 2.2.

We define

with F and I defined in (6) and (5), respectively.

In this section, we define our optimal control problems governed by (23) and prove the existence of at least an optimal solution.

We consider a control mechanism that acts on the drift function

We assume the presence of control constraints given by

Remark 3.1. The subset

Let

The term

1) Given a set of values

2) Given a square-integrable function

The norms in (26) are defined as follows

Remark 3.2. The choice of a bounded time interval I ensures that the L^{1}-norm is finite whenever

Remark 3.3. The functional

We investigate the following optimal control problem (s)

In order to discuss the existence and uniqueness of solutions to (29), we consider the control-to-state operator

The next proposition can be proved by using standard arguments [

Proposition 3.1. The mapping

where b is the drift in (1) and F is defined in (6).

The constrained optimization problem (29) can be transformed into an unconstrained one as follows

where

The solvability of (31) is ensured by the next theorem, whose proof adapts techniques given in [

Theorem 3.2. There exists at least one optimal pair

Proof. The functional

We have that

The weakly lower convergent sequence

compact, there exists a subsequence

pass to the limit in

Thanks to the estimate (18) in Proposition 2.3, the sequence

sures that

Moreover, the convexity of

and therefore the pair

Remark 3.4. The uniqueness of the control

We follow the standard approach [

Consider the reduced problem (31) and write the reduced functional

Remark 4.1. The functional

The following definitions are needed in order to determine the first-order optimality system. If

where

since

The following proposition gives a necessary condition for a local minimum of

Proposition 4.1. If

or equivalently

Proof. Since

for v sufficiently close to

Dividing by

Dividing by

we conclude that

By using results in [

Moreover, recalling the definition of

A pointwise analysis of (35), which takes into account the definition (25) of the admissible controls, ensures the existence of two nonnegative functions

Proposition 4.2. The optimal solution

We refer to the last three conditions in (37) for the pair

The differentiability of

By considering the total derivative of

Therefore, we obtain

Defining the adjoint variable p as the solution to the following adjoint problem

we obtain the following reduced gradient

After some calculation, we have that (38) can be rewritten as the following adjoint system

where

The operator

The terminal boundary-value problem (40) admits a unique solution

The reduced gradient in (39), for given u, f, and p, takes the following form

The complementarity conditions in (37) can be recast in a more compact form, as follows. We define

The complementarity conditions in (37) and the inequalities related to the Lagrange multipliers

The previous considerations can be summarized in the following propositions.

Proposition 4.3. (Optimality system for a discrete-in-time tracking functional)

A local solution

Proposition 4.4. (Optimality system for a continuous-in-time tracking functional)

A local solution

In this section, we discuss the discretization of the optimality systems given in (42) and (43). For simplicity, we focus on a one-dimensional case with

fine

are defined as follows

Notice that a cell-centered space discretization is considered with cells midpoints at

The approximation of the forward and backward FP PIDEs is based on a discretization method discussed in [

of the differential operator is carried out as follows

where

The zero-flux boundary conditions are implemented referring to the points

where

If we follow the optimize-before-discretize (OBD) approach, the optimality system has already been computed on a continuous level as in (42) and (43) and subsequently discretized. As a consequence, the OBD approach allows one to discretize the forward abd adjoint FP problems according to different numerical schemes. However, the OBD procedure might introduce an inconsistency between the discretized objective and the reduced gradient; see [

The DBO approach results in the following approximations

together with the midpoint quadrature formula applied to

The time integration of (45) is carried out with the combination of the SM splitting with a predictor corrector scheme, as in (44).

In this section, we discuss a proximal optimization scheme for solving (31). This scheme and the related theoretical discussion follow the work in [

For our discussion, we need the following definitions and properties.

Definition 6.1. Let Z be a Hilbert space and l a convex lower semi continuous function,

Proposition 6.1. Let Z be a Hilbert space and l a convex lower semi continuous function,

where

Proof. See [

Proposition 6.2. The solution

for each

Proof. From Proposition 4.7 and by using (46), we have

The relation (47) suggests that a solution procedure based on a fixed point iteration should be pursued. We discuss how such algorithm can be implemented.

In the following, we assume that

for each

for each

Inequality (49) is the starting point for the formulation of a proximal scheme, whose strategy consists of minimizing the right-hand side in (49). One can prove the following equality

Recall the definition of

Lemma 6.3. Let

where the projected soft thresholding function

Proof. See [

Based on this lemma, we conclude the following

which can be taken as starting point for a fixed-point algorithm as follows

where

with

Algorithm 1 (Inertial proximal method).

Input: initial guess

1) While

(a) Evaluate

(b) Update

where

(c) Set

(d) Compute E according to (42) or (43).

(e) If

(f)

Remark 6.1. The backtracking scheme in Algorithm 1 provides an estimation of the upper bound of the Lipschitz constant in (48), since it is not known a priori. The initial guess for L is chosen as follows. Given a small variation

Algorithm 2 (Evaluation of the gradient).

Input:

1) Compute

2) Compute

3) Evaluate

Next, we discuss the convergence of our algorithm, using some existing results [

Proposition 6.4. The sequence

・ The sequence

・ There exists a weakly convergent subsequence

Definition 6.2. The proximal residual r is defined as follows

Proposition 6.12 tells us that

Proposition 6.5. Let

In this section, we present results of numerical experiments to validate the performance of our optimal control framework. Our purpose is to determine a sparse control

We implement the discretization scheme and the algorithm described in Section 5. We take

In the first series of experiments, we consider the setting with

Also for the case

Next, we investigate the behavior of the optimal solution considering the full optimization setting, that is, the case when the L^{1}-cost actively enters in the optimization process, i.e.

In Figures 5-7, we depict the optimal controls for three different choices of values of

Finally, in the ^{2}- and L^{1}-costs are considered. For a direct comparison with the first series of experiments, we consider an unconstrained control. We find that already with a small value of

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A framework for the optimal control of probability density functions of jump-diffusion processes was discussed. In this framework, two different, discrete-in-time and continuous-in-time, tracking functionals were considered together with a sparsity promoting L^{1}-cost of the control. The resulting nonsmooth minimization problems governed by a Fokker-Planck partial integro-differential equation were investigated. The existence of at least an optimal control solution was proven. To characterize and compute the optimal controls, the corresponding first-order optimality systems were derived and their numerical approximation was discussed. These optimality systems in combination with a proximal scheme allowed to formulate an efficient solution procedure, which was also theoretically discussed. Results of numerical experiments were presented to validate the computational effectiveness of the proposed method.

Supported by the European Union under Grant Agreement Nr. 304617 “Multi-ITN STRIKE―Novel Methods in Computational Finance”. This publication was supported by the Open Access Publication Fund of the University of Würzburg. We thank very much the Referee for improving remarks.

Gaviraghi, B., Schindele, A., Annunziato, M. and Borzì, A. (2016) On Optimal Sparse-Control Problems Governed by Jump-Diffusion Processes. Applied Mathematics, 7, 1978-2004. http://dx.doi.org/10.4236/am.2016.716162