Dynamic Programming of the Stochastic Burgers Equation Driven by Lévy Noise

Abstract

In this work, we study the optimal control of stochastic Burgers equation perturbed by Gaussian and Lévy-type noises with distributed control process acting on the state equation. We use dynamic programming approach for the feedback synthesis to obtain an infinite-dimensional second-order Hamilton–Jacobi–Bellman (HJB) equation consisting of an integro-differential operator with Lévy measure associated with the stochastic control problem. Using the regularizing properties of the transition semigroup corresponding to the stochastic Burgers equation and compactness arguments, we solve the HJB equation and the resultant feedback control problem.

Appendices

Appendix A: Dynamic Programming Principle

We briefly give the key steps in deriving the stochastic Dynamic Programming Principle and HJB equation. Let us define

$$\begin{aligned} \mathfrak {L}({\textrm{X}},{\textrm{U}})=&\Vert {\textrm{D}}_\xi {\textrm{X}}\Vert ^2+\frac{1}{2}\Vert {\textrm{U}}\Vert ^2, \\ \Psi ({\textrm{X}}(T))=&\Vert {\textrm{X}}(T)\Vert ^2, \quad {\textrm{F}}({\textrm{X}},{\textrm{U}})=-\textrm{AX}+{\textrm{B}}({\textrm{X}})+{\textrm{U}}. \end{aligned}$$

Let $T>0$ be given. Then for any $t\in [0,T),$ consider the problem of minimizing the cost functional

$$\begin{aligned} {{\mathcal {J}}}(t,T;x,{\textrm{U}})= {\mathbb {E}}_t\left[ \int _t^T\mathfrak {L}({\textrm{X}}(s),{\textrm{U}}(s))ds+ \Psi ({\textrm{X}}(T)) \right] , \end{aligned}$$

over all controls ${\textrm{U}}\in {\mathscr {U}}^{t,T}_{\rho }$ and ${\textrm{X}}(\cdot )$ satisfying the following state equation

$$\begin{aligned}{} & {} {\textrm{dX}}(s)={\textrm{F}}({\textrm{X}}(s),{\textrm{U}}(s)){\textrm{d}} s+\varepsilon _1{\textrm{Q}}^{1/2}{\textrm{dW}}(s) +\varepsilon _2\int _{{\mathcal {Z}}}{\textrm{G}}(s,z) \widetilde{{\textrm{N}}}({\textrm{d}} s,{\textrm{d}} z), \quad s\in [t,T],\nonumber \\{} & {} {\textrm{X}}(t)=x, \quad x\in {\textrm{H}}. \end{aligned}$$

(A.1)

Here ${\mathbb {E}}_t[{\textrm{X}}(s)]={\mathbb {E}}[{\textrm{X}}(s) | {\textrm{X}}(t)=x]$, for $s\ge t$ and the admissible control set is defined as

$$\begin{aligned} {\mathscr {U}}^{t,T}_{\rho } =\Big \{{\textrm{U}}\in {\textrm{L}}^2(\Omega , {\textrm{L}}^2(t,T;{\textrm{H}})):\Vert {\textrm{U}}\Vert \le \rho , \ \text{ and } \ {\textrm{U}} \ \text{ is } \text{ adapted } \text{ to } \ {\mathscr {F}}_{t,s}\Big \}, \end{aligned}$$

where ${\mathscr {F}}_{t,s}$ is the $\sigma$-algebra generated by the paths of ${\textrm{W}}$ and random measures ${\textrm{N}}$ upto time s,  i.e., $\sigma \{{\textrm{W}}(r); t\le r\le s\}$ and $\sigma \{{\textrm{N}}(S); S\in {\mathscr {B}}([t,s]\times {\mathcal {Z}})\}.$ The value function of the above control problem is defined as

$$\begin{aligned} \mathscr {V}(t,x)= & {} \inf _{{\textrm{U}}\in {\mathscr {U}}^{t,T}_{\rho }} {\mathcal J}(t,T;x,{\textrm{U}}), \text { for all }t\in [0,T), \quad x\in {\textrm{H}}, \\ \mathscr {V}(T,x)= & {} \Vert x\Vert ^2. \end{aligned}$$

Following the dynamic programming strategy, any admissible control ${\textrm{U}}\in {\mathscr {U}}^{t,T}_{\rho }$ is the combination of controls in ${\mathscr {U}}^{t,\tau }_{\rho }$ and ${\mathscr {U}}^{\tau ,T}_{\rho }$ for any $0\le t< \tau < T.$ More precisely, suppose the processes ${\textrm{U}}_1(s)$ and ${\textrm{U}}_2(s)$ be the restriction of the control process ${\textrm{U}}(s)$ to the time intervals $[t,\tau ]$ and $[\tau ,T]$, respectively, i.e.,

$$\begin{aligned} {\textrm{U}}(s)= ({\textrm{U}}_1\oplus {\textrm{U}}_2)(s) =\left\{ \begin{array}{ll} {\textrm{U}}_t(s),&{} s\in [t,\tau ], \\ {\textrm{U}}_\tau (s),&{} s\in [\tau , T]. \\ \end{array}\right. \end{aligned}$$

Accordingly, the admissible control set is written as ${\mathscr {U}}^{t,T}_{\rho }={\mathscr {U}}^{t,\tau }_{\rho }\oplus {\mathscr {U}}^{\tau ,T}_{\rho }.$ Note also that the controls ${\textrm{U}}_1(s)$ and ${\textrm{U}}_2(s)$ are adapted to ${\mathscr {F}}_{t,s}$ and ${\mathscr {F}}_{\tau ,s}$, respectively.

The system state ${\textrm{X}}(\cdot )$ is determined by (A.1) with ${\textrm{U}}(s)=({\textrm{U}}_1\oplus {\textrm{U}}_2)(s)\in {\mathscr {U}}^{t,T}_{\rho }.$ We decompose the system state as ${\textrm{X}}(s)=({\textrm{X}}_1\oplus {\textrm{X}}_2)(s),$ where ${\textrm{X}}_1$ and ${\textrm{X}}_2$ satisfy

$$\begin{aligned} {\textrm{dX}}_1(s)= & {} {\textrm{F}}({\textrm{X}}_1(s),{\textrm{U}}_1(s)){\textrm{d}} s+\varepsilon _1{\textrm{Q}}^{1/2}{\textrm{dW}}(s) +\varepsilon _2\int _{{\mathcal {Z}}}{\textrm{G}}(s,z) \widetilde{{\textrm{N}}}({\textrm{d}} s,{\textrm{d}} z), \quad s\in [t,\tau ], \\ {\textrm{X}}_1(t)= & {} {\textrm{X}}(t)=x\in {\textrm{H}}, \end{aligned}$$

and

$$\begin{aligned} {\textrm{dX}}_2(s)= & {} {\textrm{F}}({\textrm{X}}_2(s),{\textrm{U}}_2(s)){\textrm{d}} s+\varepsilon _1{\textrm{Q}}^{1/2}{\textrm{dW}}(s) +\varepsilon _2\int _{{\mathcal {Z}}}{\textrm{G}}(s,z) \widetilde{{\textrm{N}}}({\textrm{d}} s,{\textrm{d}} z), \ s\in [\tau ,T],\\ {\textrm{X}}_2(\tau )= & {} {\textrm{X}}_1(\tau )={\textrm{X}}(\tau ). \end{aligned}$$

By the tower property of conditional expectation

$$\begin{aligned} {\mathbb {E}}_t\Big [{\mathbb {E}}_t\big (\mathfrak {L}({\textrm{X}}_2(s),{\textrm{U}}_2(s))\ | {\mathscr {F}}_{t,\tau }\big )\Big ] ={\mathbb {E}}_t\big [\mathfrak {L}({\textrm{X}}_2(s),{\textrm{U}}_2(s))\big ], \tau \le s\le T, \end{aligned}$$

we write

$$\begin{aligned} \mathscr {V}(t,x)&= \inf _{{\textrm{U}}\in {\mathscr {U}}^{t,T}_{\rho } }{\mathbb {E}}_t\left\{ \int _t^\tau \mathfrak {L}({\textrm{X}}(s),{\textrm{U}}(s)){\textrm{d}} s + \int _\tau ^T\mathfrak {L}({\textrm{X}}(s),{\textrm{U}}(s)){\textrm{d}} s+\Psi ({\textrm{X}}(T))\right\} \\&= \inf _{\begin{array}{c} {\textrm{U}}_1\in {\mathscr {U}}^{t,\tau }_{\rho }, {\textrm{U}}_2\in {\mathscr {U}}^{\tau ,T}_{\rho } \\ {\textrm{X}}_2(\tau )={\textrm{X}}_1(\tau ) \end{array}} {\mathbb {E}}_t\left\{ \int _t^\tau \mathfrak {L}({\textrm{X}}_1(s),{\textrm{U}}_1(s)){\textrm{d}} s \right. \\&\left. \quad +\, {\mathbb {E}}_t\left[ \int _\tau ^T\mathfrak {L}({\textrm{X}}_2(s),{\textrm{U}}_2(s)){\textrm{d}} s+\Psi ({\textrm{X}}_2(T))\big |{\mathscr {F}}_{t,\tau }\right] \right\} . \end{aligned}$$

Using the Markovian property of the process ${\textrm{X}}(\cdot )$ one can get for $\tau \le s\le T,$

$$\begin{aligned} {\mathbb {E}}_t\Big [\mathfrak {L}({\textrm{X}}_2(s),{\textrm{U}}_2(s))\ | {\mathscr {F}}_{t,\tau }\Big ] ={\mathbb {E}}_{\tau }\Big [\mathfrak {L}({\textrm{X}}_2(s),{\textrm{U}}_2(s))\ | \ {\textrm{X}}_2(\tau )={\textrm{X}}(\tau ) \Big ] \end{aligned}$$

and the same reasoning is true for $\Psi ({\textrm{X}}_2(T))$ as well. It leads to

$$\begin{aligned} {{\mathcal {J}}}(\tau ,T;{\textrm{X}}_2(\tau ),{\textrm{U}}_2)= {\mathbb {E}}_t\left\{ \int _\tau ^T\mathfrak {L}({\textrm{X}}_2(s),{\textrm{U}}_2(s)){\textrm{d}} s+\Psi ({\textrm{X}}_2(T)) \ \Big | {\mathscr {F}}_{t,\tau }\right\} . \end{aligned}$$

Besides, for more details on the case of continuous diffusion, one can refer to Yong and Zhou [37] (and also [19]) and that can be modified to this case. Hence

$$\begin{aligned} \mathscr {V}(t,x)&= \inf _{{\textrm{U}}_1\in {\mathscr {U}}^{t,\tau }_{\rho }} {\mathbb {E}}_t\left\{ \int _t^\tau \mathfrak {L}({\textrm{X}}_1(s),{\textrm{U}}_1(s)){\textrm{d}} s\right\} \nonumber \\&\quad +\, {\mathbb {E}}_t\left\{ \inf _{\begin{array}{c} {\textrm{U}}_2\in {\mathscr {U}}^{\tau ,T}_{\rho } \\ {\textrm{X}}_2(\tau )={\textrm{X}}_1(\tau ) \end{array}}{\mathbb {E}}_t\left[ \int _\tau ^T\mathfrak {L}({\textrm{X}}_2(s),{\textrm{U}}_2(s)){\textrm{d}} s+\Psi ({\textrm{X}}_2(T))\Big | {\mathscr {F}}_{t,\tau }\right] \right\} \nonumber \\&= \inf _{{\textrm{U}}_1\in {\mathscr {U}}^{t,\tau }_{\rho }} {\mathbb {E}}_t\left\{ \int _t^\tau \mathfrak {L}({\textrm{X}}_1(s),{\textrm{U}}_1(s)){\textrm{d}} s +\mathscr {V}(\tau ,{\textrm{X}}_1(\tau )) \right\} . \end{aligned}$$

Thus, we have proved the following DPP (or Bellman’s principle of optimality)

$$\begin{aligned} \mathscr {V}(t,x)=\inf _{{\textrm{U}}\in {\mathscr {U}}^{t,\tau }_{\rho }} {\mathbb {E}}\left\{ \int _t^\tau \mathfrak {L}({\textrm{X}}(s),{\textrm{U}}(s)){\textrm{d}} s +\mathscr {V}(\tau ,{\textrm{X}}(\tau )) \big | {\textrm{X}}(t)=x\right\} . \end{aligned}$$

(A.2)

1.1 A.1. Dynamic Programming Equation

Rewrite (A.2) as follows

$$\begin{aligned} \inf _{{\textrm{U}}\in {\mathscr {U}}^{t,\tau }_{\rho }} {\mathbb {E}}\left\{ \int _t^\tau \mathfrak {L}({\textrm{X}}(s),{\textrm{U}}(s))ds +\mathscr {V}(\tau ,{\textrm{X}}(\tau )) - \mathscr {V}(t,x) \big | {\textrm{X}}(t)=x\right\} =0. \end{aligned}$$

(A.3)

Now the HJB equation can be formally obtained by applying the Itô formula for $\mathscr {V}(\tau ,{\textrm{X}}(\tau )) - \mathscr {V}(t,x).$

Let us define a class of functionals

$$\begin{aligned} {{\mathcal {D}}}_{{\mathcal {A}}}^{T}=\left\{ \Phi :[0,T]\times {\textrm{H}} \rightarrow {\mathbb {R}}; \Phi (\cdot , x)\in C^{1}[0,T] \text{ and } \Phi (t,\cdot )\in \mathcal {D}(\mathscr {A}^{{\textrm{U}}})\right\} . \end{aligned}$$

Remark 7.1

(Itô formula) [20, 26] For any $t\ge 0$ and $\Phi \in {{\mathcal {D}}}_{{\mathcal {A}}}^{T}$ the following relation holds:

$$\begin{aligned} \Phi (\tau ,{\textrm{X}}(\tau ))-\Phi (t,x)=\int _t^\tau \big [{\textrm{D}}_s\Phi (s,{\textrm{X}}(s))+{\mathscr {A}}^{{\textrm{U}}}\Phi (s,{\textrm{X}}(s))\big ]{\textrm{d}} s + {\textrm{M}}_\tau , \ {\mathbb {P}} \text {-a.s.}, \end{aligned}$$

where ${\mathscr {A}}^{{\textrm{U}}}\Phi$ is the second-order partial integro-differential operator

$$\begin{aligned} {\mathscr {A}}^{{\textrm{U}}}\Phi (s,{\textrm{X}}(s))&= \left( {\textrm{D}}_x\Phi (s,{\textrm{X}}(s)), {\textrm{F}}({\textrm{X}}(s),{\textrm{U}}(s))\right) \\&\quad +\,\frac{1}{2} {\textrm{Tr}}(\varepsilon _1^2{\textrm{QD}}^2_x\Phi (s,{\textrm{X}}(s))) \\&\quad +\,\int _{{\mathcal {Z}}}\big [\Phi (s,{\textrm{X}}(s)+\varepsilon _2{\textrm{G}}(s,z))-\Phi (s,{\textrm{X}}(s))\\&\quad -\,\left( \varepsilon _2{\textrm{G}}(s,z),{\textrm{D}}_x\Phi (s,{\textrm{X}}(s))\right) \big ]\mu ({\textrm{d}} z) \end{aligned}$$

and ${\textrm{M}}_{\tau }$ is the martingale given by

$$\begin{aligned} {\textrm{M}}_\tau&=\int _t^\tau \left( {\textrm{D}}_x\Phi (s,{\textrm{X}}(s)), \varepsilon _1\sqrt{{\textrm{Q}}}{\textrm{dW}}(s)\right) \\&\quad +\, \int _t^\tau \int _{{\mathcal {Z}}} \big [\Phi (s,{\textrm{X}}(s-)+\varepsilon _2{\textrm{G}}(s,z))-\Phi (s,{\textrm{X}}(s-))\big ]\widetilde{{\textrm{N}}}({\textrm{d}} s,{\textrm{d}} z). \end{aligned}$$

Taking $\Phi (\tau ,{\textrm{X}}(\tau ))=\mathscr {V}(\tau ,{\textrm{X}}(\tau ))$ (of course by assuming required smoothness and boundedness on $\mathscr {V}$) and plug this back into (A.3), and use the fact that martingale ${\textrm{M}}_{\tau }$ has a zero mean to obtain the following

$$\begin{aligned} \inf _{{\textrm{U}}\in {\mathscr {U}}^{t,\tau }_{\rho }} {\mathbb {E}}\left\{ \int _t^\tau \big [\mathfrak {L}({\textrm{X}}(s),{\textrm{U}}(s))+{\textrm{D}}_s\mathscr {V}(s,{\textrm{X}}(s))+{\mathscr {A}}^{{\textrm{U}}}\mathscr {V}(s,{\textrm{X}}(s))\big ]{\textrm{d}} s \big | {\textrm{X}}(t)=x\right\} =0. \end{aligned}$$

Finally, take $\tau =t+h, h>0$ and divide by h. Passing $h\rightarrow 0$ and using the conditional expectation, we formally obtain the HJB equation

$$\begin{aligned}{} & {} {\textrm{D}}_t\mathscr {V}(t,x)+\inf _{\Vert {\textrm{U}}\Vert \le \rho } \big \{{\mathscr {A}}^{{\textrm{U}}}\mathscr {V}(t,x) +\mathfrak {L}(x,{\textrm{U}}(t)) \big \}=0, \quad t\in [0,T), \\{} & {} \mathscr {V}(T,x)=\Vert x\Vert ^2, \quad x\in {\textrm{H}}. \end{aligned}$$

Moreover, setting $v(t,x)=\mathscr {V}(T-t,x),$ one can obtain the following initial value problem

$$\begin{aligned}{} & {} {\textrm{D}}_tv(t,x)= \mathscr {H}(x,t,{\textrm{D}}_xv,{\textrm{D}}^2_xv), \quad t\in (0,T), \\{} & {} v(0,x)= \Vert x\Vert ^2, \quad x\in {\textrm{H}}, \end{aligned}$$

where $\mathscr {H}$ is given by

$$\begin{aligned} \mathscr {H}(x,t,{\textrm{D}}_xv,{\textrm{D}}^2_xv)&=\frac{1}{2}{\textrm{Tr}}(\varepsilon _1^2{\textrm{Q}} {\textrm{D}}_x^2v)+\left( - {\textrm{A}} x+{\textrm{B}}(x),{\textrm{D}}_xv\right) \\&\quad +\,\int _{{\mathcal {Z}}}\big [v(t,x+\varepsilon _2{\textrm{G}}(t,z))-v(t,x)-\left( \varepsilon _2{\textrm{G}}(t,z),{\textrm{D}}_xv\right) \big ]\mu ({\textrm{d}} z) \\&\quad +\,\Vert {\textrm{D}}_\xi x\Vert ^2+\inf _{\Vert {\textrm{U}}\Vert \le \rho }\left\{ \left( {\textrm{U}}(t),{\textrm{D}}_xv\right) +\frac{1}{2}\Vert {\textrm{U}}(t)\Vert ^2\right\} . \end{aligned}$$

Appendix B: Bismut–Elworthy–Li formula(BEL formula)

The BEL formula for Gaussian case has been derived in [8, 9, 16]. Further, this formula is obtained for some general stochastic evolution equations with Lévy noise in finite and infinite dimensions in [6, 23, 28]. For the sake of readers point of view, we give a formal derivation in the case of Burgers equation with Lévy noise.

Let us consider the following Kolmogorov equation associated with the uncontrolled stochastic Burger’s equation (3.9):

$$\begin{aligned}{} & {} {\textrm{D}}_tv(t,x)={\mathscr {L}}_x v(t,x), \quad t\in (0,T), \nonumber \\{} & {} v(0,x)= f(x), \quad x\in {\textrm{H}}, \end{aligned}$$

(B.1)

where ${\mathscr {L}}_x v$ is the operator defined in (3.5). For $\Phi \in {{\mathcal {D}}}_{{\mathcal {A}}}^{T},$ Itô’s formula given by Remark 7.1 yields

$$\begin{aligned} \Phi \left( t,{\textrm{Y}}(t,x)\right)&=\Phi (0,x)+\int _0^t\left[ \textrm{D}_s\Phi \left( s,{\textrm{Y}}(s,x)\right) \right. \nonumber \\&\quad \left. +\,\mathscr {L}_x\Phi \left( s,{\textrm{Y}}(s,x)\right) \right] {\textrm{d}} s+\int _0^t\left( {\textrm{D}}_x\Phi \left( s,{\textrm{Y}}(s,x)\right) ,\varepsilon _1{\textrm{Q}}^{1/2}{\textrm{dW}}(s)\right) \nonumber \\&\quad +\,\int _0^t\int _{{\mathcal {Z}}}\left[ \Phi \left( s-,{\textrm{Y}}(s-,x)+\varepsilon _2\textrm{G}(s-,z)\right) \right. \nonumber \\&\quad \left. -\,\Phi \left( s,{\textrm{Y}}(s-,x)\right) \right] \widetilde{{\textrm{N}}}({\textrm{d}} s,{\textrm{d}} z). \end{aligned}$$

(B.2)

For $f\in {\mathcal {D}}\left( \mathscr {A}^{{\textrm{U}}}\right)$, we can prove that $v(t,x)={\mathbb {E}}\left[ f({\textrm{Y}}(t,x)\right]$ is a solution of (B.1) by using Itô’s formula (B.2). Let us take $\Phi (s,{\textrm{Y}}(s,x))=v(t-s,{\textrm{Y}}(s,x))\in {{\mathcal {D}}}_{{\mathcal {A}}}^{T}$. Then applying (B.2) yields

$$\begin{aligned} f\left( {\textrm{Y}}(t,x)\right)&=v(t,x)+\int _0^t\left[ -\textrm{D}_s v\left( t-s,{\textrm{Y}}(s,x)\right) +\mathscr {L}_xv\left( t-s,{\textrm{Y}}(s,x)\right) \right] {\textrm{d}} s\nonumber \\&\quad +\,\int _0^t\left( {\textrm{D}}_x\left[ v\left( t-s,{\textrm{Y}}(s,x)\right) \right] ,\varepsilon _1{\textrm{Q}}^{1/2}{\textrm{dW}}(s)\right) \nonumber \\&\quad +\,\int _0^t\int _{{\mathcal {Z}}}\left[ v\left( t-s,{\textrm{Y}}(s-,x)+\varepsilon _2{\textrm{G}}(s-,z)\right) \right. \nonumber \\&\left. \quad -\,v\left( t-s,{\textrm{Y}}(s-,x)\right) \right] \widetilde{{\textrm{N}}}({\textrm{d}} s,{\textrm{d}} z)\nonumber \\&= v(t,x) +\int _0^t\left( {\textrm{D}}_x[v\left( t-s,{\textrm{Y}}(s,x)\right) ],\varepsilon _1{\textrm{Q}}^{1/2}{\textrm{dW}}(s)\right) \nonumber \\&\quad +\,\int _0^t\int _{{\mathcal {Z}}}\left[ v\left( t-s,{\textrm{Y}}(s-,x)+\varepsilon _2\textrm{G}(s-,z)\right) \right. \nonumber \\&\quad \left. -\,v\left( t-s,{\textrm{Y}}(s-,x)\right) \right] \widetilde{{\textrm{N}}}({\textrm{d}} s,{\textrm{d}} z). \end{aligned}$$

(B.3)

Let us take expectation on both sides of (B.3) and note that the final two terms on the right hand side of (B.3) are martingales, we obtain $v(t,x)={\mathbb {E}}\left[ f({\textrm{Y}}(t,x)\right]$. Now we multiply both sides of (B.3) by $\textrm{M}(t)={\int \limits _0^t\left( {\textrm{Q}}^{-1/2}\eta ^h(s,x),\varepsilon _1{\textrm{dW}}(s)\right) },$ where $\eta ^h(t,x):=(D_xY(t,x),h)$ is the solution of the equation:

$$\begin{aligned}{} & {} \frac{\partial }{\partial t}\eta ^h(t,x)=\left[ -{\textrm{A}}\eta ^h(t,x)+\textrm{B}(\textrm{X}(t,x),\eta ^h(t,x))+ \textrm{B}(\eta ^h(t,x),\textrm{X}(t,x))\right] ,\\{} & {} \eta ^h(0,x)=h. \end{aligned}$$

Then taking expectation and using the Markov property of semigroup, we get the Bismut–Elworthy–Li formula for the stochastic Navier–Stokes equations perturbed by Lévy noise as follows:

$$\begin{aligned} \left( {\textrm{D}}_xv(t,x), h\right) =\frac{1}{t}{\mathbb {E}}\left[ \psi \left( \textrm{X}(t,x)\right) \int _0^t\left( \textrm{Q}^{-1/2}\eta ^h(s,x),\varepsilon _1 {\textrm{W}}(s)\right) \right] . \end{aligned}$$

(B.4)

In order to get (B.4), we used Itô’s product rule to obtain ${\mathbb {E}}[\textrm{M}(t){\textrm{P}}(t)]=0$, where ${\textrm{P}}(t)$ is the final term from the right hand side of the equality (B.3). This follows from the fact that the stochastic integrals $\textrm{M}(t)$ and ${\textrm{P}}(t)$ are uncorrelated, since ${\textrm{W}}(\cdot )$ and $\widetilde{{\textrm{N}}}(\cdot ,\cdot )$ are independent.