# A change of measure formula for recursive conditional expectations

Luca Di Persio Department of Computer Science, University of Verona, via Ca’
Vignal 2, 37129 Verona, Italy. E-mail<EMAIL_ADDRESS>Alessandro
Gnoatto (Corresponding Author) Department of Economics, University of Verona,
via Cantarane 24, 37129 Verona, Italy. E-mail<EMAIL_ADDRESS>Marco Patacca Department of Economics, University of Verona, via Cantarane 24,
37129 Verona, Italy. E-mail<EMAIL_ADDRESS>
###### Abstract

We derive a representation for the value process associated to the solutions
of forward-backward stochastic differential equations in a jump-diffusion
setting under multiple probability measures. Motivated by concrete financial
problems, the latter representations are then applied to devise a
generalization of the change of numéraire technique allowing to obtain
recursive pricing formulas in the presence of non-linear funding terms due to
e.g. collateralization agreements.

Keywords Pricing $\cdot$ Change of measure $\cdot$ BSDE $\cdot$ recursive
conditional expectation $\cdot$ non-linear valuation

JEL Classification C63 $\cdot$ G12 $\cdot$ G13

##  1 Introduction and Motivation

Let $\mathbb{P}$ and $\mathbb{Q}$ be two probability measures on the
measurable space $\left(\Omega,\mathcal{F}\right)$. From the Radon-Nikodym
theorem, it is well known that $\mathbb{Q}\ll\mathbb{P}$ on
$\left(\Omega,\mathcal{F}\right)$ if and only if there exists an
$\mathcal{F}$-measurable and non negative random variable
$\mathcal{E}:\Omega\rightarrow\mathbb{R}^{+}$ with
$\mathbb{E}^{\mathbb{P}}\left[\mathcal{E}\right]=1$, such that, for any random
variable $X\in L^{1}\left(\Omega,\mathcal{F},\mathbb{Q}\right)$ we have

$\mathbb{E}^{\mathbb{Q}}\left[X\right]=\mathbb{E}^{\mathbb{P}}\left[\mathcal{E}X\right]\;,\;\mathbb{Q}-a.s.$
(1.1)

The previous result can be generalized to conditional expectations, which is
sometimes referred to as the “abstract Bayes rule”. More specifically, if
$\mathcal{G}$ is a $\sigma$-algebra with $\mathcal{G}\subseteq\mathcal{F}$ we
have

$\mathbb{E}^{\mathbb{Q}}\left[X|\mathcal{G}\right]=\frac{\mathbb{E}^{\mathbb{P}}\left[\mathcal{E}X|\mathcal{G}\right]}{\mathbb{E}^{\mathbb{P}}\left[\mathcal{E}|\mathcal{G}\right]}\;,\;\mathbb{Q}-a.s.$
(1.2)

The above relations play a relevant role in several applications, particularly
in Mathematical Finance, see e.g. Bjork (2020). Within this setting, the
likelihood ratio $\mathcal{E}$, due to the Girsanov theorem, is usually given
by the Doléans-Dade stochastic exponential driven by possibly discontinuous
semimartingales. Such tools are well understood for standard conditional
expectations, while the case when the conditional expectation has a recursive
structure is less clear. For the sake of clarity, let us first introduce such
situation in an informal way.

By recursive expectation, we mean an expression of the following form

$Y_{t}=\mathbb{E}^{\mathbb{Q}}\left[g(X_{T})+\int_{t}^{T}f(s,X_{s},Y_{s},Z_{s})\mathrm{d}s|\mathcal{F}_{t}\right],$
(1.3)

where the functions $g,f$ and the processes $X,Y$ and $Z$ are regular enough,
also having implicitly introduced the filtration
$\mathbb{F}=(\mathcal{F}_{t})_{0\leq t\leq T}$, $T>0$ being a fixed time
horizon. Such filtration is generated by the process
$X\coloneqq\\{X_{t},t\in[0,T]\\}$, typically assumed to be an Itô
semimartingale.

Expectations like (1.3) naturally arise from the study of Forward-Backward
Stochastic Differential Equations (FBSDEs) of the form

$\displaystyle\mathrm{d}X_{t}$
$\displaystyle=\mu(t,X_{t})\mathrm{d}t+\sigma(t,X_{t})\mathrm{d}W_{t}^{\mathbb{Q}}\quad,\quad
X_{0}=x$ (1.4) $\displaystyle-\mathrm{d}Y_{t}$
$\displaystyle=f(t,X_{t},Y_{t},Z_{t})\mathrm{d}t-Z_{t}\mathrm{d}Z_{t}^{\mathbb{Q}}\quad,\quad
Y_{T}=g(X_{T})\,,$ (1.5)

where all vector field functions are assumed to be sufficiently regular, while
$W^{\mathbb{Q}}$ is a standard Brownian motion under the reference measure
$\mathbb{Q}$. For a background on FBSDEs, we refer the reader to Bismut
(1973), Pardoux and Peng (1990), El Karoui et al. (1997), Ma and Yong (1999)
and Delong (2013) among others.

The aim of this work is to provide a change of measure formula for conditional
expectations of the form (1.3) associated to FBSDEs of the form (1.4) within a
Markovian Jump Diffusion setting. We derive such result in Theorem 3.3 which
constitutes the main result of the present paper.

Our motivation to consider this type of recursive conditional expectations is
mainly steered by financial applications. As a response to the financial
crisis, the valuation of contingent claim has been generalized along several
directions. For example Piterbarg (2010) obtained, starting from a Black-
Scholes type replication argument and applying the Feynman-Kač formula, the
following recursive expression for the price of a contingent claim when
collateral is exchanged:

$Y_{t}=\mathbb{E}^{\mathbb{Q}}\left[e^{-\int_{t}^{T}r_{u}^{c}\mathrm{d}u}Y_{T}-\int_{t}^{T}e^{-\int_{t}^{T}r_{s}^{c}\mathrm{d}s}(r_{u}^{f}-r_{u}^{c})(C_{u}-Y_{u})\mathrm{d}u\big{|}\mathcal{F}_{t}\right]\,,$
(1.6)

where the collateral $C$ and the short rates $r^{c},r^{f}$ are sufficiently
regular, such that the above expression is well posed: in applications, $C$ is
assumed to be a Lipschitz function of $Y$. Expressions of the aforementioned
form have been then justified via BSDE-techniques in, e.g., Bielecki and
Rutkowski (2015), Bichuch et al. (2018), Brigo et al. (2019), Biagini et al.
(2021) and Gnoatto and Seiffert (2021). Such recursive expectations are now
well understood, and are at least approximately implemented in several
industry-standard software packages via a combination of quadrature formulas
for the time integral, and Monte Carlo simulations for the expectation. To the
best of our knowledge, a comprehensive study of the change of measure/change
of numéraire technique applied to such conditional expectations, is still
missing. It is worth mentioning that in Cohen and Elliott (2015) the authors
observed that linear BSDEs can be used to represent a measure change, also
providing the formulation of a linear BSDE under two probability measures. The
conditional expectation they obtained does not exhibit a recursive structure
since the integrand is a predictable process not depending on the value
process nor on the controls. We take this idea one step further. Indeed, we
consider a fairly general structure of the driver which is relevant for the
aforementioned applications, allowing us to generalize the “change of
numéraire” technique, see Geman et al. (1995), so as to also cover the
recursive conditional expectation case.

The paper is organised as follows: in Section 2 we present the probabilistic
setup, whereas in Section 3 we derive our main result, i.e. Theorem 3.3,
subsequently, in Section 4, we provide some financial applications. Concerning
these latter, we exploit our Theorem 3.3 to analyse the market model
introduced in Bielecki and Rutkowski (2015) focusing on the change from the
physical to the risk-neutral measure, allowing us also to emphasize the link
between our results and the benchmark approach proposed by Platen and Heath
(2006). We also show how our method can be used to revisit the valuation of
exchange options in a setting including collateral, permitting us to deduce a
general procedure to perform changes of numéraire in non-linear market models.
A simple example of market driven by a jump process is also provided.

##  2 Notations and setting

Let $T\in\mathbb{R}_{+}$, $T<\infty$, be a fixed horizon time. We consider a
filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$, where
the filtration $\mathbb{F}\coloneqq\\{\mathcal{F}_{t},t\in\left[0,T\right]\\}$
is assumed to be complete and right continuous. For all semimartingales
considered in the sequel, we consider a version with right continuous paths
and left limits.

Let $d\in\mathbb{N}^{+}$. We assume that the filtered probability space
supports an $\mathbb{R}^{d}$-valued Brownian motion
$W^{\mathbb{P}}\coloneqq\\{W_{t}^{\mathbb{P}},t\in[0,T]\\}$, for a finite
positive $T$, and a Poisson random measure $N$, with compensator
$\vartheta(\mathrm{d}t,\mathrm{d}z)$ on $\mathbb{R}^{d}\setminus\\{0\\}$,
while $\vartheta$ is $\sigma$-finite and such that
$\int_{0}^{T}\int_{\mathbb{R}^{d}\setminus\\{0\\}}(1\wedge|z|^{2})\vartheta(\mathrm{d}t,\mathrm{d}z)<\infty$.
Moreover, we define the compensated random measure $\tilde{N}$, associated to
$N$, as follows

$\tilde{N}(\mathrm{d}t,\mathrm{d}z)\coloneqq
N(\mathrm{d}t,\mathrm{d}z)-\vartheta(\mathrm{d}t,\mathrm{d}z),$ (2.1)

and we introduce the following space of processes

  * •

$\mathcal{P}$: the predictable $\sigma$-field on
$\Omega\times\left[0,T\right]$ generated by all left continuous and
$\mathbb{F}$-adapted processes;

  * •

$\mathbb{L}_{T}^{2}(\mathbb{R}^{d})$: the space of all
$\mathcal{F}_{T}$-measurable random variables
$X:\Omega\rightarrow\mathbb{R}^{d}$ satisfying
$||X||^{2}\coloneqq\mathbb{E}^{\mathbb{P}}(|X|^{2})<+\infty$;

  * •

$\mathbb{H}_{T}^{2}(\mathbb{R}^{d})$: the space of all predictable process
$\phi:\Omega\times\left[0,T\right]\rightarrow\mathbb{R}^{d}$ such that
$||\phi||^{2}\coloneqq\mathbb{E}^{\mathbb{P}}\left[\int_{0}^{T}|\phi_{t}|^{2}\mathrm{d}t\right]<+\infty$.

As previously mentioned, the filtration $\mathbb{F}$ supports both the
Brownian motion $W$ and the Poisson random measure $N$. Let us provide the
statement of the predictable representation property in the present setting.

######  Lemma 2.1.

[See Lemma 3, 4.24 in Jacod and Shiryaev (2003)] Any $\mathbb{F}$-local
martingale $M$ has the representation

$M_{t}=M_{0}+\int_{0}^{t}Z_{s}\mathrm{d}W_{s}^{\mathbb{P}}+\int_{0}^{t}\int_{\mathbb{R}^{d}}U_{s}(z)\tilde{N}(\mathrm{d}s,\mathrm{d}z)\;,\;0\leq
t\leq T\;,$ (2.2)

where $Z$ and $U$ are $\mathcal{F}$-predictable processes integrable with
respect to both $W$ and $\tilde{N}$.

For the sake of completeness, let us also further specify the structure of the
random measure:

######  Assumption 2.2.

$N$ is an integer-valued random measure with compensator

$\vartheta(\mathrm{d}t,\mathrm{d}z)=Q(t,\mathrm{d}z)\eta(t)\mathrm{d}t$ (2.3)

where $\eta:\Omega\times[0,T]\rightarrow[0,\infty)^{d}$ is a predictable
process and $Q$ is a kernel from $(\Omega\times[0,T],\mathcal{P})$ into
$(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$ satisfying

$\int_{0}^{T}\int_{\mathbb{R}^{d}}|z|^{2}Q(t,\mathrm{d}z)\eta(t)\mathrm{d}t<\infty\;,$
(2.4)

and we set
$N(\\{0\\},\mathbb{R}^{d})=N((0,T],\\{0\\})=\vartheta((0,T],\\{0\\})=0$.

Following Delong (2013), we then introduce the standard spaces that allow us
to properly define the concept of solution to FBSDEs with jumps. We define

  * •

$\mathbb{H}_{T,N}^{2}(\mathbb{R}^{d})$, the space of all predictable process
$U:\Omega\times\left[0,T\right]\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$
satisfying
$\mathbb{E}^{\mathbb{P}}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}|U_{t}(z)|^{2}Q(t,\mathrm{d}z)\eta(t)\mathrm{d}t\right]<+\infty$,
the integral being considered w.r.t. to the aforementioned predictable
compensator $\tilde{N}$;

  * •

$\mathbb{S}_{T}^{2}(\mathbb{R}^{d})$ the space of $\mathbb{F}$-adapted càdlàg
processes $Y$, $Y:\Omega\times[0,T]\rightarrow\mathbb{R}^{d}$ satisfying
$\mathbb{E}^{\mathbb{P}}\left[\sup\limits_{t\in[o,T]}|Y_{t}|^{2}\right]<+\infty$.

##  3 BSDE with Jumps

Let us outline the FBSDE we consider in the present work.

###  3.1 Forward SDE

Let $T\in\mathbb{R}^{+}$, $T<\infty$. We consider the following
$\mathbb{R}^{d}$-valued forward SDE

$X_{s}=x+\int_{t}^{s}\mu(X_{r-})\mathrm{d}r+\int_{t}^{s}\sigma(X_{r-})\mathrm{d}W_{r}^{\mathbb{P}}\\\
+\int_{s}^{t}\int_{\mathbb{R}^{d}}\gamma(x_{r-},z)\tilde{N}(\mathrm{d}r,\mathrm{d}z)\;,\;0\leq
t\leq s\leq T\;,$ (3.1)

meaning that we are considering the following system of $\mathbb{R}$-valued
SDEs

$X_{s}^{i}=x^{i}+\int_{t}^{s}\mu^{i}(X_{r-})\mathrm{d}r+\int_{t}^{s}\sigma^{i}(X_{r-})\mathrm{d}W_{r}^{\mathbb{P}}\\\
+\int_{s}^{t}\int_{\mathbb{R}^{d}}\gamma^{i}(x_{r-},z)\tilde{N}(\mathrm{d}r,\mathrm{d}z)\;,\quad
0\leq t\leq s\leq T\;,\;1\leq i\leq d.$ (3.2)

Here, the mappings $\mu^{i}:\mathbb{R}^{d}\rightarrow\mathbb{R}$,
$\sigma^{i}:\mathbb{R}^{d}\rightarrow\mathbb{R}$,
$\gamma^{i}:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ are all
assumed to be measurable for any $i\in\\{1,\ldots,n\\}$.

Moreover, for any $i\in\\{1,\ldots,n\\}$ and $K$ denoting, for better
readability, a universal constant such that all below definitions are well
posed, we assume that:

  * (A1)

the functions $\mu^{i}$ and $\sigma^{i}$ are Lipschitz continuous;

  * (A2)

the functions $\gamma^{i}$ are measurable and satisfy

$\displaystyle|\gamma^{i}(x,z)|$ $\displaystyle\leq
K(1\wedge|z|)\;,\;(x,z)\in\mathbb{R}^{d}\times\mathbb{R}^{d}$
$\displaystyle|\gamma^{i}(x,z)-\gamma^{i}(x^{\prime},z)|$ $\displaystyle\leq
K|x-x^{\prime}|(1\wedge|z|)\;,\;(x,z),(x^{\prime},z)\in\mathbb{R}^{d}\times\mathbb{R}^{d};$

  * (A3)

$N$ indicates a Poisson random measure generated by a Lévy process with Lévy
measure $\nu$, namely: $\nu(\\{0\\})=0$ and
$\int_{\mathbb{R}^{d}}1\wedge|z|^{2}\nu(\mathrm{d}z)<\infty$.

It is worth noting that Assumption (A3) allows us to simplify Assumption 2.2.
Indeed, setting: $\eta(t)\equiv 1$ and
$Q(t,\mathrm{d}z)=\nu^{\mathbb{P}}(\mathrm{d}z)$ the jumps are those of a
time-homogeneous Lévy process. Furthermore, we denote with
$(X_{s}^{t,x})_{t\leq s\leq T}$ the solution to (3.1) with initial state
$(t,x)$.

######  Theorem 3.1.

Under Assumptions (A1)-(A3)

  * a)

For each $(t,x)\in[0,T]\times\mathbb{R}^{d}$ there exists a unique adapted,
càdlàg solution $X^{t,x}\coloneqq(X_{s}^{t,x})_{t\leq s\leq T}$ to (3.1).

  * b)

The solution $X^{t,x}$ is a homogeneous Markov process.

###### Proof.

Case a), resp. case b), follows from Theorem 6.2.9., resp. from Theorem 6.4.6,
in Applebaum (2009). ∎

###  3.2 Related BSDE

Let us consider the following backward SDE (BSDE) characterized by a terminal
condition and a generator depending on the state process solving the FSDE:
(3.1)

$Y_{s}^{t,x}=g(X_{T}^{t,x})+\int_{s}^{T}f\left(r,X_{r-}^{t,x},Y_{r-}^{t,x},Z_{r}^{t,x},\int_{\mathbb{R}^{d}}U_{r}^{t,x}(z)\delta(z)\nu^{\mathbb{P}}(\mathrm{d}z)\right)\mathrm{d}r\\\
-\int_{s}^{T}\alpha(X_{r}^{t,x})Y_{r}^{t,x}\mathrm{d}r+\int_{s}^{T}\beta(X_{r}^{t,x})Z_{r}^{t,x}\mathrm{d}r+\int_{s}^{T}\int_{\mathbb{R}^{d}}U_{r}^{t,x}(z)\delta(z)\nu^{\mathbb{P}}(\mathrm{d}z)\mathrm{d}r\\\
-\int_{s}^{T}Z_{r}^{t,x}\mathrm{d}W_{r}^{\mathbb{P}}-\int_{s}^{T}\int_{\mathbb{R}^{d}}U_{r}^{t,x}(z)\tilde{N}(\mathrm{d}r,\mathrm{d}z).$
(3.3)

Again considering a suitable constant $K>0$, we assume that the function:

  * (A4)

$f:[0,T]\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$
is Lipschitz continuous, i.e.

$|f(t,x,y,z,u)-f^{\prime}(t,x^{\prime},y^{\prime},z^{\prime},u^{\prime})|\\\
\leq
K\left(|x-x^{\prime}|+|y-y^{\prime}|+|z-z^{\prime}|+|u-u^{\prime}|\right)$
(3.4)

for all
$(t,x,y,z,u),(t,x^{\prime},y^{\prime},z^{\prime},u^{\prime})\in[0,T]\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}$,
uniformly in $t$;

  * (A5)

$g:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is measurable and satisfies

$|g(x)-g(x^{\prime})|\leq
K\left(|x-x^{\prime}|\right)\;,\;x,x^{\prime}\in\mathbb{R}^{d};$ (3.5)

  * (A6)

$\alpha:\mathbb{R}^{d}\rightarrow\mathbb{R}$ satisfies
$|\alpha(X_{r}^{t,x})|\leq K,\;t\leq r\leq T$;

  * (A7)

$\beta:\mathbb{R}^{d}\rightarrow\mathbb{R}$ satisfies
$|\beta(X_{r}^{t,x})|\leq K,\;t\leq r\leq T$;

  * (A8)

$\delta:\mathbb{R}^{d}\rightarrow\mathbb{R}$ satisfies
$\int_{\mathbb{R}^{d}}|\delta(z)|^{2}Q^{\mathbb{P}}(t,\mathrm{d}z)\leq
K,\;t\leq r\leq T$ and $\delta(z)>-1$, $z\in\mathbb{R}^{d}$.

######  Theorem 3.2.

[See Theorem 4.1.3 in Delong (2013)] Under assumptions (A1)–(A5), there exists
a unique solution
$((X_{s}^{t,x}),(Y_{s}^{t,x}),(Z_{s}^{t,x}),(U_{s}^{t,x}))\in\mathbb{S}_{T}^{2}(\mathbb{R}^{d})\times\mathbb{S}_{T}^{2}(\mathbb{R})\times\mathbb{H}_{T}^{2}(\mathbb{R}^{d})\times\mathbb{H}_{T,N}^{2}(\mathbb{R}^{d})$
to the forward-backward SDE (FBSDE) (3.1), (3.3).

In what follows we provide the main result of the paper, namely a
representation for the value process of the FBSDE (3.1 \- 3.3) under different
probability measures.

######  Theorem 3.3.

Under the preceeding assumptions, let $T\in\mathbb{R}^{+}$, $T<\infty$ and
consider $0<t\leq T$. Let us define

$\displaystyle\begin{aligned}
\mathcal{E}_{t}^{T}&\coloneqq\exp\left\\{-\frac{1}{2}\int_{t}^{T}\beta(X_{r}^{t,x})^{2}\mathrm{d}r+\int_{t}^{T}\beta(X_{r}^{t,x})\mathrm{d}W^{\mathbb{P}}\right.\\\
&\quad\left.+\int_{t}^{T}\int_{\mathbb{R}^{d}}\ln(1+\delta(z))-\delta(z)\nu^{\mathbb{P}}(\mathrm{d}z)\mathrm{d}r+\int_{t}^{T}\int_{\mathbb{R}^{d}}\ln(1+\delta(z))\tilde{N}(\mathrm{d}r,\mathrm{d}z)\right\\}\\\
&=\frac{H_{T}}{H_{t}}\,,\end{aligned}$ (3.6)

i.e. $H=\\{H_{r}^{\mathbb{P}},r\in[t,T]\\}$ solves the FSDE

$\mathrm{d}H_{r}=H_{r}\left(\beta(X_{r}^{t,x})\mathrm{d}W_{r}^{\mathbb{P}}+\int_{\mathbb{R}^{d}}\delta(z)\tilde{N}(\mathrm{d}r,\mathrm{d}z)\right)\;,\;0<t\leq
r\leq T\;,$ (3.7)

then the process $Y^{t,x}$ in (3.3) admits the representation

$Y_{t}^{t,x}=\mathbb{E}^{\mathbb{P}}\left[g(X_{T}^{t,x})e^{-\int_{t}^{T}\alpha(X_{r}^{t,x})\mathrm{d}r}\mathcal{E}_{t}^{T}+\int_{t}^{T}e^{-\int_{t}^{r}\alpha(X_{s}^{t,x})\mathrm{d}s}\mathcal{E}_{t}^{r}\right.\\\
\left.\cdot
f\left(r,X_{r-}^{t,x},Y_{r-}^{t,x},Z_{r}^{t,x},\int_{\mathbb{R}^{d}}U_{r}^{t,x}(z)\delta(z)\nu^{\mathbb{P}}(\mathrm{d}z)\right)\mathrm{d}r\,\big{|}\,\mathcal{F}_{t}\right]\;\mathbb{P}\,\mbox{a.s.}\,$
(3.8)

and, $\mathcal{E}^{T}$ defined in (3.6) is a true
$(\mathbb{P},\mathcal{F})$-martingale.

Define

  * •

$\frac{\partial\mathbb{P}^{2}}{\partial\mathbb{P}}\bigg{|}_{\mathcal{F}_{r}}\coloneqq\mathcal{E}_{t}^{r}\;;$

  * •

$W_{r}^{\mathbb{P}^{2}}\coloneqq
W_{r}^{\mathbb{P}}-\int_{t}^{r}\beta(X_{s}^{t,x}))\mathrm{d}s\;;$

  * •

$\nu^{\mathbb{P}^{2}}(\mathrm{d}z)\coloneqq(1+\delta(z))\nu^{\mathbb{P}}(\mathrm{d}z)\;,$

then the above defined FBSDE, resp. the process $Y_{s}^{t,x}$, admits the
equivalent representation

$Y_{s}^{t,x}=g(X_{T}^{t,x})+\int_{s}^{T}f\left(r,X_{r-}^{t,x},Y_{r-}^{t,x},Z_{r}^{t,x},\int_{\mathbb{R}^{d}}U_{r}^{t,x}(z)\frac{\delta(z)}{1+\delta(z)}\nu^{\mathbb{P}^{2}}(\mathrm{d}z)\right)\mathrm{d}r\\\
-\int_{s}^{T}\alpha(X_{r}^{t,x})Y_{r}^{t,x}\mathrm{d}r+\int_{s}^{T}\int_{\mathbb{R}^{d}}U_{r}^{t,x}(z)\frac{\delta(z)}{1+\delta(z)}\nu^{\mathbb{P}^{2}}(\mathrm{d}z)\mathrm{d}r\\\
-\int_{s}^{T}Z_{r}^{t,x}\mathrm{d}W_{r}^{\mathbb{P}^{2}}-\int_{s}^{T}\int_{\mathbb{R}^{d}}U_{r}^{t,x}(z)\tilde{N}(\mathrm{d}r,\mathrm{d}z)\;,$
(3.9)

resp.

$Y_{t}^{t,x}=\mathbb{E}^{\mathbb{P}^{2}}\left[g(X_{T}^{t,x})e^{-\int_{t}^{T}\alpha(X_{r}^{t,x})\mathrm{d}r}+\int_{t}^{T}e^{-\int_{t}^{r}\alpha(X_{s}^{t,x})\mathrm{d}s}\right.\\\
\left.\cdot
f\left(r,X_{r-}^{t,x},Y_{r-}^{t,x},Z_{r}^{t,x},\int_{\mathbb{R}^{d}}U_{r}^{t,x}(z)\frac{\delta(z)}{1+\delta(z)}\nu^{\mathbb{P}^{2}}(\mathrm{d}z)\right)\mathrm{d}r\,\big{|}\,\mathcal{F}_{t}\right]\;\mathbb{P}^{2}\,\mbox{a.s.}\,.$
(3.10)

under the probability measure $\mathbb{P}^{2}$,.

###### Proof.

Thanks to Assumptions (A7) and (A8), the process $\mathcal{E}^{T}$ is a true
$(\mathbb{P},\mathcal{F})$-martingale by Proposition 2.5.1 in Delong (2013).
Let us define
$B_{r}\coloneqq\exp{\int_{0}^{r}\alpha(X_{s}^{t,x})\mathrm{d}s}$. We consider
the square integrable martingale

$\displaystyle
M_{s}=\mathbb{E}^{\mathbb{P}}\left[H_{T}\frac{g(X_{T}^{t,x})}{B_{T}}+\int_{t}^{T}\frac{H_{r}}{B_{r}}f\left(r,X_{r-}^{t,x},Y_{r-}^{t,x},Z_{r}^{t,x},\int_{\mathbb{R}^{d}}U_{r}^{t,x}(z)\delta(z)\nu^{\mathbb{P}}(\mathrm{d}z)\right)\mathrm{d}r\,\big{|}\,\mathcal{F}_{s}\right],$

for $t\leq s\leq T$. The martingale representation theorem implies the
existence of stochastic processes
$\tilde{Z}\in\mathbb{H}_{T}^{2}(\mathbb{R}^{d})$,
$\tilde{U}\in\mathbb{H}_{T,N}^{2}(\mathbb{R}^{d})$, such that

$\displaystyle
M_{s}=M_{t}+\int_{t}^{s}\tilde{Z}_{r}^{t,x}\mathrm{d}W_{r}^{\mathbb{P}}+\int_{t}^{s}\int_{\mathbb{R}^{d}}\tilde{U}_{r}^{t,x}(z)\tilde{N}(\mathrm{d}r,\mathrm{d}z)\;.$
(3.11)

Setting
$\tilde{Z}_{r}^{t,x}\coloneqq\frac{H_{r}}{B_{r}}\left(Z_{r}^{t,x}+Y_{r}^{t,x}\beta(X_{r}^{t,x})\right)$
and
$\tilde{U}_{r}^{t,x}(z)\coloneqq\frac{H_{r}}{B_{r}}\left(U_{r}^{t,x}(z)(1+\delta(z))+Y_{r}^{t,x}\delta(z)\right)$,
the Itô-Döblin lemma implies

$\displaystyle\mathrm{d}\left(\frac{Y_{s}^{t,x}}{B_{s}}\right)=$
$\displaystyle\frac{1}{H_{s}}\left(\mathrm{d}\left(H\frac{Y_{s}^{t,x}}{B_{s}}\right)-\frac{Y_{s}^{t,x}}{B_{s}}\mathrm{d}H_{s}-\mathrm{d}\left[H,\frac{Y_{s}^{t,x}}{B_{s}}\right]\right)$
(3.12) $\displaystyle=$
$\displaystyle\frac{1}{H_{s}}\left(-\frac{H_{s}}{B_{s}}f\left(s,X_{s-}^{t,x},Y_{s-}^{t,x},Z_{s}^{t,x},\int_{\mathbb{R}^{d}}U_{r}^{t,x}(z)\delta(z)\nu^{\mathbb{P}}(\mathrm{d}z)\right)\mathrm{d}s\right.$
(3.13)
$\displaystyle+\tilde{Z_{s}}\mathrm{d}W_{s}^{\mathbb{P}}-\frac{Y_{s}^{t,x}}{B_{s}}H_{s}\beta(X_{s}^{t,x})\mathrm{d}W_{s}^{\mathbb{P}}-\frac{Z_{s}^{t,x}}{B_{s}}H_{s}\beta(X_{s}^{t,x})\mathrm{d}s$
(3.14)
$\displaystyle+\int_{\mathbb{R}^{d}}\frac{H_{s}}{B_{s}}\left(U_{s}^{t,x}(z)(1+\delta(z))+Y_{s}^{t,x}\delta(z)\right)\tilde{N}(\mathrm{d}s,\mathrm{d}z)$
(3.15)
$\displaystyle-\int_{\mathbb{R}^{d}}\frac{Y_{s}^{t,x}}{B_{s}}H_{s}\delta(z)\tilde{N}(\mathrm{d}s,\mathrm{d}z)$
(3.16)
$\displaystyle\left.-\int_{\mathbb{R}^{d}}\frac{1}{B_{s-}}H_{s-}\delta(z)U_{s-}^{t,x}N(\mathrm{d}s,\mathrm{d}z)\pm\int_{\mathbb{R}^{d}}\frac{1}{B_{s-}}H_{s-}\delta(z)U_{s-}^{t,x}\nu^{\mathbb{P}}(\mathrm{d}z)\mathrm{d}s\right)$
(3.17) $\displaystyle=$
$\displaystyle-\frac{1}{B_{s}}f\left(s,X_{s-}^{t,x},Y_{s-}^{t,x},Z_{s}^{t,x},\int_{\mathbb{R}^{d}}U_{s}^{t,x}(z)\delta(z)\nu^{\mathbb{P}}(\mathrm{d}z)\right)\mathrm{d}s+\frac{Z_{s}^{t,x}}{B_{s}}\mathrm{d}W_{s}^{\mathbb{P}}$
(3.18)
$\displaystyle-\frac{Z_{s}^{t,x}}{B_{s}}\beta(X_{s}^{t,x})\mathrm{d}s-\frac{1}{B_{s}}\int_{\mathbb{R}^{d}}\delta(z)U_{s}^{t,x}(z)\nu^{\mathbb{P}}(\mathrm{d}z)\mathrm{d}s+\int_{\mathbb{R}^{d}}\frac{1}{B_{s}}U_{s}^{t,x}(z)\tilde{N}(\mathrm{d}s,\mathrm{d}z),$
(3.19)

and since we also have

$\mathrm{d}\left(\frac{Y_{s}^{t,x}}{B_{s}}\right)=\frac{1}{B_{s}}\mathrm{d}Y_{s}^{t,x}-\frac{Y_{s}^{t,x}}{B_{s}^{2}}\mathrm{d}B_{s},$
(3.20)

then

$\displaystyle\mathrm{d}Y_{s}^{t,x}$
$\displaystyle=B_{s}\mathrm{d}\left(\frac{Y_{s}^{t,x}}{B_{s}}\right)+\frac{Y_{s}^{t,x}}{B_{s}}\mathrm{d}B_{s}$
(3.21) $\displaystyle=$
$\displaystyle-f\left(s,X_{s-}^{t,x},Y_{s-}^{t,x},Z_{s}^{t,x},\int_{\mathbb{R}^{d}}U_{s}^{t,x}(z)\delta(z)\nu^{\mathbb{P}}(\mathrm{d}z)\right)\mathrm{d}s$
(3.22)
$\displaystyle+Z_{s}^{t,x}\mathrm{d}W_{s}^{\mathbb{P}}-Z_{s}^{t,x}\beta(X_{s}^{t,x})\mathrm{d}s$
(3.23)
$\displaystyle-\int_{\mathbb{R}^{d}}\delta(z)U_{s}^{t,x}(z)\nu^{\mathbb{P}}(\mathrm{d}z)\mathrm{d}s+\int_{\mathbb{R}^{d}}U_{s}^{t,x}(z)\tilde{N}(\mathrm{d}s,\mathrm{d}z)+\alpha(X_{s}^{t,x})Y_{s}^{t,x}\mathrm{d}s\;\,.$
(3.24)

Summing up the previous points we recover (3.3), namely:

$-\mathrm{d}Y_{s}^{t,x}=\left(f\left(s,X_{s-}^{t,x},Y_{s-}^{t,x},Z_{s}^{t,x},\int_{\mathbb{R}^{d}}U_{s}^{t,x}(z)\delta(z)\nu^{\mathbb{P}}(\mathrm{d}z)\right)+Z_{s}^{t,x}\beta(X_{s}^{t,x})\right.\\\
\left.+\int_{\mathbb{R}^{d}}\delta(z)U_{s}^{t,x}(z)\nu^{\mathbb{P}}(\mathrm{d}z)-\alpha(X_{s}^{t,x})Y_{s}^{t,x}\right)\mathrm{d}s-Z_{s}^{t,x}\mathrm{d}W_{s}^{\mathbb{P}}-\int_{\mathbb{R}^{d}}U_{s}^{t,x}(z)\tilde{N}(\mathrm{d}s,\mathrm{d}z)\;.$
(3.25)

Finally, since $\mathcal{E}^{T}$ is a true
$(\mathbb{P},\mathcal{F})$-martingale, we can apply the Girsanov Theorem as in
Theorem 2.5.1 in Delong (2013). By repeating the previous steps under the
equivalent measure $\mathbb{P}^{2}$, we have that the new representation of
the process $Y_{s}^{t,x}$ under $\mathbb{P}^{2}$ reads as follows:

$Y_{t}^{t,x}=\mathbb{E}^{\mathbb{P}^{2}}\left[g(X_{T}^{t,x})e^{-\int_{t}^{T}\alpha(X_{r}^{t,x})\mathrm{d}r}+\int_{t}^{T}e^{-\int_{t}^{r}\alpha(X_{s}^{t,x})\mathrm{d}s}\right.\\\
\left.\cdot
f\left(r,X_{r-}^{t,x},Y_{r-}^{t,x},Z_{r}^{t,x},\int_{\mathbb{R}^{d}}U_{r}^{t,x}(z)\frac{\delta(z)}{1+\delta(z)}\nu^{\mathbb{P}^{2}}(\mathrm{d}z)\right)\mathrm{d}r\,\big{|}\,\mathcal{F}_{t}\right]\;\mathbb{P}^{2}\,\mbox{a.s.}\,.$
(3.26)

∎

We notice that changing the measure in (3.10) results in the appearance of the
density process $\mathcal{E}^{T}$ both next to the terminal condition (which
is standard) and inside the integral term which is an uncommon feature of the
non-linear and recursive context we are considering.

##  4 Financial Applications

In what follows, we adopt the setup of Bielecki and Rutkowski (2015), see also
Gnoatto and Seiffert (2021). In particular, we fix a positive but finite time
horizon $T$. All stochastic processes introduced in the sequel are defined on
a filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$
where the filtration
$\mathbb{F}\coloneqq\\{\mathcal{F}_{t},t\in\left[0,T\right]\\}$ satisfies
usual assumptions, while $\mathcal{F}_{0}$ is the trivial $\sigma-$algebra at
initial time. All processes introduced in the sequel are assumed to be
$\mathbb{F}$-adapted and we work with càdlàg versions.

We denote by $S^{i}$ the price (ex-dividend price) of the i-th risky asset
with cumulative dividend stream $A^{i}$ with $i=1,\ldots,d$. We take the
perspective of the hedger, i.e., the entity that sells a certain contract and
constructs a hedging portfolio against the liability stemming from the
contract. Let $\xi_{t}^{i}$ denote the number of the hedger’s positions in
asset $S^{i}$ at time $t$. We let $B^{0}=B$ be the cash account for unsecured
funding and introduce $B^{1},\ldots,B^{d}$ cash accounts allowing for the
financing of repurchase agreements (repo trading) on the risky assets. We
assume that:

  * i)

$S^{i}$ for $i=1,\ldots,d$ are semimartingales;

  * ii)

$A^{i}$ for $i=1,\ldots,d$ are finite variation processes with $A_{0}^{i}=0$;

  * iii)

$B^{j}$ for $j=1,\ldots,d$ are strictly positive and continuous processes of
finite variation with $B_{0}^{j}=1$.

We introduce the cumulative dividend price $S^{i,cld}$, given by

$S_{t}^{i,cld}\coloneqq
S_{t}^{i}+B_{t}^{i}\int_{(0,t]}(B_{u}^{i})^{-1}\mathrm{d}A_{u}^{i}\;,\quad
t\in[0,T]\;,$ (4.1)

together with its discounted version

$\hat{S}_{t}^{i,cld}\coloneqq\hat{S}_{t}^{i}+\int_{(0,t]}(B_{u}^{i})^{-1}\mathrm{d}A_{u}^{i}\;,$
(4.2)

with $\hat{S}_{t}^{i,cld}=(B^{i})^{-1}S^{i}$.

A trading strategy, or dynamic portfolio, is composed of positions in risky
assets $\xi^{i}$, $i=1,\ldots,d$, positions in unsecured funding repo accounts
$\psi^{j}$, $i=0,\ldots,d$, and the cash account related to lending and
borrowing collateral $\psi^{c,l}$, $\psi^{c,b}$

$\varphi=(\xi^{1},\ldots,\xi^{d},\psi^{0},\ldots,\psi^{d},\psi^{c,l},\psi^{c,b}).$
(4.3)

Processes $(\xi^{1},\ldots,\xi^{d})$ are assumed to be
$\mathbb{F}$-predictable, while
$(\psi^{0},\ldots,\psi^{d},\psi^{c,l},\psi^{c,b})$ are $\mathbb{F}$-adapted,
allowing the stochastic integrals we then introduce to be well defined.
Trading strategies are subject to admissibility constraints. A contract is
represented by means of its cumulative stream of cash flows, hence by a
process $A$ of finite variation with càdlàg paths, such that $A_{0^{-}}=0$. As
to provide an example, if the hedger sells a standard call option written on
$S^{i}$, with maturity time $T$ and strike price $\mathcal{K}$, at a price
$p$, then
$A_{t}=p\mathbbm{1}_{[0,T]}(t)-(S_{T}^{i}-\mathcal{K})^{+}\mathbbm{1}_{[T]}(t)$.

To prevent arbitrage opportunities via trivial positions in the cash accounts,
we impose the repo constraint

$\psi_{t}^{i}B_{t}^{i}+\xi_{t}^{i}S_{t}^{i}=0\;,\;\mathrm{d}\mathbb{P}\otimes\mathrm{d}t-a.s.$
(4.4)

meaning that the holdings in the i-th cash account is instrumental to the
purpose of financing the position in the i-th risky asset only.

In the setting of Bielecki and Rutkowski (2015), the meaning of a martingale
measure is that of a measure such that all processes $\hat{S}^{i,cld}$ are
local martingales.

######  Proposition 4.1.

[See Proposition 3.1 in Bielecki and Rutkowski (2015)] Assume that all
strategies available to the hedger are admissible and satisfy the repo
constraint (4.4). If $\exists$ a probability measure $\mathbb{Q}$ on
$(\Omega,\mathcal{F}_{T})$ such that $\mathbb{Q}\sim\mathbb{P}$ and the
processes $\hat{S}^{i,cld}$, $i=1,\ldots,d$ are local
$(\mathbb{Q},\mathbb{F})$ martingales, then the market model is free of
arbitrage opportunities for the hedger.

Let us now specialize the above introduced setting by means of the following
assumptions:

  1. 1.

All processes $B^{j}$ are absolutely continuous, meaning that they can be
written as $\mathrm{d}B_{t}^{j}=r_{t}^{j}B_{t}^{j}\mathrm{d}t$ for some
$\mathbb{F}$-adapted processes $r^{j}$, $j=0,\ldots,d$. To simplify the
treatment of existence and uniqueness of BSDEs, we assume that such rates are
bounded.

  2. 2.

We let $W^{1,\mathbb{P}},\ldots,W^{d,\mathbb{P}}$ be standard mutually
independent Brownian motions and we define
$W^{\mathbb{P}}=\begin{pmatrix}W^{1,\mathbb{P}}\\\ \vdots\\\
W^{d,\mathbb{P}}\end{pmatrix}$.

  3. 3.

We introduce vector fields
$\mu^{1},\ldots,\mu^{d},\sigma^{1},\ldots,\sigma^{d}$ which are assumed to be
bounded stochastic process, and we write

$\mu=\begin{pmatrix}\mu^{1}\\\ \vdots\\\
\mu^{d}\end{pmatrix}\;,\;\Sigma=\begin{pmatrix}\sigma^{1}&&\\\ &\ddots&\\\
&&\sigma^{d}\end{pmatrix}\;.$ (4.5)

Each asset is then characterized by the following dividend process

$A_{t}^{i}=\int_{0}^{t}k_{u}^{i}S_{u}^{i}\mathrm{d}u\;,$ (4.6)

where $k^{i}$, $i=1,\ldots,d$ are $\mathbb{F}$-adapted bounded stochastic
processes,

while the risky assets evolve according to

$\mathrm{d}S=\mathrm{d}\begin{pmatrix}S_{t}^{1}\\\ \vdots\\\
S_{t}^{d}\end{pmatrix}=diag(S)\left(\mu_{t}\mathrm{d}t+\Sigma_{t}\bar{\rho}_{t}\mathrm{d}W_{t}^{\mathbb{P}}\right)\;,$
(4.7)

where $\bar{\rho}_{t}$ is a matrix such that
$\rho_{t}=\bar{\rho}_{t}\bar{\rho}_{t}^{\intercal}$ represents the correlation
matrix between $W^{1},\ldots,W^{d}$. Moreover, $\bar{\rho}$ is such that
$\Sigma_{t}\bar{\rho}_{t}$ is invertible, for every $t\in[0,T]$. Given the
previous setting, let us recall the shape of the martingale measure
$\mathbb{Q}$ as stated in Lemma 5.2 in Bielecki and Rutkowski (2015).

######  Lemma 4.2.

Under the measure $\mathbb{Q}$, risky assets evolve according to

$\mathrm{d}\begin{pmatrix}S_{t}^{1}\\\ \vdots\\\
S_{t}^{d}\end{pmatrix}=diag(S)\left(\begin{pmatrix}r_{t}^{1}-k_{t}^{1}\\\
\vdots\\\
r_{t}^{d}-k_{t}^{d}\end{pmatrix}\mathrm{d}t+\Sigma_{t}\bar{\rho}_{t}\mathrm{d}W_{t}^{\mathbb{Q}}\right),$
(4.8)

where the multivariate Brownian motion $W^{\mathbb{Q}}$ is given as

$W^{\mathbb{Q}}\coloneqq
W_{t}^{\mathbb{P}}+\int_{0}^{t}\left(\Sigma_{u}\bar{\rho}_{u}\right)^{-1}\begin{pmatrix}\mu_{u}^{1}-r_{u}^{1}+k_{u}^{1}\\\
\vdots\\\ \mu_{u}^{d}-r_{u}^{d}+k_{u}^{d}\end{pmatrix}\mathrm{d}u.$ (4.9)

Given the assumptions 1 and 3 and denoting

$\displaystyle\theta_{u}\coloneqq\left(\Sigma_{u}\bar{\rho}_{u}\right)^{-1}\begin{pmatrix}\mu_{u}^{1}-r_{u}^{1}+k_{u}^{1}\\\
\vdots\\\ \mu_{u}^{d}-r_{u}^{d}+k_{u}^{d}\end{pmatrix},$ (4.10)

it is immediate to conclude that the stochastic exponential

$\mathcal{E}_{0}^{t}=\exp{-\int_{0}^{t}\theta_{u}^{\intercal}\mathrm{d}W_{u}-\frac{1}{2}\int_{0}^{t}\theta_{u}^{\intercal}\theta_{u}\mathrm{d}u}$
(4.11)

is a true martingale such that $W^{\mathbb{Q}}$ defined in (4.9) is a Brownian
motion according to the Girsanov theorem.

###  4.1 Wealth Dynamics and Pricing under Different Measures

In what follows, we analyse the situation where collateral is exchanged in
continuous time between the hedger and the counterparty to reduce possibly
outstanding credit exposures.

Under the previously introduced probability space, we denote by $C$ an
$\mathbb{F}$-adapted stochastic process representing the exchanged collateral.
$C^{+}$ represents the collateral received while $C^{-}$ is the collateral
posted by the hedger. Collateral posted, resp. received, earns, resp. pays, an
interest at a rate $r^{c,l}$, resp. $r^{c,b}$, and we set $C_{T}=0$, meaning
that all collateral is returned at the terminal time when trading stops. The
interest rates $r^{c,l},r^{c,b}$ are assumed to be $\mathbb{F}$-adapted and
bounded processes. When collateral is posted via cash and the receiver is
allowed to use it to fund own trading activity, then, according to equations
5.23 and 5.24 in Bielecki and Rutkowski (2015), the wealth dynamics are as
follows:

$\mathrm{d}V_{t}(\varphi)=r_{t}V_{t}(\varphi)\mathrm{d}t+\sum_{i=1}^{d}\xi_{t}^{i}(\mathrm{d}S_{t}^{i}-r_{t}^{i}S_{t}^{i}\mathrm{d}t+k_{t}^{i}S_{t}^{i}\mathrm{d}t)+\mathrm{d}\hat{F}_{t}^{h}+\mathrm{d}A_{t},$
(4.12)

where

$\hat{F}_{t}^{h}\coloneqq\int_{0}^{t}C_{u}^{+}(r_{u}-r_{u}^{c,b})\mathrm{d}u-\int_{0}^{t}C_{u}^{-}(r_{u}-r_{u}^{c,l})\mathrm{d}u.$
(4.13)

We are then in a position to apply the representation stated in Section 3.2 so
as to obtain pricing formulas under different measures.

######  Assumption 4.3.

$A_{t}=p\mathbbm{1}_{[0,T]}(t)+X\mathbbm{1}_{[T]}(t)$ for
$X\in\mathcal{F}_{T}=\mathcal{F}_{T}^{S}$, where by
$\mathbb{F}=\mathbb{F}^{S}$ we mean that the filtration is generated by the
risky assets, and $p$ is the initial price.

The collateral process is of the form $C=c(V)$, where $c$ is a Lipschitz
function.

Let us now apply Theorem 3.3. We can substitute the $\mathbb{P}$ dynamics in
(4.7) inside (4.12) as

$\displaystyle\begin{split}\mathrm{d}V_{t}(\varphi)&=r_{t}V_{t}(\varphi)\mathrm{d}t+\sum_{i=1}^{d}\xi_{t}^{i}(\mu_{t}^{i}S_{t}^{i}\mathrm{d}t+S_{t}^{i}\left(\Sigma_{t}\bar{\rho}_{t}\mathrm{d}W_{t}^{\mathbb{P}}\right)_{i}-r_{t}^{i}S_{t}^{i}\mathrm{d}t+k_{t}^{i}S_{t}^{i}\mathrm{d}t)\\\
&\qquad\qquad\qquad\qquad\qquad\qquad+\mathrm{d}\hat{F}_{t}^{h}+\mathrm{d}A_{t}\end{split}$
(4.14)
$\displaystyle\begin{split}&=r_{t}V_{t}(\varphi)\mathrm{d}t+\left\langle\xi_{t}diag(S_{t})\Sigma_{t}\bar{\rho}_{t},\left(\Sigma_{u}\bar{\rho}_{u}\right)^{-1}\begin{pmatrix}\mu_{u}^{1}-r_{u}^{1}+k_{u}^{1}\\\
\vdots\\\
\mu_{u}^{d}-r_{u}^{d}+k_{u}^{d}\end{pmatrix}\right\rangle\mathrm{d}t\\\
&\qquad\qquad\qquad\qquad\qquad\qquad+\left\langle\xi_{t}diag(S_{t})\Sigma_{t}\bar{\rho}_{t},\mathrm{d}W_{t}^{\mathbb{P}}\right\rangle+\mathrm{d}\hat{F}_{t}^{h}+\mathrm{d}A_{t}\;.\end{split}$
(4.15)

We set $Z\coloneqq\xi diag(S)\Sigma\bar{\rho}\,\in\mathbb{R}^{d}$ and $\theta$
as in (4.10),therefore, we are left with a BSDE providing a non-linear pricing
rule, i.e.:

$\mathrm{d}V_{t}(\varphi)=r_{t}V_{t}(\varphi)\mathrm{d}t+Z_{t}^{\intercal}\theta_{t}\mathrm{d}t+Z_{t}^{\intercal}\mathrm{d}W_{t}^{\mathbb{P}}+(r_{t}-r_{t}^{c,b})C_{t}^{+}\mathrm{d}t\\\
-(r_{t}-r_{t}^{c,l})C_{t}^{-}\mathrm{d}t+\mathrm{d}A_{t}.$ (4.16)

Therefore, by the results in Section 3.2, we obtain the following version of
Proposition 5.4 in Bielecki and Rutkowski (2015) under $\mathbb{P}$.

######  Lemma 4.4.

Let $X$ be $\mathcal{F}_{T}^{S}$-measurable with
$X\in\mathbb{L}_{T}^{2}(\Omega,\mathcal{F}_{T}^{S})$, then there exists a
unique solution
$(V,Z)\in\mathbb{S}^{2}_{T}(\mathbb{R})\times\mathbb{H}^{2}_{T}(\mathbb{R})$
to the BSDE (4.16). The contract
$A_{t}=p\mathbbm{1}_{[0,T]}(t)+X\mathbbm{1}_{[T]}(t)$ with collateral
specification b) can be replicated by an admissible trading strategy $\xi$. We
have $p_{t}=V_{t}$ and the process $V$ admits the following representation
under $\mathbb{P}$:

$p_{t}=V_{t}(\varphi)=-\mathbb{E}^{\mathbb{P}}\left[\frac{B_{t}}{B_{T}}X\mathcal{E}_{t}^{T}+\int_{t}^{T}\frac{B_{t}}{B_{u}}\mathcal{E}_{t}^{u}\right.\\\
\left.\left[(r_{u}-r_{u}^{c,b})C_{u}^{+}-(r_{u}-r_{u}^{c,l})C_{u}^{-}\right]\mathrm{d}u|\mathcal{F}_{t}\right].$
(4.17)

######  Remark 4.5.

In the formula above, we recognize the term
$\frac{B_{t}}{B_{T}}\mathcal{E}_{t}^{T}$ as the stochastic discount factor or
state price density in the terminology of financial economics, see e.g. Duffie
(2001), Barucci and Fontana (2017).

Next, by exploiting the result derived in Theorem 3.3, see eq.(3.10), we can
equivalently rewrite the BSDE under the measure $\mathbb{Q}$ as

$\mathrm{d}V_{t}(\varphi)=r_{t}V_{t}(\varphi)\mathrm{d}t+Z_{t}^{\intercal}\mathrm{d}W_{t}^{\mathbb{Q}}+(r_{t}-r_{t}^{c,b})C_{t}^{+}\mathrm{d}t-(r_{t}-r_{t}^{c,l})C_{t}^{-}\mathrm{d}t+\mathrm{d}A_{t}\;,$
(4.18)

allowing us to reformulate Proposition 5.4 in Bielecki and Rutkowski (2015).

######  Lemma 4.6.

Let $X$ be $\mathcal{F}_{T}^{S}$-measurable with
$X\in\mathbb{L}_{T}^{2}(\Omega,\mathcal{F}_{T}^{S})$, then $\exists!$ solution
$(V,Z)\in\mathbb{S}^{2}_{T}(\mathbb{R})\times\mathbb{H}^{2}_{T}(\mathbb{R})$
to the BSDE (4.18). The contract
$A_{t}=p\mathbbm{1}_{[0,T]}(t)+X\mathbbm{1}_{[T]}(t)$ with collateral
specification b) in Assumption 4.3 can be replicated by an admissible trading
strategy $\xi$. We have $p_{t}=V_{t}(\varphi)$ and the process $V_{t}$ admits
the representation under $\mathbb{Q}$:

$p_{t}=V_{t}(\varphi)=-\mathbb{E}^{\mathbb{Q}}\left[\frac{B_{t}}{B_{T}}X+\int_{t}^{T}\frac{B_{t}}{B_{u}}\left[(r_{u}-r_{u}^{c,b})C_{u}^{+}-(r_{u}-r_{u}^{c,l})C_{u}^{-}\right]\mathrm{d}u|\mathcal{F}_{t}\right].$
(4.19)

######  Remark 4.7.

A classical example for $X\in\mathbb{L}_{T}^{2}(\Omega,\mathcal{F}_{T}^{S})$
is given by a Lipschitz function of the risky assets $S^{1},\ldots,S^{d}$
which covers e.g., European call and put options.

###  4.2 Benchmark approach interpretation

In what follows we further analyse (4.17), from the Benchmark Approach point
of view. Let us explicitly state the form of the product of the ratio of cash-
accounts times the density process

$\displaystyle\frac{B_{t}}{B_{T}}\mathcal{E}_{t}^{T}$
$\displaystyle=\exp{-\int_{t}^{T}r_{s}\mathrm{d}s-\frac{1}{2}\int_{t}^{T}\theta_{s}^{\intercal}\theta_{s}\mathrm{d}s-\int_{t}^{T}\theta_{s}^{\intercal}\mathrm{d}W^{\mathbb{P}}}$
(4.20)
$\displaystyle=\frac{\exp{\int_{0}^{t}r_{s}\mathrm{d}s+\frac{1}{2}\int_{0}^{t}\theta_{s}^{\intercal}\theta_{s}\mathrm{d}s+\int_{0}^{t}\theta_{s}^{\intercal}\mathrm{d}W^{\mathbb{P}}}}{\exp{\int_{0}^{T}r_{s}\mathrm{d}s+\frac{1}{2}\int_{0}^{T}\theta_{s}^{\intercal}\theta_{s}\mathrm{d}s+\int_{0}^{T}\theta_{s}^{\intercal}\mathrm{d}W^{\mathbb{P}}}}.$
(4.21)

We recognize in this ratio the solution to the SDE satisfied by the Growth
Optimal Portfolio (GOP) in the benchmark approach of Platen and Heath (2006),
namely

$\frac{\mathrm{d}S_{t}^{\delta^{\text{*}}}}{S_{t}^{\delta^{\text{*}}}}=r_{t}\mathrm{d}t+\langle\theta_{t},\theta_{t}\mathrm{d}t+\mathrm{d}W_{t}^{\mathbb{P}}\rangle.$
(4.22)

It is worth recalling that the GOP is the portfolio maximizing the expected
logarithmic utility. In Platen and Heath (2006), the GOP plays a central role
in financial evaluation. Indeed, it has the role of the numéraire for the
physical probability measure $\mathbb{P}$, allowing for the derivation of a
pricing formula for contingent claims under $\mathbb{P}$, i.e. the so-called
real-world pricing formula. The latter generalizes the risk-neutral valuation
formula since it can be applied also when a risk-neutral measure does not
exist. Exploiting our previous calculations, we deduce that (4.17) can be
written as in the following corollary, hence extending the real world pricing
formula derived in Platen and Heath (2006) in the presence of collateral and
multiple interest rates.

######  Corollary 4.7.1.

[The real World Pricing Formula - Case with collateral] The price of the claim
$A_{t}=p\mathbbm{1}_{[0,T]}(t)+X\mathbbm{1}_{[T]}(t)$ with
$X\in\mathbb{L}_{T}^{2}(\Omega,\mathcal{F}_{T}^{S})$ is given by

$p_{t}=V_{t}(\varphi)=-\mathbb{E}^{\mathbb{P}}\left[\frac{S_{t}^{\delta^{\text{*}}}}{S_{T}^{\delta^{\text{*}}}}X+\int_{t}^{T}\frac{S_{t}^{\delta^{\text{*}}}}{S_{u}^{\delta^{\text{*}}}}\left[(r_{u}-r_{u}^{c,b})C_{u}^{+}-(r_{u}-r_{u}^{c,l})C_{u}^{-}\right]\mathrm{d}u|\mathcal{F}_{t}\right]\;,$
(4.23)

where $S_{t}^{\delta^{\text{*}}}$ is the growth optimal portfolio with
dynamics

$\frac{\mathrm{d}S_{t}^{\delta^{\text{*}}}}{S_{t}^{\delta^{\text{*}}}}=r_{t}\mathrm{d}t+\langle\theta_{t},\theta_{t}\mathrm{d}t+\mathrm{d}W_{t}^{\mathbb{P}}\rangle.$
(4.24)

###  4.3 Exchange option example

In this subsection we reconsider the classical example of an exchange option,
see Geman et al. (1995), but in the presence of collateral and multiple
interest rates. We define $\tilde{V}=\frac{V}{B}$ the discounted wealth
process. From Equation (4.18), the dynamics under the measure $\mathbb{Q}$ are
given by

$\mathrm{d}\tilde{V}_{t}(\varphi)=\frac{Z_{t}^{\intercal}}{B_{t}}\mathrm{d}W_{t}^{\mathbb{Q}}+\frac{1}{B_{t}}\left((r_{t}-r_{t}^{c,b})C_{t}^{+}-(r_{t}-r_{t}^{c,l})C_{t}^{-}\right)\mathrm{d}t+\frac{1}{B_{t}}\mathrm{d}A_{t}.$
(4.25)

From Proposition 4.1 we know that the ratio $\frac{S^{2}}{B^{2}}$ is a
$(\mathbb{Q},\mathbb{F})$-local martingale. Such ratio can be used to
construct a new probability measure. Let us first compute the dynamics of
$\frac{B^{2}}{S^{2}}$ under $\mathbb{Q}$ as

$\displaystyle\mathrm{d}\left(\frac{B_{t}^{2}}{S_{t}^{2}}\right)={}$
$\displaystyle\frac{B_{t}^{2}r_{t}^{2}\mathrm{d}t}{S_{t}^{2}}+B_{t}^{2}\left(-\frac{1}{(S_{t}^{2})^{2}}\left(r_{t}^{2}S_{t}^{2}\mathrm{d}t+S_{t}^{2}\left(\Sigma_{t}\bar{\rho}_{t}\mathrm{d}W_{t}^{\mathbb{Q}}\right)_{2}\right.\right.$
(4.26)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.\left.+\frac{1}{2}\frac{2}{(S_{t}^{2})^{3}}(S_{t}^{2})^{2}(\sigma^{2})^{2}\mathrm{d}t\right)\right)$
(4.27) $\displaystyle=$
$\displaystyle\frac{B_{t}^{2}}{S_{t}^{2}}\left((\sigma_{t}^{2})^{2}\mathrm{d}t-\sigma_{t}^{2}\sum_{j=1}^{d}\bar{\rho}_{2,j}\mathrm{d}W_{t}^{\mathbb{Q},j}\right)=\frac{B_{t}^{2}}{S_{t}^{2}}\left((\sigma_{t}^{2})^{2}\mathrm{d}t-\sigma_{t}^{2}\bar{\rho}_{2,\cdot,t}\mathrm{d}W_{t}^{\mathbb{Q}}\right)\;,$
(4.28)

where $\left(\Sigma_{t}\bar{\rho}_{t}\mathrm{d}W_{t}^{\mathbb{Q}}\right)_{2}$
refers to the second coordinate of the vector
$\left(\Sigma_{t}\bar{\rho}_{t}\mathrm{d}W_{t}^{\mathbb{Q}}\right)$ and
$\bar{\rho}_{2,\cdot,t}$ is the second row of $\bar{\rho}_{t}$. We can write

$\displaystyle\mathrm{d}\left(\frac{B_{t}^{2}}{S_{t}^{2}}\right)={}$
$\displaystyle\frac{B_{t}^{2}}{S_{t}^{2}}\left((\sigma_{t}^{2})^{2}\frac{\bar{\rho}_{2,\cdot,t}\bar{\rho}_{\cdot,2,t}}{\bar{\rho}_{2,\cdot,t}\bar{\rho}_{\cdot,2,t}}\mathrm{d}t-\sigma_{t}^{2}\bar{\rho}_{2,\cdot,t}\mathrm{d}W_{t}^{\mathbb{Q}}\right)$
(4.29) $\displaystyle={}$
$\displaystyle\frac{B_{t}^{2}}{S_{t}^{2}}\left(-\sigma_{t}^{2}\bar{\rho}_{2,\cdot,t}\left(\mathrm{d}W_{t}^{\mathbb{Q}}-\sigma_{t}^{2}\frac{\bar{\rho}_{\cdot,2,t}}{\bar{\rho}_{2,\cdot,t}\bar{\rho}_{\cdot,2,t}}\mathrm{d}t\right)\right)\;\,,$
(4.30)

where we have $\bar{\rho}_{2,\cdot,t}\bar{\rho}_{\cdot,2,t}=\rho_{2,2}=1$,
since $\bar{\rho}$ is a Cholesky factorization of the correlation matrix.

Define

  * •

$\frac{\partial\mathbb{Q}^{S^{2}}}{\partial\mathbb{Q}}\bigg{|}_{\mathcal{F}_{T}}\coloneqq\frac{S_{T}^{2}}{B_{T}^{2}}\frac{B_{0}^{2}}{S_{0}^{2}}\;;$

  * •

$\mathrm{d}W_{t}^{\mathbb{Q}^{S^{2}}}\coloneqq\mathrm{d}W_{t}^{\mathbb{Q}}-\sigma_{t}^{2}\frac{\left(\bar{\rho}_{2,\cdot,t}\right)^{\intercal}}{\bar{\rho}_{2,\cdot,t}\bar{\rho}_{\cdot,2,t}}\mathrm{d}t\;.$

Since

$\mathrm{d}\tilde{V}_{t}(\varphi)=\frac{1}{B_{t}}\sum_{i=1}^{d}\sum_{j=1}^{d}\xi_{t}^{1}S_{t}^{i}\sigma_{t}^{i}\bar{\rho}_{i,j}\mathrm{d}W_{t}^{\mathbb{Q},j}\\\
+\frac{1}{B_{t}}\left((r_{t}-r_{t}^{c,b})C_{t}^{+}-(r_{t}-r_{t}^{c,l})C_{t}^{-}\right)\mathrm{d}t+\frac{1}{B_{t}}\mathrm{d}A_{t}\;,$
(4.31)

then the dynamics of the product $\frac{V}{B}\frac{B^{2}}{S^{2}}$ is given by:

$\displaystyle\mathrm{d}\left(\frac{V}{B}\frac{B^{2}}{S^{2}}\right)_{t}$
$\displaystyle=\frac{B_{t}^{2}}{S_{t}^{2}}\left(\frac{1}{B_{t}}\sum_{i=1}^{d}\sum_{j=1}^{d}\xi_{t}^{1}S_{t}^{i}\sigma_{t}^{i}\bar{\rho}_{i,j}\mathrm{d}W_{t}^{\mathbb{Q},j}\right.$
(4.32)
$\displaystyle\qquad\quad\left.+\frac{1}{B_{t}}\left((r_{t}-r_{t}^{c,b})C_{t}^{+}-(r_{t}-r_{t}^{c,l})C_{t}^{-}\right)\mathrm{d}t+\frac{1}{B_{t}}\mathrm{d}A_{t}\right)$
(4.33)
$\displaystyle\qquad\quad+\frac{V_{t}}{B_{t}}\frac{B_{t}^{2}}{S_{t}^{2}}\left((\sigma_{t}^{2})^{2}\mathrm{d}t-\sigma_{t}^{2}\sum_{j=1}^{d}\bar{\rho}_{2,j}\mathrm{d}W_{t}^{\mathbb{Q},j}\right)$
(4.34)
$\displaystyle\qquad\quad+\mathrm{d}\left\langle\sum_{i,j=1}^{d}\int_{0}^{\cdot}\xi_{u}^{1}S_{u}^{i}\sigma_{u}^{i}\bar{\rho}_{i,j,u}\mathrm{d}W_{u}^{\mathbb{Q},j},-\sum_{j=1}^{d}\int_{0}^{\cdot}\sigma_{u}^{2}\bar{\rho}_{2,j,u}\mathrm{d}W_{u}^{\mathbb{Q},j}\right\rangle_{t}\frac{B_{t}^{2}}{B_{t}S_{t}^{2}}\;.$
(4.35)

Concerning the quadratic covariation term we have

$\displaystyle\mathrm{d}\left\langle\sum_{i,j=1}^{d}\int_{0}^{\cdot}\xi_{u}^{1}S_{u}^{i}\sigma_{u}^{i}\bar{\rho}_{i,j,u}\mathrm{d}W_{u}^{\mathbb{Q},j},-\sum_{j=1}^{d}\int_{0}^{\cdot}\sigma_{u}^{2}\bar{\rho}_{2,j,u}\mathrm{d}W_{u}^{\mathbb{Q},j}\right\rangle_{t}\frac{B_{t}^{2}}{B_{t}S_{t}^{2}}$
(4.36)
$\displaystyle=-\sum_{i=1}^{d}\xi_{t}^{1}S_{t}^{i}\sigma_{t}^{i}\sigma_{t}^{2}\sum_{j=1}^{d}\sum_{k=1}^{d}\bar{\rho}_{i,j,t}\bar{\rho}_{2,k,t}\mathrm{d}\langle
W_{\cdot}^{\mathbb{Q},j},\mathrm{d}W_{\cdot}^{\mathbb{Q},k}\rangle_{t}\frac{B_{t}^{2}}{B_{t}S_{t}^{2}}$
(4.37)
$\displaystyle=-\frac{B_{t}^{2}}{B_{t}S_{t}^{2}}\sum_{i=1}^{d}\xi_{t}^{1}S_{t}^{i}\sigma_{t}^{i}\sigma_{t}^{2}\sum_{k=1}^{d}\bar{\rho}_{i,k,t}\bar{\rho}_{2,k,t}\mathrm{d}t$
(4.38)
$\displaystyle=-\frac{B_{t}^{2}}{B_{t}S_{t}^{2}}\left(\xi_{t}diag(S_{t})\Sigma_{t}\bar{\rho}_{t}\right)\sigma_{t}^{2}(\bar{\rho}_{2,\cdot,t})^{\intercal}\mathrm{d}t\;,$
(4.39)

where we recall that $\xi_{t}diag(S_{t})\Sigma_{t}\bar{\rho}_{t}=Z_{t}$ and
$\bar{\rho}_{2,\cdot,t}$ denotes the second row of $\bar{\rho}_{t}$.
Collecting all terms we have

$\displaystyle\begin{aligned}
\mathrm{d}\left(\frac{V}{B}\frac{B^{2}}{S^{2}}\right)_{t}&=\frac{B_{t}^{2}}{S_{t}^{2}}\left(\frac{1}{B_{t}}Z_{t}^{\intercal}\mathrm{d}W_{t}^{\mathbb{Q}}+\frac{1}{B_{t}}\left((r_{t}-r_{t}^{c,b})C_{t}^{+}-(r_{t}-r_{t}^{c,l})C_{t}^{-}\right)\mathrm{d}t+\frac{1}{B_{t}}\mathrm{d}A_{t}\right)\\\
&\qquad\quad+\frac{V_{t}}{B_{t}}\frac{B_{t}^{2}}{S_{t}^{2}}\left(-\sigma_{t}^{2}\bar{\rho}_{2,\cdot,t}\left(\mathrm{d}W_{t}^{\mathbb{Q}}-\sigma_{t}^{2}\bar{\rho}_{\cdot,2,t}\mathrm{d}t\right)\right)-\frac{B_{t}^{2}}{B_{t}S_{t}^{2}}Z_{t}^{\intercal}\sigma_{t}^{2}(\bar{\rho}_{2,\cdot,t})^{\intercal}\mathrm{d}t\;,\end{aligned}$
(4.40)

therefore, after rearranging, we obtain

$\displaystyle\begin{aligned}
\mathrm{d}\left(\frac{V}{B}\frac{B^{2}}{S^{2}}\right)_{t}&=\frac{B_{t}^{2}}{S_{t}^{2}}\left[\frac{1}{B_{t}}Z_{t}^{\intercal}\left(\mathrm{d}W_{t}^{\mathbb{Q}}-\sigma_{t}^{2}(\bar{\rho}_{2,\cdot,t})^{\intercal}\mathrm{d}t\right)\right.\\\
&\qquad\quad\left.+\frac{1}{B_{t}}\left((r_{t}-r_{t}^{c,b})C_{t}^{+}-(r_{t}-r_{t}^{c,l})C_{t}^{-}\right)\mathrm{d}t+\frac{1}{B_{t}}\mathrm{d}A_{t}\right]\\\
&\qquad\quad-\frac{V_{t}}{B_{t}}\frac{B_{t}^{2}}{S_{t}^{2}}\sigma_{t}^{2}\bar{\rho}_{2,\cdot,t}\left(\mathrm{d}W_{t}^{\mathbb{Q}}-\sigma_{t}^{2}(\bar{\rho}_{2,\cdot,t})^{\intercal}\mathrm{d}t\right)\;.\end{aligned}$
(4.42)

If we now define $\tilde{Z}\coloneqq
Z_{t}-V_{t}\sigma_{t}^{2}\bar{\rho}_{2,\cdot,t}$, then, by the Girsanov
Theorem, we have:

$\mathrm{d}\left(\frac{V_{t}}{B_{t}}\frac{B_{t}^{2}}{S_{t}^{2}}\right)=\frac{B_{t}^{2}}{S_{t}^{2}B_{t}}\tilde{Z}^{\intercal}\mathrm{d}W_{t}^{\mathbb{Q}^{S^{2}}}+\frac{B_{t}^{2}}{S_{t}^{2}B_{t}}\left[\left((r_{t}-r_{t}^{c,b})C_{t}^{+}-(r_{t}-r_{t}^{c,l})C_{t}^{-}\right)\mathrm{d}t+\mathrm{d}A_{t}\right]\;.$
(4.44)

We omit the conditional expectation representation of the BSDE above, since it
is immediate. Now, under the further assumption $r^{2}=r$, by applying Theorem
3.3 we recover, the usual change of Numéraire, but in the presence of a non-
linear, recursive integral term:

$p_{t}=-S_{t}^{2}\mathbb{E}^{\mathbb{Q}^{S^{2}}}\left[\frac{X}{S_{T}^{2}}+\int_{t}^{T}\frac{1}{S_{s}^{2}}\left[(r_{s}-r_{s}^{c,b})C_{s}^{+}-(r_{s}-r_{s}^{c,l})C_{s}^{-}\right]\mathrm{d}s|\mathcal{F}_{t}\right]\;.$
(4.45)

In the present example, $X$ is the payoff of the exchange option, i.e.

$X=(S_{T}^{1}-S_{T}^{2})^{+}.$ (4.46)

In summary, we obtain

$p_{t}=-S_{t}^{2}\mathbb{E}^{\mathbb{Q}^{S^{2}}}\left[\frac{(S_{T}^{1}-S_{T}^{2})^{+}}{S_{T}^{2}}+\int_{t}^{T}\frac{1}{S_{s}^{2}}\left[(r_{s}-r_{s}^{c,b})C_{s}^{+}-(r_{s}-r_{s}^{c,l})C_{s}^{-}\right]\mathrm{d}s|\mathcal{F}_{t}\right]\;,$
(4.47)

i.e.

$p_{t}=-S_{t}^{2}\mathbb{E}^{\mathbb{Q}^{S^{2}}}\left[\left(\frac{S_{T}^{1}}{S_{T}^{2}}-1\right)^{+}+\int_{t}^{T}\frac{1}{S_{s}^{2}}\left[(r_{s}-r_{s}^{c,b})C_{s}^{+}-(r_{s}-r_{s}^{c,l})C_{s}^{-}\right]\mathrm{d}s\big{|}\mathcal{F}_{t}\right]\;.$
(4.48)

When there is no collateral, we recover the classical Margrabe’s formula under
the standard Black-Scholes multivariate model.

###  4.4 Pure jump setting example

We assume that the risky asset evolves now according to

$\frac{\mathrm{d}S_{t}^{i}}{S_{t-}^{i}}=\mu^{i}\mathrm{d}t+\int_{\mathbb{R}}(e^{z}-1)(N-\nu^{\mathbb{P}})(\mathrm{d}t,\mathrm{d}z)\;.$
(4.49)

which corresponds to the SDE associated to the class of exponential Lévy
models. For the sake of the present example, we assume that $N$ has
compensator
$\nu^{\mathbb{P}}(\mathrm{d}z)\mathrm{d}t=\lambda\mathbbm{1}_{\alpha}(\mathrm{d}z)\mathrm{d}t$,
i.e. we consider a Poisson process with constant intensity $\lambda$ and fixed
jump size $\alpha$. We construct the martingale measure along the following
steps. In line with Bielecki and Rutkowski (2015), our objective is to
guarantee that the process

$\tilde{S}_{t}^{i}\coloneqq\left(\frac{S^{i}}{B^{i}}\right)_{0\leq t\leq T}$
(4.50)

is a local martingale. Consider next

$\displaystyle\mathrm{d}\tilde{S}_{t}^{i}$
$\displaystyle=\tilde{S}_{t-}^{i}\left((\mu^{i}-r^{i})\mathrm{d}t+\int_{\mathbb{R}}(e^{z}-1)(N-\nu^{\mathbb{P}})(\mathrm{d}t,\mathrm{d}z)\right)$
(4.51)
$\displaystyle=\tilde{S}_{t-}^{i}\left((e^{\alpha}-1)N(\mathrm{d}t,\\{\alpha\\})-\int_{\mathbb{R}}(e^{z}-1)\lambda\mathbbm{1}_{\alpha}(z)\mathrm{d}z\mathrm{d}t)+(\mu^{i}-r^{i})\mathrm{d}t\right)$
(4.52)
$\displaystyle=\tilde{S}_{t-}^{i}\left((e^{\alpha}-1)N(\mathrm{d}t,\\{\alpha\\})-\left((e^{\alpha}-1)\lambda-(\mu^{i}-r^{i})\right)\mathrm{d}t\right)$
(4.53)
$\displaystyle=\tilde{S}_{t-}^{i}\left((e^{\alpha}-1)N(\mathrm{d}t,\\{\alpha\\})-(e^{\alpha}-1)\left(\lambda-\frac{\mu^{i}-r^{i}}{e^{\alpha}-1}\right)\mathrm{d}t\right)$
(4.54)
$\displaystyle=\tilde{S}_{t-}^{i}\left((e^{\alpha}-1)N(\mathrm{d}t,\\{\alpha\\})-(e^{\alpha}-1)\lambda\left(1-\frac{\mu^{i}-r^{i}}{(e^{\alpha}-1)\lambda}\right)\mathrm{d}t\right)\;.$
(4.55)

Recalling the notation of Theorem 3.3, this means that we have

$\delta(z)=\delta=-\frac{\mu-t}{(e^{\alpha}-1)\lambda}\;,$ (4.56)

which is the Girsanov kernel in the present setting. Under $\mathbb{Q}$, we
have

$\frac{\mathrm{d}S_{t}^{i}}{S_{t-}^{i}}=r^{i}\mathrm{d}t+\int_{\mathbb{R}}(e^{z}-1)(N-\nu^{\mathbb{P}})(\mathrm{d}t,\mathrm{d}z)\;,$
(4.57)

with

$\displaystyle\nu^{\mathbb{Q}}(\mathrm{d}z)$
$\displaystyle=(1+\delta(z))\nu^{\mathbb{P}}(\mathrm{d}z)$ (4.58)
$\displaystyle=\left(1-\frac{\mu-r}{(e^{\alpha}-1)\lambda}\right)\lambda\mathbbm{1}_{\alpha}(\mathrm{d}z)\;.$
(4.59)

In summary we have:

Under $\mathbb{P}$

$\frac{\mathrm{d}S_{t}^{i}}{S_{t-}^{i}}=\mu^{i}\mathrm{d}t+\int_{\mathbb{R}}(e^{z}-1)(N-\nu^{\mathbb{P}})(\mathrm{d}t,\mathrm{d}z)\;,\;i.e.$
(4.60) $S_{t}^{i}=S_{0}^{i}\exp{\mu
t-\int_{\mathbb{R}}(e^{z}-1-z)\nu(\mathrm{d}z)t+\int_{\mathbb{R}}z(N-\nu^{\mathbb{P}})(\mathrm{d}t,\mathrm{d}z)}\;.$
(4.61)

Under $\mathbb{Q}$

$\frac{\mathrm{d}S_{t}^{i}}{S_{t-}^{i}}=r^{i}\mathrm{d}t+\int_{\mathbb{R}}(e^{z}-1)(N-\nu^{\mathbb{Q}})(\mathrm{d}t,\mathrm{d}z)\;,\;$
(4.62)
$\nu^{\mathbb{Q}}(\mathrm{d}z)=(1+\delta(z))\nu^{\mathbb{P}}(\mathrm{d}z)=\left(1-\frac{\mu-r}{(e^{\alpha}-1)\lambda}\right)\lambda\mathbbm{1}_{\alpha}(\mathrm{d}z)\;.$
(4.63)

Density process $\mathcal{E}_{t}^{T}$

$\mathcal{E}_{t}^{T}=\exp\left\\{\int_{t}^{T}\int_{\mathbb{R}^{d}}\ln(1+\delta(z))\right.\\\
\left.-\delta(z)\nu^{\mathbb{P}}(\mathrm{d}z)\mathrm{d}r+\int_{t}^{T}\int_{\mathbb{R}^{d}}\ln(1+\delta(z))\tilde{N}(\mathrm{d}r,\mathrm{d}z)\right\\}\,.$
(4.64)

####  4.4.1 Portfolio dynamics with jumps

We start again from the wealth dynamics

$\mathrm{d}V_{t}(\varphi)=r_{t}V_{t}(\varphi)\mathrm{d}t+\sum_{i=1}^{d}\xi_{t}^{i}(\mathrm{d}S_{t}^{i}-r_{t}^{i}S_{t}^{i}\mathrm{d}t+k_{t}^{i}S_{t}^{i}\mathrm{d}t)+\mathrm{d}\hat{F}_{t}^{h}+\mathrm{d}A_{t}\;,$
(4.65)

then, substituting the $\mathbb{P}$ dynamics of $S$, and setting $d=1$, we
have

$\mathrm{d}V_{t}(\varphi)=r_{t}V_{t}(\varphi)\mathrm{d}t+\xi_{t}^{1}\left(S_{t-}^{1}\left(\mu^{1}\mathrm{d}t+\int_{\mathbb{R}}(e^{z}-1)(N-\nu^{\mathbb{P}})(\mathrm{d}t,\mathrm{d}z)\right)-r^{1}S^{1}\mathrm{d}t\right)\\\
+\mathrm{d}\hat{F}_{t}^{h}+\mathrm{d}A_{t}\;,$ (4.66)

it follows that, setting

$U^{1}\coloneqq\xi^{1}S_{t-}^{1}$ (4.67)

we obtain

$\displaystyle\mathrm{d}V_{t}(\varphi)$
$\displaystyle=r_{t}V_{t}(\varphi)\mathrm{d}t+U_{t}^{1}(\mu^{1}-r^{1})\mathrm{d}t+U_{t-}^{1}\int_{\mathbb{R}}(e^{z}-1)(N-\nu^{\mathbb{P}})(\mathrm{d}t,\mathrm{d}z)$
(4.68)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\mathrm{d}\hat{F}_{t}^{h}+\mathrm{d}A_{t}$
$\displaystyle=r_{t}V_{t}(\varphi)\mathrm{d}t+\int_{\mathbb{R}}U_{t-}^{1}\frac{\mu^{1}-r^{1}}{(e^{\alpha}-1)\lambda}(e^{z}-1)\nu^{\mathbb{P}}(\mathrm{d}z)\mathrm{d}t$
$\displaystyle\qquad\qquad\qquad\qquad\
+\int_{\mathbb{R}}U_{t-}^{1}(e^{z}-1)(N-\nu^{\mathbb{P}})(\mathrm{d}t,\mathrm{d}z)+\mathrm{d}\hat{F}_{t}^{h}+\mathrm{d}A_{t}\;,$

allowing us to apply Theorem 3.3 in the present pure jump setting.

######  Lemma 4.8.

Let $X$ be $\mathcal{F}_{T}^{S}$-measurable with
$X\in\mathbb{L}_{T}^{2}(\Omega,\mathcal{F}_{T}^{S})$, then there exists a
unique solution
$(V,Z)\in\mathbb{S}^{2}_{T}(\mathbb{R})\times\mathbb{H}^{2}_{T,N}(\mathbb{R})$
to the BSDE (4.68). The contract
$A_{t}=p\mathbbm{1}_{[0,T]}(t)+X\mathbbm{1}_{[T]}(t)$ with collateral
specification b) in Assumption 4.3, can be replicated by an admissible trading
strategy $\xi$. We have $p_{t}=V_{t}(\varphi)$ and the process $V$ admits the
representation under $\mathbb{P}$.

$p_{t}=V_{t}(\varphi)=-\mathbb{E}^{\mathbb{P}}\left[\frac{B_{t}}{B_{T}}X\mathcal{E}_{t}^{T}+\int_{t}^{T}\frac{B_{t}}{B_{u}}\mathcal{E}_{t}^{u}\right.\\\
\left.+\left[(r_{u}-r_{u}^{c,b})C_{u}^{+}-(r_{u}-r_{u}^{c,l})C_{u}^{-}\right]\mathrm{d}u|\mathcal{F}_{t}\right].$
(4.69)

Next, again using the change of measure formula of Theorem 3.3, we can rewrite
the BSDE under the measure $\mathbb{Q}$ as

$\mathrm{d}V_{t}(\varphi)=r_{t}V_{t}(\varphi)\mathrm{d}t+\int_{\mathcal{R}}U_{t-}^{1}(e^{z}-1)(N-\nu^{\mathbb{P}})(\mathrm{d}t,\mathrm{d}z)+\hat{F}_{t}^{h}+\mathrm{d}A_{t}\;.$
(4.70)

Finally, we can reformulate Proposition 5.4 in Bielecki and Rutkowski (2015)
in a pure jump setting. We omit the statement since it is analogous to our
Lemma 4.6.

###  4.5 A generalization of the change of numéraire technique

The reasoning we illustrated in Section 4.1 provides us with a generalization
of the well-known change of numéraire technique that covers recursive
conditional expectations. The technique involves the following steps:

  1. 1.

Start by considering wealth dynamics under a starting probability measure
$\mathbb{Q}$, for example as in Equation 4.18 for $\tilde{V}$;

  2. 2.

Consider a traded asset, suitably discounted so as to treat a
$(\mathbb{Q},\mathbb{F})$-martingale (the ratio $\frac{S^{2}}{B^{2}}$ in
Section 4.3) and compute its (inverse) dynamics via the Itô formula;

  3. 3.

Identify the stochastic drivers under the new measure $\mathbb{Q}^{\prime}$;

  4. 4.

Compute the dynamics of the product between $\tilde{V}$ and the (inverse)
density process;

  5. 5.

Apply Theorem 3.3 to obtain the expression for the price under the new
probability measure $\mathbb{Q}^{\prime}$.

## Acknowledgement

We are grateful to Samuel Cohen for drawing our attention to useful
references.

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# Exploring the origins of perpendicular magnetic anisotropy in amorphous Tb-
Co via changes in medium-range ordering

Ellis Kennedy  Department of Materials Science and Engineering, University of
California Berkeley, Berkeley, California, 94720, USA  Emily Hollingworth
Department of Physics, University of California Berkeley, Berkeley,
California, 94720, USA  Alejandro Ceballos  Department of Materials Science
and Engineering, University of California Berkeley, Berkeley, California,
94720, USA  Daisy O’Mahoney  Department of Materials Science and Engineering,
University of California Berkeley, Berkeley, California, 94720, USA  Colin
Ophus  NCEM, Molecular Foundry, Lawrence Berkeley National Laboratory,
Berkeley, California, 94720, USA  Frances Hellman  Department of Materials
Science and Engineering, University of California Berkeley, Berkeley,
California, 94720, USA  Department of Physics, University of California
Berkeley, Berkeley, California, 94720, USA  Materials Science Division,
Lawrence Berkeley National Laboratory, Berkeley, California, 94720, USA  M.C.
Scott<EMAIL_ADDRESS>Department of Materials Science and
Engineering, University of California Berkeley, Berkeley, California, 94720,
USA  NCEM, Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley,
California, 94720, USA

###### Abstract

Amorphous thin films of Tb17Co83 (a-Tb-Co) grown by magnetron co-sputtering
exhibit changes in magnetic anisotropy with varying growth and annealing
temperatures. The magnetic anisotropy constant increases with increasing
growth temperature, which is reduced or vanishes upon annealing at
temperatures above the growth temperature. The proposed explanation for this
growth-induced anisotropy in high orbital moment Tb-based transition metal
alloys such as a-Tb-Co is an amorphous phase texturing with preferential in-
plane and out-of-plane local bonding configurations for the rare-earth and
transition metal atoms. Scanning nanodiffraction performed in a transmission
electron microscope (TEM) is applied to a-Tb17Co83 films deposited over a
range of temperatures to measure relative changes in medium-range ordering
(MRO). These measurements reveal an increase in MRO with higher growth
temperatures and a decrease in MRO with higher annealing temperatures. The
trend in MRO indicates a relationship between the magnetic anisotropy and
local atomic ordering. Tilting select films between 0∘ and 40∘ in the TEM
measures variations in the local atomic structure a function of orientation
within the films. The findings support claims that preferential ordering along
the growth direction results from temperature-mediated adatom configurations
during deposition, and that oriented MRO correlates with the larger anisotropy
constants.

††preprint: Ain-plane/123-QED

## Introduction

Amorphous rare earth – transition metal (RE-TM) alloys have tunable magnetic
properties, such as compensation temperature and perpendicular magnetic
anisotropy (PMA) Ding and Poon (2013); Andreenko and Nikitin (1997); Hasegawa
_et al._ (1974); Hellman and Gyorgy (1992); Gambino (1998); Harris  _et al._
(1994, 1993). These properties make them desirable as novel spintronic
materials and for ultrafast magneto-optical recording devices. For REs with
non-zero total orbital angular momentum, such as Tb, their magnetic anisotropy
is believed to originate from the single ion anisotropy of the RE atoms as
well as complex pair-bonding and preferential atomic configurations induced
during film growth and subsequent thermal treatments Hellman and Gyorgy
(1992). Out-of-plane RE-TM bonding and in-plane TM-TM bonding have been
correlated to bulk perpendicular magnetic anisotropy (PMA) in a-RE-TM systems
Hellman and Gyorgy (1992); Harris  _et al._ (1993, 1994). The amorphous nature
of these systems complicates their analysis, as variations in magnetic
anisotropy must instead be attributed to subtle variations in local atomic
ordering Cochrane  _et al._ (1978). Previous work shows that PMA is
independent of film thickness, surface layer magnetic interactions, and
macroscopic growth-induced strain Hellman and Gyorgy (1992).

The succinct description of crystalline structures with defined unit cells and
permitted symmetry operations does not extend to amorphous materials. Their
lack of translational and rotational symmetry require a statistical approach
for analysis Voyles and Muller (2002); Treacy  _et al._ (2005). Two-body and
multi-body distribution functions are applied to the study of amorphous
structures to determine the probability that two or more atoms will be
separated by a specific distance. Short-range ordering (SRO) is probed through
two-body statistical analysis, such as radial distribution functions. However,
SRO is limited to the first coordination shell of the constituent atom.
Looking out a little further, medium-range ordering (MRO) on the 1 – 5 nm
length scale, can be probed with transmission electron microscopy (TEM)
Williamson (1995); Voyles and Muller (2002). Statistical analysis of MRO can
determine the degree and type of ordering that exists between the extremes of
long-range and short-range order Nakhmanson  _et al._ (2001); Treacy  _et al._
(2005).

In this work, a-Tb17Co83 is used as a representative a-RE-TM system. Tb (RE)
atoms have more than half-filled 4f-electron orbitals and magnetic moments
that align antiferromagnetically to the moments of the Co (TM) atoms Hansen
(1991). The Curie temperature (TC), saturation moment, and compensation
temperature (Tcomp) are solely dependent on composition, while PMA and
coercivity depend as well on growth temperature, and other deposition
parameters Hellman and Gyorgy (1992); Hasegawa  _et al._ (1974). A series of
a-Tb17Co83 films were deposited at 20∘C, 200∘C, and 300∘C. The films then
either received no further heat treatment or were annealed at 200∘C or 300∘C.
In-plane and out-of-plane magnetization vs magnetic field measurements were
collected from the films. Congruent with previous studies, the PMA was found
to increase with growth temperature and decrease with annealing
temperatureMergel  _et al._ (1993); Hasegawa  _et al._ (1974); Ceballos  _et
al._ (2021); Hellman and Gyorgy (1992). The TEM method of scanning
nanodiffraction was used to probe the underlying structural mechanisms
responsible for variations in the magnetic properties of a-Tb17Co83 as a
function of film deposition and annealing temperature. The relative MRO across
the series of samples is measured with fluctuation electron microscopy (FEM),
a specialized application of TEM scanning nanodiffraction that is sensitive to
changes in diffracted intensity related to variations in atomic configurations
within an amorphous system Voyles and Muller (2002). We determine that MRO in
the films increases with higher deposition temperatures and decreases with
higher annealing temperatures. The results support the model of magnetism in
a-RE-TM systems in which PMA is related to atom-specific preferential local
atomic ordering.

## Experimental Details

a-Tb17Co83 films were produced using magnetron co-sputtering at 1.8 mTorr Ar
pressure from separate Tb and Co targets. The base pressure of the chamber was
7 x 10-8 Torr. The a-Tb17Co83 films were sputtered at room temperature (20∘C),
200∘C, and 300∘C with a capping layer of a-SiNx sputtered at 3.0 mTorr of Ar
pressure. The films were deposited on Norcada grids with 10 nm thick a-SiNx
membranes. Samples consisted of 30 nm a-Tb17Co83 films capped with 10 nm of
a-SiNx to prevent oxidation. The 30 nm thickness was selected for the
a-Tb17Co83, as it was experimentally determined to produce a strong speckle
intensity in scanning nanodiffraction. A control sample of 10 nm of a-SiNx was
sputtered at 3.0 mTorr of Ar pressure to determine the influence of the
capping layer on the diffraction data. After deposition at room temperature,
some films were annealed at 200∘C or 300∘C for an hour under high vacuum.

The magnetization as a function of applied magnetic field in-plane and out-of-
plane was measured in a Quantum Design Magnetic Properties Measurement System
(MPMS) at room temperature for a series of samples grown and annealed under
identical conditions to those detailed above (Supplemental Materials). The
films grown at room temperature were annealed at 150∘C, 200∘C, 275∘C, and
350∘C to establish how magnetic anisotropy depends on annealing temperature.
The applied field was varied between - 1 and 1 T, a range exceeding the
magnetic saturation values for all samples. The intrinsic uniaxial anisotropy
constant, Kui, is calculated using the relation below, where Ms is the
saturation magnetization and Hk is the anisotropy field required to fully
magnetize the films in-plane.Hellman  _et al._ (1999); Ceballos  _et al._
(2021)

$\displaystyle K_{ui}=\frac{H_{k}M_{s}}{2}+2\pi M_{s}^{2}$ (1)

For all samples, Tc = 190∘C, Tcomp = -73∘C, and Ms is approximately 150
emu/cc. All samples exhibited uniaxial PMA.

A subset of the films measured with magnetometry were analyzed using FEM. To
measure changes in the MRO of the ferrimagnetic films, FEM experiments were
carried out using scanning transmission electron microscopy (STEM) in an FEI
TitanX operated at an accelerating voltage of 200 kV. The third condenser lens
was set to produce a convergence angle of 0.51 mrad resulting in a measured
2.2 nm diameter probe and a 15.5 pA probe current. Images were collected on an
Orius charge coupled device (CCD) detector with an exposure time of 0.3
seconds and a camera length of 300 mm. The images were binned by a factor of
four, to a final size of 512x512 pixels. Nanodiffraction data were collected
as 12x12 image stacks (144 images per stack) for each film with 5 nm step
sizes between regions of analysis. Collection was repeated four times per film
(576 images in total per film) for statistical averaging over different
regions of the film and to avoid excess contamination. During the experiment,
all diffraction patterns were taken with the same imaging conditions to ensure
that the effects of microscope misalignment were the same across all patterns.
The small feature that appears around 4 nm-1 in all spectra is from a defect
on the CCD. Additionally, the films deposited at 20∘C and 300∘C were tilted at
an angle varied between 0∘ and 40∘ in 5∘ increments to determine whether MRO
varies with orientation through the film. These two films were selected for
tilting because they had the greatest difference in MRO as determined from FEM
variance measurements.

FEM analysis was performed following the method outlined by Kennedy et
al.Kennedy  _et al._ (2020) Custom scripts were used to fit the radial
profiles of the scanning nanodiffraction data using least squares fitting of
the first diffracted ring as described by Gammer et al. Gammer  _et al._
(2018) The fitting parameters are used to correct for any ellipticity in the
patterns before calculating the variance in intensity as a function of
scattering vector. For patterns collected from tilted films, the ellipticity
is not corrected. Instead, changes in the major elliptical axes are used as a
measure of relative changes in mean bond lengths. The procedure for collecting
scanning nanodiffraction data and FEM analysis is outlined in the Supplemental
Materials. For FEM, the variance is calculated with respect to scattering
angle k, position on the sample r, and resolution (which is a function of r);
however, in STEM FEM only the scattering vector varies and thus the equation
for variance  Vσ in diffracted intensity I becomesTreacy  _et al._ (2005):

$\displaystyle V_{\sigma}(\textbf{k},r)=\frac{\langle
I^{2}(\textbf{k},r)\rangle-\langle I(\textbf{k},r)\rangle^{2}}{\langle
I(\textbf{k},r)\rangle^{2}},$ (2)

Additionally, Lorentz TEM images were collected from the as-deposited film
grown at room temperature and the films grown at room temperature and annealed
at 200∘C and 300∘C to probe the effect of annealing on domain structure.
Lorentz TEM was carried out in a FEI ThemIS operated at 300 kV in Lorentz
mode. All films had been previously subjected to a magnetic field of
approximately 2 T during FEM image acquisition, but the magnetic field of the
objective lens was minimized to less that 0.1 T for Lorentz imaging. The films
were tilted 15∘ and the beam was defocused to produce visible magnetic
domains.

## Results

Figure 1:  Magnetic anisotropy constant as a function of deposition and
annealing temperatures derived from in-plane and out-of-plane magnetization
measurements. Ms is approximately the same for all films.

The magnetic anisotropy constant $K_{ui}$ was found to be 8.33 x 105 J/m3 for
films grown at room temperature, and increases to 9.26 x 105 J/m3 for films
grown at 200∘C and to 10.30 x 105 J/m3 for films grown at 300∘C. For films
grown at room temperature and subsequently annealed at temperatures between
150∘C and 350∘C, the magnetic anisotropy decreases with temperature. The
relationships between deposition temperature, annealing temperature, and
magnetic anisotropy are shown in Fig. 1. Similar trends have been shown in
other amorphous RE-TM thin film systems Harris  _et al._ (1994). The out-of-
plane magnetic saturation $M_{s}$ remains constant (150 emu/cc) for all
samples in the series.

High resolution TEM imaging and the mean converged beam electron diffraction
(CBED) patterns confirm that the samples are amorphous (shown in Supplemental
Materials). The uniformity of the diffracted speckle in the scanning
nanodiffraction patterns and lack of strong Bragg scattering indicates that
the films lack long range order and have no nanocrystals.

In Fig. 2, the normalized variance in intensity V(k) for films in their as-
deposited state (a) and after annealing (b) is plotted as a function of
scattering vector using mean statistics. The MRO increases with increasing
growth temperature, as indicated by an increase in V(k). The variance for the
300∘C growth is significantly sharper than the 20∘C and 200∘C variances. There
is an increase in the V(k) and a decrease in the full width at half maximum
(FWHM) across the three films as a function of deposition temperature.

Figure 2:  Variance as a function of scattering vector k for a-Tb17Co83 films (a) deposited at 20∘C, 200∘C, and 300∘C and (b) subsequently annealed at 200∘C and 300∘C following deposition at 20∘C. The height of the peak centered between 3.30 nm-1 and 3.33 nm-1 corresponds to the degree of relative MRO in the films. The magnitude of the peak increases with growth temperature, indicating that MRO is greater for films deposited at higher temperatures. The position of the peaks along the scattering vector axis indicates the average size of the bonds resulting in detectable MRO. Features located around 2.4 nm-1 are attributable to the a-SiNx layers. Table 1: Summarized peak heights and positions along the scattering vector k axis from FEM curves in Fig. 2 Temperature parameters | Relative Height111The peak height of the film grown at 20∘C is used as the reference. Other peak heights are given relative to it. | Position (nm-1)
---|---|---
20∘C deposition | 1 | 3.1
200∘C deposition | 1.10 | 3.0 to 3.2
300∘C deposition | 1.23 | 3.3
20∘C deposition, 200∘C anneal | 1.01 | 3.1
20∘C deposition, 300∘C anneal | 0.81 | 3.2

The peak heights correspond to relative MRO and the peak positions correspond
to the mean interatomic distance of the MRO. The peaks in V(k) are typically
attributable to oriented clusters of atoms (MRO) and not to individual bond
types (SRO) Voyles and Muller (2002); Li  _et al._ (2014). Two main peak
features emerge in the variance plots in Fig. 2. The first peak is centered
around 2.4 nm-1 (4.2 Å) and is attributable to a-SiNx, as confirmed through
FEM analysis of 10 nm of a-SiNx deposited onto a-SiNx membranes. The second
peak is centered between 3.0 and 3.3 nm-1 (3.3 and 3.0 Å) and originates from
the a-Tb17Co83. The peak positions of the a-Tb17Co83 growth series shift to
larger k values with increasing deposition temperature. The change in the
magnitude of the a-Tb17Co83 peak variance relates to degree of MRO in the
films. Peak positions and relative heights are provided in Table 1.

The FEM diffracted signal varies slightly as a function of orientation through
the individual films, as shown in Fig. 3, suggesting directional dependency in
bonding structure. The most significant difference in MRO as a function of
tilt angle was observed between the films grown at 20∘C and 300∘C. These films
were tilted in the TEM to 40∘C to check for directional dependency of the MRO.
Changes in the ellipticity of the FEM patterns as a function of tilt angle
reveal that the films deposited at 300∘C exhibit a greater change in the
length of their major axes, compared to the major axes of the films grown at
20∘C. The mean bond length of both films decreases as a function of
orientation away from the film normals. The decreases is substantially more
pronounced for the film deposited at 300∘C, indicating greater orientation-
dependency of the MRO bond lengths. By assigning the changes in the FEM
diffraction rings to changes in bond length in the in- and out-of-plane
directions, we can analyze the tilted data using strain relationships
(Supplemental Materials). From this, we determine that the maximum change in
mean bond length is -1.43% for the film deposited at 20∘C and -3.14% for the
film deposited at 300∘C. These values assume the trend continues past the 40∘C
tilt angle, to the out-of-plane orientation. This implies that the films grown
at 300∘C have greater directional dependence of interatomic ordering.

## Discussion

Many models have been proposed for the origin of PMA in amorphous RE-TM thin
films. Postulations include the formation of columnar microstructures,
anelastic strains, surface layer anisotropy, growth-induced anisotropy
resulting from magnetic interactions, microlite formation, and temperature-
mediated subtle alignments of atoms around the RE atoms Leamy and Dirks
(1978); Dirks and Gijsbers (1979); Yan  _et al._ (1991); Fu and Mansuripur
(1992); Clark (1973); Hellman and Gyorgy (1992). The mechanisms described in
the latter of these models is commonly referred to as amorphous phase
texturing Hellman and Gyorgy (1992). Many of these models have been disproven
by varying growth parameters and measuring the resulting variations in PMA.
This work supports the model of texturing in which local adatom configurations
arrange themselves to minimize surface energy during the deposition process.

In a-Tb17Co83, the itinerant 3d-electrons of Co influence the magnetic
ordering of the localized 4f-electrons of Tb (RE) through an antiferromagnetic
exchange interaction Brooks  _et al._ (1989). The magnetic moments of the Tb
4f-electron orbitals, which are more than half-filled, align
antiferromagnetically to the moments of the Co (TM) atoms. The combined effect
of the resulting negative exchange together with local uniaxial anisotropy,
which acts primarily on the Tb moments, is to produce non-colinear
ferrimagnetism that is Co-dominant above Tcomp and Tb-dominant below Tcomp
Vas’kovskiy  _et al._ (2015); Finley and Liu (2016). a-Tb17Co83 has a Tcomp of
approximately -73∘C, making it Co-dominant at room temperature. The Curie
temperature (TC) of a-Tb17Co83 is approximately 190∘C. The PMA increases with
increasing growth temperature in the a-Tb17Co83 system, including for films
are grown above TC Hansen (1988). The same trend has been independently
observed in other amorphous RE-TM systems Andreenko and Nikitin (1997);
Hellman and Gyorgy (1992); Hajjar  _et al._ (1990); Kobayashi  _et al._
(1983); Zhang  _et al._ (2010).

Figure 3:  Relative % change in mean bond distances as a function of tilt
angle for the films deposited at 20∘C and 300∘C. The 20∘C data are shown in
the bottom row with blue data markers and the 300∘C data are shown in the top
row with magenta markers. The 0∘ tilt relative mean bond length is set to zero
for all orientations. The z’-oriented strain changes with tilt angle as the
orientation changes from in-pane (0∘ tilt) to out-of-plane (extrapolated to
90∘ tilt). The film deposited at 300∘C exhibits a greater degree of change in
relative bond lengths compared to the film deposited at 20∘C. The measured %
changes in mean bond length are modelled by the infinitesimal strain theory
applied to a rotated system, as shown by the orange curves. The fit models
used to generate the orange curves for each component are described in the
Supplemental Materials. The green arrows in the insets show the orientation of
each l component at 90∘C (experimentally inaccessible).

FEM measurements indicate that varying the growth temperature also impacts
local atomic ordering on the length scale of a few nanometers. The in-plane
MRO of the system increases with increasing deposition temperature. The
increased peak height of the FEM V(k) curve shows the effect of growth
temperature on relative MRO in the films. Changes in peak height are
summarized in Table 1. Additionally, the full-width at half maximum height
(FWHM) decreases as a function of growth temperature, indicating that higher
growth temperatures produce more uniform bonding on the medium-range length
scale. The V(k) peak for the film grown at 20∘C is located at 3.1 nm-1 (3.2
Å). The V(k) peak position increases to 3.3 nm-1 (3.0 Å) for the film grown at
300∘C. At the intermediate growth temperature of 200∘C, the V(k) has a broader
peak top, indicating the possible existence of overlapping populations with
similar MRO length scales or a decrease in the homogeneity of bond lengths.
Overall, the MRO bond length decreases at higher growth temperatures. This is
likely the result of denser atomic packing in-plane at higher growth
temperatures and larger distances out-of-plane, i.e. amorphous structure
texturing to minimize surface energy during growth, similar to what is called
texturing of polycrystalline films.

There is an inverse trend between PMA and annealing temperature. Similarly, as
the annealing temperature increases, the MRO decreases, as shown in Fig. 2 and
summarized in Table 1. Thus, increased deposition temperature increases MRO
and PMA, while increased annealing temperature reduces MRO and PMA in
a-Tb17Co83. The V(k) curves for the room temperature growth and the film
annealed at 200∘C are similar with nearly identical FWHM, position, and
relative height. The similarity suggests that there is a minimum temperature
required for MRO atomic rearrangement during annealing. Overcoming the energy
barrier for atomic re-orientation detectable with FEM requires a long enough
anneal time and high enough temperature. At 300∘C and one hour, the annealing
parameters are sufficient to allow energy-minimizing atomic rearrangement in
the bulk of the film.

Above TC, the system is paramagnetic and atomic interactions are not governed
by magnetic interactions. Thus, the trend in MRO for the a-Tb17Co83 growth
series cannot be ascribed to magnetic interactions. Instead, the relationship
between MRO and growth temperature-mediated PMA supports models in which local
adatom configurations arrange such that they minimize the surface energy
during growth Hellman (1994). As layers build, the preferential configuration
is preserved which leads to the oriented local magnetic anisotropy that
produces an overall macroscopic PMA.

The films grown at room temperature and 300∘C were selected for tilted FEM
analysis to understand the relationship between temperature-mediated atomic
texturing during deposition and orientationally anisotropic MRO Kennedy  _et
al._ (2020). In this measurement, FEM data is acquired at varying stage tilt
angles, which enables us to deduce the relative change in bond lengths
associated with the MRO in the in- and out-of-plane directions. Because the
diffraction patterns contain information about d-spacings perpendicular to the
direction of the beam, as the effective angle between the beam and the sample
is varied, any changes in bond length between the in- and out-of plane
directions in the film will cause an ellipticity in diffraction pattern
(Supplemental Materials). This effect is similar to ellipticity in diffraction
data caused by anisotropic strain, although in our case, decoupling the
influences of strain and changes in preferential atomic orientations is not
possible. For this reason, we consider changes in the major axis of the fitted
ellipse to correspond broadly to changes in the average bond length, $l$, as a
function of orientation through the film. lxx is along the x-axis, lzz in the
y-z plane along the z’-axis, and lxx bisects the x’ and z’ directions.

The stage tilt is limited to 40∘ in the TitanX, but the out-of-plane bond
length can be extrapolated from measurements at lower angles by considering
the geometry of the experiment and the influence of infinitesimal shifts in
bond length on the diffraction data, which gives a $\sin^{2}$ dependence to
the projected out-of-plane bond length. As shown in Fig. 3, lxx is modeled
with a linear fit and lxz is modeled with a $\sin^{2}$$/\sqrt{2}$ dependence.
The geometric origins of these terms are illustrated in the Supplemental
Materials.

Fig. 3 shows the trend in average relative % change in mean bond length as a
function of tilt angle for the films grown at room temperature and 300∘C.
Relative % change below the $\pm 0.2\%$ margin of error indicates that the
films are isotropic within the limit of the technique. Above this margin of
error, the films exhibit changes in relative mean bond distances dependent on
orientation through the films.

From the lzz fits, the extrapolated maximum bond lengths are calculated. For
the film deposited at room temperature, the extrapolated maximum relative %
change in mean bond length is 1.4%. For the film deposited at 300∘C, the
extrapolated maximum relative % change in mean bond length is 3.1%. These
values are determined by extending the fitting curves to a theoretical 90∘
tilt. Both films exhibit shorter mean bond lengths in the in-plane direction
compared to the out-of-plane direction. The film grown at 300∘C exhibits a
larger change in mean bond length between the in-plane and out-of-plane
orientations compared with the film grown at room temperature. The in-plane
bond lengths in the film grown at 300∘C are larger, on average, than the bonds
in the out-of-plane orientation. Assuming the texturing model for PMA, this
corresponds to a greater proportion of Co-Tb bonds in the out-of-plane
orientation relative to Co-Co bonds in the in-plane orientations. The greater
relative % change in mean bond length for 300∘C indicates that at higher
deposition temperatures adatoms with higher energies at impingement orient
such that Co-Tb bonds form preferentially in the out-of-plane direction,
consistent with EXAFS data on a-Tb-Fe filmsHarris  _et al._ (1992, 1993).

The dramatic decrease in PMA observed between the film grown at room
temperature and the films annealed at higher temperatures and the
corresponding change in MRO prompted the use of Lorentz TEM. The Lorentz
images, shown in Supplemental Materials, show the typical high-contrast domain
patterns seen in films with strong PMA and corroborate the trends observed in
variance and magnetization for a-Tb17Co83 annealed at increasing temperatures
Hellman  _et al._ (1999). Higher PMA causes a high energy cost of forming
domain walls and narrower domain walls, with well defined domains, while lower
PMA causes wider domain walls and less defined domains. The result is sharper
domain walls and better defined domains in films deposited at higher
temperatures, and less defined domains in films grown at lower temperatures
after annealing.

The changes in mean bond length with varying orientation through the films,
the increase in MRO as the deposition temperature increases, and the
subsequent decrease in MRO after annealing, collectively indicate the presence
of MRO bond length anisotropy in the a-Tb17Co83 films. During deposition,
structural variations are introduced as a function of growth temperature. A
higher deposition temperature favors a higher proportion of Tb-Co bonds in the
out-of-plane direction, which results in greater MRO and increased PMA.
Annealing, on the other hand, allows for structural relaxation in the films,
and atom rearrangement. Taken together, the observed trends in MRO with
respect to growth conditions suggest that the evolution of PMA in the films is
due to structural changes.

## Conclusions

Numerous studies on amorphous RE-TM films have noted the relationships between
growth temperature parameters and PMA, but the structural origin remains under
debate. This work established a relationship between thermal growth
parameters, PMA, and local atomic ordering using the TEM technique of scanning
nanodiffraction. MRO is found to increase with increasing film deposition
temperature and decrease with increasing anneal temperature. Thus, there is a
correlation between the degree of MRO and PMA in a-Tb17Co83 films grown via
magnetron co-sputtering. These results support an amorphous phase texturing
model in which adatom configuration varies as a function of deposition
temperature and annealing allows for subsequent relaxations of preferential
configurations within the films. Tb-Co (RE-TM) pairs prefer to form vertically
during deposition, but annealing allows for rearrangement of atomic pairs such
that there is greater uniformity across orientations. Tilted scanning
nanodiffraction probes for changes in the mean MRO bond length as a function
of orientation through the films. Fitting the tilted FEM data shows a greater
local ordering anisotropy in the film exhibiting the highest degree of MRO
(300 ∘C growth) compared to the the film grown at room temperature with less
MRO.

###### Acknowledgements.

This work was supported by National Science Foundation STROBE grant
DMR-1548924. Work at the Molecular Foundry was supported by the Office of
Science, Office of Basic Energy Sciences, of the U.S. Department of Energy
under Contract No. DE-AC02-05CH11231. Growth and non-electron microscopy
characterization of experimental a-Tb17Co83 and a-SiNx were performed by D.O.
and E.H. and supported by the U.S. Department of Energy, Office of Science,
Basic Energy Sciences, Materials Sciences and Engineering Division under
Contract No. DE-AC02-05-CH11231, under the Nonequilibrium Magnetic Materials
Program (KC2204). C.O. acknowledges support from the US Department of Energy
Early Career Research Program.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author, M.C.S., upon reasonable request. The scripts used to
process the FEM data are available at https://github.com/ScottLabUCB/FEM.

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