Optimal consumption and investment with Epstein–Zin recursive utility

We study continuous-time optimal consumption and investment with Epstein–Zin recursive preferences in incomplete markets. We develop a novel approach that rigorously constructs the solution of the associated Hamilton–Jacobi–Bellman equation by a fixed point argument and makes it possible to compute both the indirect utility and, more importantly, optimal strategies. Based on these results, we also establish a fast and accurate method for numerical computations. Our setting is not restricted to affine asset price dynamics; we only require boundedness of the underlying model coefficients.

1. Our analysis imposes no structural conditions on the underlying model coefficients, but requires them to be bounded; see (A1) and (A2) in Sect. 4 and (A1′) in Sect. 7.

2. In particular, [42] covers specifications with (untruncated) affine dynamics as in Kim and Omberg [24] and Heston [23].

3. Condition (3.1) holds if and only if one of the conditions (a), (b), (c), and (d) in [26, Proposition 3.2] is satisfied; see also [40, (2)]. We are not aware of rigorous results that ensure (E1) and (E2) for parametrizations not subsumed by (3.1).

4. Typically, the pair \((X,Z)\) would be referred to as a solution of the BSDE (6.4). For simplicity of notation, and since \(Z\) is not required for our further analysis, here and in the following, we also refer to \(X\) alone as a solution of (6.4).

5. Machine: Intel® Core™ i3-540 Processor (4M Cache, 3.06 GHz), 4 GB RAM.

6. Here we slightly abuse notation since \(\langle u\rangle^{q}_{x}\) has only been defined for functions on \([0,T]\times \mathbb{R}^{d}\). Of course, for \(u:\ \mathbb{R}^{d}\to \mathbb{R}\) and \(q\in(0,1)\), we understand that \(\langle u\rangle^{q}_{x} :=\sup_{x,x' \in \mathbb{R}^{d},\ |x-x'| \leq 1} \frac {|u(x) - u(x')|}{|x-x'|^{q}}\).

All authors wish to thank Jakša Cvitanić (editor), the Associate Editor and two referees (anonymous) for very helpful comments. We thank Darrell Duffie, Bernard Dumas, Francis Longstaff, Claus Munk, Lukas Schmid, and Carsten Sørensen for very helpful discussions, comments and suggestions. We also thank the participants of the Bachelier Finance Society 8th World Congress, the 11th German Probability and Stochastics Days, the 9th Bachelier Colloquium and seminar participants at ETH Zürich, Copenhagen Business School, the University of Copenhagen, and the University of Southern Denmark for many helpful comments and suggestions. Holger Kraft gratefully acknowledges financial support from Deutsche Forschungsgemeinschaft (DFG) and the Center of Excellence SAFE, funded by the State of Hessen initiative for research LOEWE. Thomas Seiferling gratefully acknowledges financial support from Studienstiftung des Deutschen Volkes.

Authors and Affiliations

1. Faculty of Economics and Business Administration, Goethe University, Theodor-W.-Adorno-Platz 3, 60323, Frankfurt am Main, Germany

   Holger Kraft

2. Department of Mathematics, University of Kaiserslautern, 67653, Kaiserslautern, Germany

   Thomas Seiferling

3. Department IV—Mathematics, University of Trier, Universitätsring 19, 54296, Trier, Germany

   Frank Thomas Seifried

Corresponding author

Correspondence to Frank Thomas Seifried.

Appendix A: Proofs omitted from the main text

Proof of Lemma 4.6

Since \(h\) solves the reduced HJB equation (4.7), we have

where \(z:=(t,x,y, w_{x}, w_{y}, w_{x_{y}} w_{xx}, w_{yy})\). Separating the terms in the function \(H\) as \(H(z, \pi ,c) :=u(z, \pi) + s(z, c) + q(z) \), it is easy to see that the candidate solutions \(\hat{\pi}\) and \(\hat{c}\) defined in (4.6) are the unique solutions of the associated first-order conditions

Concavity of \(u\) and \(s\) implies that \(H(z, \hat{\pi}, \hat{c}) = { \sup_{\pi\in \mathbb{R},\, c\in(0,\infty)}} H(z, \pi ,c)\). □

Proof of Lemma 4.7

By (A1) and (A2), \(\tilde{\alpha}\) and \(\tilde{r}\) are bounded. Moreover,

so that \(\tilde{\alpha}\) is Lipschitz-continuous. Finally,

 □

Proof of Lemma 5.3

The candidate optimal wealth process \(\hat{X}\) has dynamics

Put \(a_{t}:=r_{t} + \frac{1}{\gamma}\frac{\lambda_{t}^{2}}{\sigma_{t}^{2}}+ \frac{k}{\gamma}\frac{\lambda_{t}\beta_{t} \rho}{\sigma_{t}} \frac{h_{y}}{h} - \delta^{\psi}h^{q-1}\) and \(b_{t}:=\frac{1}{\gamma}\frac{\lambda_{t}}{\sigma_{t}} + \frac{k}{\gamma}\beta_{t} \rho \frac {h_{y}}{h}\). Our assumptions on the coefficients and on \(h_{y}\) and \(h\) imply that both \(a\) and \(b\) are bounded. By Itô’s formula,

where \(\mathcal {E}_{t}({\,\cdot \,})\) denotes the stochastic exponential. Choose the constant \(M>0\) such that \(|p a_{t}| + |p(p-1) b_{t}^{2}|, |p b_{t}| < M\) for all \(t\in[0,T]\). By Novikov’s condition, \({\mathcal {E}}_{t} (p\int_{0}^{{\,\cdot \,}}b_{s} \,\mathrm{d} W_{s} )\) is then an \(L^{2}\)-martingale; so using Doob’s \(L^{2}\)-inequality, we get

 □

Proof of Lemma 5.4

Lemma 5.3 and the boundedness of \(\delta^{\psi}h(t,Y_{t})^{q-1}\) imply that \(\operatorname{E}[ {\sup_{t\in[0,T]}} |\hat{c}_{t}|^{p} ] < \infty\) for all \(p \in \mathbb{R}\). In particular, \(\hat{c} \) is in \(\mathcal{C}\). By Itô’s formula,

where \(M\) is a local martingale. Hence \(\mathrm{d} V_{t} = - f(\hat{c}_{t}, V_{t}) \,\mathrm{d} t + \,\mathrm{d} M_{t}\) by Lemma 4.6. Moreover, exploiting the special form of \(w\), we get

Here \(V_{t}\) can be rewritten as \(V_{t} = w(t, \hat{X}_{t}, Y_{t}) = \frac{1}{1-\gamma} \hat{X}_{t}^{1-\gamma} h(t,Y_{t})^{k}\). By (4.8), the function \(h\) is bounded and bounded away from zero. Thus we have for all \(p \in \mathbb{R}\) by Lemma 5.3 that \(\operatorname{E}[{\sup_{t\in[0,T]}} |V_{t}|^{p} ] <\infty\). Hence \(V\) is a utility process associated with \(\hat{c}\); by (E1), it follows that \(V=V^{\hat{c}}\). Finally, the first-order condition (A.1) for the optimal consumption implies that \(w_{x}(t, \hat{X}_{t}, Y_{t}) = f_{c}(t, w(t, \hat{X}_{t}, Y_{t})) = f_{c}(\hat{c}_{t}, \hat{V}_{t})\). □

Proof of Lemma 5.5

For simplicity of notation, we set \(r_{t}:=r(Y_{t})\), \(\lambda_{t}:=\lambda(Y_{t})\) and \(\sigma_{t}:=\sigma(Y_{t})\). We have \(\mathrm{d} Z^{\pi, c}_{t} = \hat{m}_{t} c_{t} \,\mathrm{d} t + \hat{m}_{t} \,\mathrm{d} X_{t}^{\pi,c} + X_{t}^{\pi ,c} \,\mathrm{d} \hat{m}_{t} + \,\mathrm{d} [\hat{m},X^{\pi,c}]_{t}\) by the product rule. Inserting the dynamics of \(X^{\pi ,c}\) from (4.1), we get

By Lemma 5.4, \(\hat{V}_{t}=w(t,\hat{X}_{t},Y_{t})\) and \(\hat{m}_{t} = e^{\int_{0}^{t} f_{v}(\hat{c}_{s}, \hat{V}_{s}) \,\mathrm{d} s} w_{x}(t,\hat{X}_{t},Y_{t})\). From here on, we abbreviate \(f_{v} = f_{v}(\hat{c}_{t}, \hat{V}_{t})\), \(w_{x} = w_{x}(t, \hat{X}_{t}, Y_{t})\) etc. Clearly, we have \(\mathrm{d} \hat{m}_{t} = \hat{m}_{t} ( f_{v} \,\mathrm{d} t + \frac{\mathrm{d} w_{x}}{w_{x}} )\).

From the explicit expression \(f_{v}(c, v) = \delta \frac {\phi -\gamma}{1-\phi} c^{1 -\phi} ((1-\gamma)v)^{\frac{\phi-1}{1-\gamma}} - \delta \theta\), we obtain \(f_{v}(\hat{c}_{t}, w(t, \hat{X}_{t}, Y_{t}))= \frac{\phi-\gamma}{1-\phi} \delta^{\psi} h^{q -1} - \delta\theta\). By Itô’s formula,

Substituting for \(w\), we find

Plugging in the candidate \(\hat{\pi}\) from (5.1) and the dynamics of \(\hat{X}\) and \(Y\) yields

where

For the sum of the \(\frac {h_{y}^{2}}{h^{2}}\)-terms, we have

by our choice of \(k\). Combining the above, we obtain

and it follows that \(\mathrm{d} [\hat{m}, X^{\pi , c}]_{t} = - \lambda_{t} \pi_{t} \hat{m}_{t} X_{t}^{\pi,c} \,\mathrm{d} t\). Since \(h\) solves (4.7), we get

where \(\mathrm{d} M_{t}:=\hat{m}_{t} X_{t}^{\pi,c} ((\pi_{t} \sigma_{t} - \frac {\lambda_{t}}{\sigma _{t}} ) \,\mathrm{d} W_{t} + k\sqrt{1- \rho^{2}}\beta_{t} \frac {h_{y}}{h} \,\mathrm{d} \bar{W}_{t} )\) defines a local martingale \(M\). A direct calculation using the definition of \(\hat{\pi}\) yields the statement for \(Z^{\hat{\pi}, \hat{c}}\). □

Proof of Lemma 5.6

Recall that \(\underline{h}\leq h\leq \overline{h}\) so that

and we get \(0 \leq \exp(p{\int_{0}^{T}} f_{v}(\hat{c}_{s}, \hat{V}_{s}) \,\mathrm{d} s ) \leq e^{Tp m_{1}}\). On the other hand, Lemma 5.4 implies that \(\operatorname{E}[{\sup_{t \in [0,T]}} f_{c}(\hat{c}_{t}, \hat{V}_{t})^{p}] < \infty\) for all \(p \in \mathbb{R}\). It follows that

To show that \(Z^{\hat{\pi}, \hat{c}}\) is a martingale, note that \(\frac {1- \gamma}{\gamma}\frac {\lambda_{t}}{\sigma_{t}} + \frac{k}{\gamma}\beta_{t} \rho \frac {h_{y}}{h}\) is uniformly bounded by some \(c>0\). Hence by Lemma 5.3, we have

Analogously, we obtain that \(\int_{0}^{T} \operatorname{E}[ \hat{m}_{t}^{2} \hat{X}_{t}^{2} (k\sqrt{1- \rho^{2}}\beta_{t} \frac {h_{y}}{h})^{2}] \,\mathrm{d} t <\infty\). From this and Lemma 5.5, we conclude that \(Z^{\hat{\pi},\hat{c}}\) is an \(L^{2}\)-martingale. □

Proof of Proposition 6.4

For any fixed \(\kappa > c + \varrho\), define a metric \(d\) equivalent to \(\|{\,\cdot \,}\|_{\infty}\) by \(d(X,Y):=\mathop {\mathrm {ess}\,\mathrm {sup}}_{\,\mathrm{d} t \otimes \,\mathrm{d} \mathrm{P}} e^{-\kappa (T-t)} |X_{t} - Y_{t}|\). Then \((A,d)\) is a complete metric space. By definition, \(|X_{s}-Y_{s}| \leq e^{\kappa (T-s)} d(X,Y) \,\mathrm{d} t \otimes \,\mathrm{d}\mathrm{P}\)-a.e., so

and we conclude that \(d(SX,SY) \leq \frac {c}{\kappa- \varrho} d(X,Y)\), where \(\frac {c}{\kappa- \varrho}<1\). Hence \(S\) is a contraction on \((A,d)\). Thus by Banach’s fixed point theorem, there is a unique \(X\in A\) with \(S X = X\), and we have \(d(X_{(n)},X) \leq ( \frac {c}{\kappa-\varrho})^{n} d(X_{(0)},X)\) for all \(n\in \mathbb{N}\). Hence it follows that

and thus \(\|X_{(n)}-X\|_{\infty}\le e^{\kappa T} (\|X_{(0)}\|_{\infty}+ \|X\|_{\infty}) (\frac {c}{\kappa - \varrho} )^{n}\), for every \(n\in \mathbb{N}\) and every choice of \(\kappa>c+\varrho\). Setting \(\kappa=\frac{n+T\varrho}{T}\) for \(n>cT\), we obtain the asserted error bound. □

Appendix B: Stochastic Gronwall inequality

This appendix provides a ramification of the stochastic Gronwall–Bellman inequality which is required for the proofs in this article. Related results can be found in [17, 1, 36]. We work on a general probability space \((\varOmega, \mathcal {F},\mathrm{P})\) that is endowed with a filtration \((\mathcal {F}_{t})_{t \geq 0}\) that is right-continuous and complete.

Proposition B.1

Suppose \(A=(A_{t})_{t\in[0,T]}\) is bounded and progressively measurable, \(Z \in L^{p}(\mathrm{P})\) and \(B=(B_{t})_{t\in[0,T]}\) is a progressively measurable process in \(L^{p}(\,\mathrm{d} t \otimes \,\mathrm{d}\mathrm{P})\) for some \(p>1\). Moreover, let \(X = (X_{t})_{t \in [0,T]}\) be right-continuous and adapted with \(\operatorname{E}[\sup_{t\in [0,T]} |X_{t}|]<\infty\). If

for every stopping time \(\tau\) and \(X_{T} \geq Z\), then

Proof

We set

Since \(A\) is bounded above, \(Z\in L^{p}(\mathrm{P})\) and \(B \in L^{p}(\,\mathrm{d} t \otimes \,\mathrm{d}\mathrm{P})\), it follows from Doob’s \(L^{p}\)-inequality that \(\operatorname{E}[\sup_{t\in [0,T]} |M_{t}|^{p}]<\infty\). In particular, \(M\) is well defined as a uniformly integrable martingale. Now set

Since \(A\) is bounded below, we have \(\operatorname{E}[\sup_{t\in[0,T]}|Y_{t}|^{p}]<\infty\), and integration by parts yields

where \(N_{t}:=\int_{0}^{t} e^{-\int_{0}^{s} A_{u}\,\mathrm{d} u}\,\mathrm{d} M_{s}\) is a uniformly integrable martingale. For an arbitrary stopping time \(\tau\), we obtain

so that

Since \((1_{\{\tau >t \}}(N_{t} - N_{\tau}))_{s\in[t,T]}\) is a martingale, it follows that

We set \(\Delta_{t}:=X_{t} -Y_{t}\) and obtain \(\Delta_{T} = X_{T}-Z \ge 0\) and \(\operatorname{E}[\sup_{t\in[0,T]}|\Delta_{t}|]<\infty\). Moreover, (B.1) and (B.2) imply that for any stopping time \(\tau\),

Thus Proposition C.2 in [40] applies to yield \(\Delta_{t} \geq 0\) for all \(t\in[0,T]\) a.s., i.e.,

 □

Appendix C: Some facts on parabolic partial differential equations

This appendix collects the relevant results on linear and semilinear parabolic partial differential equations that are used in this article. Following [29], we first introduce the Hölder spaces \(H^{r/2,r}([0,T] \times \mathbb{R}^{d})\) for \(r\in \mathbb{R}_{+}\). For a continuous function \(u: [0,T]\times \mathbb{R}^{d} \to \mathbb{R}, (t,x) \mapsto u(t,x)\), and \(q\in(0,1)\), we define the Hölder coefficient \(\langle u\rangle^{q}_{x}\) in space via

and the Hölder coefficient \(\langle u\rangle^{q}_{t}\) in time via

The space \(H^{r/2,r}([0,T]\times \mathbb{R}^{d})\) consists of all functions \(u: [0,T]\times \mathbb{R}^{d}\to \mathbb{R}\) that are continuous along with all derivatives \(D^{\alpha}_{t} D^{\beta}_{x} u\) with “order” \(2|\alpha| + |\beta| \le r\) and satisfy \(\|u\|_{H}^{r/2,r} < \infty\). Here the norm \(\|u\|_{H}^{r/2,r}\) of \(u\) is given by

where the mixed space-time Hölder coefficient \(\langle u\rangle^{r/2,r}_{\bullet}\) of \(u\) is given by

Thus \(\|u\|_{H}^{r/2,r}\) sums up the \(L^{\infty}\)-norms of all relevant derivatives plus the Hölder coefficients of the highest-order derivatives. Analogously, for \(r\in \mathbb{R}_{+}\), the space \(H^{r}(\mathbb{R}^{d})\) is defined as the collection of all \(\lfloor r \rfloor\) times continuously differentiable functions \(u: \mathbb{R}^{d} \to \mathbb{R}\) with \(\|u\|_{H}^{r} < \infty\), where^{Footnote 6}

Linear Cauchy problem

Consider a linear second-order differential operator

where the coefficients \(a,b,c\) are defined on \([0,T]\times \mathbb{R}^{d}\) and \((a_{i,j}(t,x))_{i,j}\) is a symmetric matrix for all \((t,x) \in [0,T]\times \mathbb{R}^{d}\). The main existence and uniqueness result for linear Cauchy problems in \(\mathbb{R}^{d}\), Theorem C.1 below, relies on the following conditions:

\((\mathrm{P1})\) :

The operator \(L\) is uniformly parabolic, i.e., there exist \(0< c_{1}< c_{2}<\infty\) such that for every \((t,x)\in[0,T]\times \mathbb{R}^{d}\), we have

$$c_{1} |y|^{2} \leq { \sum_{i,j=1}^{d}} a_{ij}(t,x) y_{i} y_{j} \leq c_{2} |y|^{2} \quad \text{for all }y\in \mathbb{R}^{d}. $$

\((\mathrm{P2})^{r}\) :

For all \(i,j=1,\dots,d\), we have \(a_{i,j}, b_{i}, c \in H^{r/2,r}([0,T]\times \mathbb{R}^{d})\).

Theorem C.1

Suppose \((\mathrm{P1})\) and \((\mathrm{P2})^{r}\) are satisfied with \(r\in \mathbb{R}_{+}\), \(r\notin \mathbb{N}\), and let \(\varphi\in H^{r+2}(\mathbb{R}^{d})\) and \(f\in H^{r/2,r}([0,T]\times \mathbb{R}^{d})\). Then there exists a unique function \(u\in H^{(r+2)/2,r+2}([0,T]\times \mathbb{R}^{d})\) such that

Moreover, \(u\) satisfies

where \(c>0\) is a global constant that is independent of \(\varphi\) and \(f\).

Proof

See [29, Theorem IV.5.1]. □

As a special case, we obtain the result we have used in the proof of Lemma 7.1.

Corollary C.2

Suppose that

(C1):

\(a, b, c: \mathbb{R}\to \mathbb{R}\) are bounded and Lipschitz-continuous;

(C2):

the function \(a\) has a bounded Lipschitz-continuous derivative and satisfies \(\inf_{y \in \mathbb{R}} a (y) >0\);

(\(\mathrm{C3}^{\prime}\)):

\(\hat{\varepsilon}\in H^{r+2}(\mathbb{R})\) for some \(r \in (0,1)\).

Then for each bounded and Lipschitz-continuous function \(f: [0,T] \times \mathbb{R}\to \mathbb{R}\), there exists a unique \(g\in C_{b}^{1,2}([0,T] \times \mathbb{R})\) that solves

Proof

Consider the second-order differential operator

By assumptions (C1) and (C2), the differential operator \(L\) satisfies (P1) and \((\mathrm{P2})^{r}\) for \(r \in (0,1)\). Moreover, \(f\) is in \(H^{r/2,r}([0,T] \times \mathbb{R})\) since it is Lipschitz-continuous. Hence Theorem C.1 yields \(u\in H^{(r+2)/2,r+2}([0,T] \times \mathbb{R})\) such that

Thus defining \(g\in C_{b}^{1,2}([0,T]\times \mathbb{R})\) by \(g(t,y) :=u(T-t, y)\), we obtain

 □

Quasilinear Cauchy problem

Next consider the nonlinear differential operator

with principal part in divergence form. We set

We now state the conditions required for Theorem C.3.

1. (Q1)

   For all \(t\in(0,T]\), \(x,p\in \mathbb{R}^{d}\) and \(u\in \mathbb{R}\), we have

   $${ \sum_{i,j=1}^{d}} a_{ij}(t,x,u,p) y_{i} y_{j} \geq 0\quad \text{for all } y \in \mathbb{R}^{d}. $$
2. (Q2)

   There exist \(b_{1}, b_{2} \geq 0\) such that for all \(t \in (0,T]\), \(x \in \mathbb{R}^{d}\) and \(u \in \mathbb{R}\), we have

   $$A(t,x,u,0) \geq - b_{1} u^{2} - b_{2}. $$
3. (Q3)

   The functions \(a\) and \(a_{i}\) are continuous, and \(a_{i}\) is differentiable with respect to \(x\), \(u\) and \(p\). Moreover, there exist \(c_{1}, c_{2} >0\) such that for all tuples \(v =(t,x,u,p)\) in \([0,T]\times \mathbb{R}^{d} \times \mathbb{R}\times \mathbb{R}^{d}\), we have

   $$c_{1}|y|^{2} \leq { \sum_{i,j=1}^{n}} a_{ij}(v) y_{i} y_{j} \leq c_{2} |y|^{2}\quad \text{for all } y\in \mathbb{R}^{d} $$

   and, with \(a_{ij}\) given by (C.1),

\((\mathrm{Q4})^{\beta}\) :

There exists \(\beta\in(0,1)\) such that for all compact sets \(K\subset \mathbb{R}\), \(\bar{K} \subset \mathbb{R}^{d}\), the functions

$$a_{i}, a, a_{ij}, \frac{\partial a_{i}}{\partial u}, \frac{\partial a_{i}}{\partial x_{i}}: [0,T] \times \mathbb{R}^{d} \times K \times \bar{K} \to \mathbb{R}$$

are Hölder-continuous in \(t,x,u\) and \(p\) with exponents \(\beta/2\), \(\beta\), \(\beta\) and \(\beta\), respectively.

Here we say that \(f:[0,T] \times \mathbb{R}^{d} \times K \times \bar{K} \to \mathbb{R}, z= (z^{1},z^{2}, z^{3}, z^{4})\mapsto f(z)\) is \(\beta\)-Hölder-continuous in \(z^{i}\) if

Theorem C.3

Suppose \(\psi_{0}\) is in \(H^{2+\beta}(\mathbb{R}^{d})\) and (Q1), (Q2), (Q3) and (Q4)^β are satisfied for some \(\beta \in (0, 1)\). Then there exists a solution \(u \in H^{(2+ \beta)/2, 2+ \beta}([0,T] \times \mathbb{R}^{d})\) of the Cauchy problem

Proof

See [29, Theorem V.8.1]. □

In the proof of Theorem 6.10, we require the following ramification of this result.

Corollary C.4

Suppose that

(C1):

\(a, b, c: \mathbb{R}\to \mathbb{R}\) are bounded and Lipschitz-continuous;

(C2):

the function \(a\) has a bounded Lipschitz-continuous derivative and satisfies \(\inf_{y \in \mathbb{R}} a (y) >0\);

(\(\mathrm{C3}^{\prime}\)):

\(\hat{\varepsilon}\in H^{r+2}(\mathbb{R})\) for some \(r \in (0,1)\);

and let \(f\in C^{1}_{b}(\mathbb{R})\). Then the semilinear PDE

has a solution \(g\in C_{b}^{1,2}([0,T]\times \mathbb{R})\).

Proof

After setting \(a_{1}(t,x,u,p):=p a(x)\) and

we can represent the relevant differential operator as

Hence if \(u \in C_{b}^{1,2}([0,T]\times \mathbb{R})\) solves \(Lu=0\), \(u(0,\cdot)=\hat{\varepsilon}\), then \(g(t,x):=u(T-t,x)\) defines a member of \(C_{b}^{1,2}([0,T]\times \mathbb{R})\) that satisfies

We now verify the assumptions of Theorem C.3 for \(L\). Note that

so (Q1) holds since

Next observe that

Thus (Q2) is satisfied since

with \(b_{1}:=\|c\|_{\infty}\) and \(b_{2}:=\|c\|_{\infty}+ \|f\|_{\infty}\). To check (Q3), note that by (C1) and (C2), the functions \(a_{1}\) and \(\bar{a}\) are continuous, and \(a_{1}\) is differentiable; moreover,

for all \(t \in [0,T]\) and \(x,u,p,y \in \mathbb{R}\). For all \(v = (t,x,u,p) \in [0,T] \times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\), we further have

since \(a,b,c,f\) and \(a'\) are bounded. Thus (Q3) holds with

and \(c_{1}:=\inf_{x \in \mathbb{R}} a(x) >0\). Finally, for any compact set \(K \subset \mathbb{R}\), the functions

restricted to \([0,T] \times \mathbb{R}\times K \times K\) are Lipschitz-continuous in \(x\), \(u\) and \(p\), because \(a\), \(a'\), \(b\), \(c\) and \(f\) are bounded and Lipschitz by (C1), (C2) and since \(f \in C_{b}^{1}(\mathbb{R})\). Hence (Q4)\(^{\frac{1}{2}}\) holds as well. Thus by Theorem C.3, the Cauchy problem

has a solution \(u \in H^{5/4,5/2}([0,T] \times \mathbb{R}^{d}) \subset C_{b}^{1,2} ([0,T] \times \mathbb{R}^{d})\). Uniqueness follows from standard BSDE arguments; see e.g. [20, Proposition 4.3]. □

Reprints and permissions